Slide 1

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 2

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 3

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 4

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 5

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 6

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 7

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 8

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 9

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 10

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 11

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 12

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 13

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 14

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 15

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 16

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 17

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 18

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 19

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 20

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 21

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 22

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 23

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 24

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 25

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 26

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 27

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 28

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 29

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 30

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 31

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 32

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 33

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 34

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 35

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 36

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 37

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 38

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2


Slide 39

copyright Sautter 2003

Measurement
• All measurement is comparison to a standard. Most
often that standard is an excepted standard such as a
foot of length, liter of volume or gram of mass.
• Unusual standards may be used in obtaining
measurements but this is rarely done since few
people would be familiar with the standard used. For
example, someone measuring a distance can pace off
that distance but since the length of one’s step is
variable and this would give a very unreliable
measure.
• We generally work with two systems of
measurement, English and metric. The metric system
is used more frequently in science although the
English system can be used.

Measurement
• Basic metric units are systematically subdivided
using a series of prefixes. Each prefix multiplies the
basic unit by a specific value. For example the
prefix “centi” multiplies by 0.01 (one hundredth –
100 cents in a dollar), “deci” multiplies by 0.10 (one
tenth - 10 dimes in a dollar) and so on.
• The prefix or multiplier may be applied to any basic
measurement, grams, liters or meters and others yet
to be discussed. The prefix may subdivide the unit
or enlarge it. For example, “milli” divides the unit
into a 1000 parts (0.001 or one thousandth) while
“kilo” multiplies the unit by 1000 (a thousand
times).

UNIT CONVERSIONS
• Quantities can be converted from one type of unit
to another. This conversion may occur within the
same system (metric or English) or between
systems (metric to English or English to metric).
• Conversions cannot be made between measures of
different properties, that is, mass units to length
units for example.
• A method of unit conversion commonly used is
called Dimensional Analysis or Unit Analysis. In
this procedure, units are used to decide when to
multiply or divide in order to obtain the correct
answer.

1 cm

1 milliliter
1 cm

1 cm

Volume = length x width x height
1 cm3 = 1 cm x 1 cm x 1 cm
cc means cubic centimeter
1.00 ml = 1.00 cc
1000 ml = 1000 cc = 1.0 liter

Unit Analysis
• Let’s apply Unit Analysis to a sample problem. In
order to use this method we must have available a
list of conversion factors from English to metric and
vice versa. Some have been provided on the
previous slides.
• To begin we will examine a metric to metric
conversion problem.

FROM THE CONVERSION TABLE

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

PLACE THE NUMBERS IN THE SPOTS
INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.
• Solution: First, we decide that units we are starting with (km) and the
units we want to find (mm). Km.  mm.
• Next, we will examine the metric relationships that are available to be
used for the conversion.
• Millimeter means 0.001 meters or 1000 mm = 1m
• Kilometer means 1000 meter so 1000 m = 1km
• Now, we will set up unit fractions so that all units will cancel out
leaving only the unit for the answer (mm)
• We are starting with km
• Km x (m / km) x (mm / m) = mm (the units for our answer)
• Km will cancel and m will cancel leaving just mm in our set up.
• Now place the numbers in the positions indicated by the units
• 5.35 x (1000/1) x (1000/1) = 5.35 x 106 mm
• Mental check: since mm are very small and km are large there
should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English  metric
• Problem: How many milligrams are contained in 25 lbs?
• Solution: We are starting with pounds and want to find
milligrams. Lbs  mg
• We need an English – metric weight (mass) conversion.
We will use 454 grams = 1.0 lbs. We will also use 1000
mg = 1.0 grams
• Set up the units: lbs x (g / lb) x (mg / g) = mg
• 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 107 mg
• Check: there are lots of grams in a pound and lots to
milligrams in a gram, therefore expect a large number and
11.34 million is a large number!

In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier

Scientific numbers use powers of 10

RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved

Any number to the
Zero power = 1

RULE 3
When scientific numbers are multiplied
The powers of 10 are added

RULE 4
When scientific numbers are divided
The powers of 10 are subtracted

RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied

RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied

RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term.
Powers of 10 are
Different. Values
Cannot be added !
Move the decimal
And change the power
Of 10

Power are now the
Same and values
Can be added.

DENSITY
• Density is a fundamental property of all matter. It measures
the quantity of matter in a given volume of space.
• For elements and compounds, density can be an identifying
characteristic. For example, the density of gold is 19.5
grams per milliliter. Any substance with appearances similar
to gold cannot be gold unless it has the density of 19.5
grams per milliliter.
• The density of solids is usually greater than that of liquids
and the density of gases is always significantly lower than
that of liquids or solids.A notable exception to the density
relationships of solids and liquids is that for water. Water in
the solid state (ice) has a lower density than liquid water
(ice floats on water). Most substances do not have this
inverted density relationship.

