Slide 1

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 2

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 3

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 4

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 5

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 6

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 7

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 8

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 9

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 10

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 11

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 12

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 13

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 14

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 15

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 16

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 17

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 18

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 19

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10


Slide 20

Mathematics
Math in Physics

This set of slides may be updated this weekend. I have given you my
current slides to get you started on the PreAssignment. Check back here
frequently to see if updates have been made. I will post here a record of
updates that have been made.

1

Mathematics
Math in Physics

What is a physical quantity?
A physical quantity is any quantity that can be measured with a
certain mathematical precision.
Example: Force
What is a dimension?
The product or quotient of fundamental physical quantities, raised
to the appropriate powers, to form a derived physical quantity.
Example: mass x length / time2 (ML/T2)
What is a unit?

A precisely defined (standard) value of physical quantity against
which any measurements of that quantity can be compared.
Example: Newton = kilogram x meters / second2
2

Mathematics
Math in Physics

Sytème International

3

temperature
Kelvin (K)

length
meters (m)

current

time
seconds (s)

amount of substance
Mole (mol)

mass
kilograms (kg)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

US Customary System

4

temperature
Fahrenheit (F)

length
inches (in, ")

current

time
seconds (s)

amount of substance
Mole (mol)

mass
pounds (lb)

luminous intensity
candela (cd)

Amperes (A)

Mathematics
Math in Physics

Other units
length
Feet (ft, '), mile (m, mi), furlong, hand
time
minute (m, min), hour (hr), second (s), fortnight, while
energy
force
Joule (J)
Newton (N)
power
temperature
Watt (W)
Celsius (C)
pressure
Pascal (P)
magnetic field
Tesla (T), Gauss (G)
5

Mathematics
Math in Physics

Common conversion factors
length
1 in = 2.54 cm
time
60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr
force
1 N = 0.2248 lb

energy
1 J = 107 erg, 1 eV = 1.602x10-19 J
power
1 hp = 746 W
pressure
1 atm = 101.3 kPa

6

magnetic field
1 T = 104 G

Mathematics
Math in Physics

How to convert units

1 day  1 day

24 hr 60 min 60 sec
1 day

1 hr

1 day  24  60  60 sec

1 day  86 , 400 sec

7

1 min

Mathematics
Math in Physics

Most commonly used prefixes for powers of 10

tera
giga
mega
kilo
centi
milli
micro
nano

8

T
G
M
k
c
m
μ
n

1,000,000,000,000
1,000,000,000
1,000,000
1000
0.01
0.001
0.000001
0.000000001

1012
109
106
103
10-2
10-3
10-6
10-9

Mathematics
Math in Physics

Common physical quantities
Mass
Distance
Time
Speed / Velocity
Acceleration
Force

9

Mathematics
Math in Physics

What is scientific notation?
1,562,788.
0.0012789

10

 1.562788x106
 1.2789x10-3

Mathematics
Math in Physics

Using scientific notation in calculators
When putting numbers in a calculator, it is best to
covert them to non-prefixed units first (e.g. 1mm 
1x10-3 m) and then convert back to the desired units
when the problem is complete.
When using a calculator, it is also better to put in the
full number (e.g. 1350x10-3 m instead of 1.35 m for 1350
mm). In this way you will avoid many of the “decimal
place errors” so common in this class.

11

Mathematics
Math in Physics

Significant figures defined

Significant figures in a number indicate the certainty to
which a number is known.
(For example 1350 mm is known to within ~0.5 mm.)
Other than leading zeros, all digits in a number are
significant.
(For example 0.003450900000 has 10 significant figures.)
Numbers are rounded up or down to the nearest
significant figure.

12

Mathematics
Math in Physics

Significant figures are used because any further