DENSITY
• Density units can be any mass unit divided by any volume unit.
Usually grams / ml are used for liquids and solids. A volume
measurement of cubic centimeters is also often used. One cubic
centimeter (cc) equals one milliliter (ml). When measuring the density
of gases grams per liter (g / l) are generally used.
• The term specific gravity is also used to measure density. It is a ratio
of the density of substance divided by the density of water. The
density of water is generally considered to be 1.0 grams per ml.
(Actually, the density of water like all substances varies with
temperature and is really 1.0 g/ml at 4 0C)
• Since the density of water is used as 1.0 g/ml dividing it into the
density of any substance gives back the density of that substance
however the units are divided out and therefore the specific gravity
value has no units associated with it.

A lead
Fishing
sinker

A room full
of air

THE AIR! WHY?
BECAUSE THERE IS MUCH MORE OF IT !
LEAD HAS A GREATER DENSITY BUT NOT
NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY
• Problem: A rectangular block to substance X measures
3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams.
Find its density.
• Solution: Density = mass / volume
• Volume = length x width x height (rectangular solid)
• Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to
cm so that the volume is calculated in cubic centimeters)
• Volume = 300 cc
• Density = 150 grams / 300 cc = 0.50 g/cc

Significant Figures
• Significant figures are used to distinguish truly measured values from
those simply resulting from calculation. Significant figures determine
the precision of a measurement. Precision refers to the degree of
subdivision of a measurement.
• As an example, suppose we were to ask how much money you had
and you replied “About one hundred dollars”. This would be written
as $100 with no decimal point included.This is shown with one
significant figure the “1”, the zeros don’t count and it tells us that you
have about $100 but it could be $90 or even $110. If we continued to
inquire you might say “ OK, ninety seven dollars. This would be
written as $97. It contains two significant figures, the 9 and the 7.
Now we know that you have somewhere between $96.50 and $97.49.
• If we continue to ask you may eventually say, “Ninety seven dollars
and twenty cents”. This is written as $97.20 and in this case the zero is
significant because it say that you have exactly 20 cents, not 19 or 21,
in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one
approximated value. The pencil is 3.6 cm long.
1

3

2

3

4

4

5

7

With more calibration a more
precise measurement is possible
The pencil is 3.64 cm long!
3.6

The calibration of the instrument
determines measurement precision

6

3.7

Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT
IS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OF
SUBBDIVISION OF THE MEASUREMENT
FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND
YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR
MEASUREMENT IS VERY PRECISE BUT INACCURATE !
MEASUREMENTS SHOULD BE ACCURATE AND AS
PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures
• In working with significant figures, zeros are the most
problematic. Non zero numbers are always significant.
Zeros are sometimes significant and other times not as
we saw in the previous frame. To work successfully with
significant figures a set of rules are required. Here they
are:
• (1) Zeros to the right of non zeros and left of the decimal
are significant. In 300 the zero are to the right of a non
zero but not left of a decimal and are not significant. The
number contains only one sig fig. Zeros to the right of the
decimal and to the right of non zeros are significant. In
0.02300 the zeros are to the right of non zeros and the
decimal and are significant. The zeros not preceded by
non zeros are not significant. The number has four sig
figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures
• (2) Zeros between non zeros are always significant. In the
number 4009 the zeros are significant being in between
the 4 and the 9. The number has four sig figs.
• (3) Zeros with no decimal to the right are not significant.
In 4500 the zeros have no decimal to the right and are not
significant. The number has two sig figs, the 4 and the 5.
• (4) Zeros with a decimal to the right are significant when
preceded by non zeros. In 4500. the zeros are significant.
The number has four sig figs, the 4, 5, and both zeros.
• (5) A significant zero means a value was measured and
found to be zero in that position. A non significant zero
(called a place holding zero) means that no measurement
was taken in that position. The value is unknown!

No decimal
point
2 sig figs

Zeros are not
significant!

Decimal
Point
All digits including
zeros to the left of
The decimal are
significant.
6 sig figs

All figures are
Significant
4 sig figs

All figures are
Significant
5 sig figs

Zeros between
Non zeros are
significant

Zero to the
Right of the
Decimal are
significant

3 sig figs
Zeros to the right of
The decimal with no
Non zero values
Before the decimal
Are not significant

5 sig figs
Zeros to the right of the decimal
And to the right of non zero values
Are significant

Exact equivalences have an unlimited number of
significant figures
There are exactly 3 feet in exactly 1 yard.
Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.
and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !
The same is true for:

Mathematics and Significant Figures
• (5) Multiplying and dividing with significant figures.
The result of multiplication or division can have no more sig figs than
the term with the least number. For example, 9 x 2 = 20 since the 9
has one sig fig and the 2 has one sig fig, the answer 20 must have only
one and is written without a decimal to show that fact. By contrast, 9.0
x 2.0 = 18 each term has two sig figs and the answer must also have
two.
• (6) Adding and subtracting with significant figures.
The position, not the number, of the significant figures is important in
adding and subtracting. For example,
12.03
(the last sig fig is in hundredth place (0.01))
+ 2.0205 (the last sig fig is in ten thousandth (0.0001))
14.05 (the answer is rounded off to the least significant position
hundredths place)

The numbers in
these positions are
not zeros, they are
unknown

The answer is rounded to the
position of least significance

The sum of an
unknown number
and a 6 is not valid.
The same is true
For the 2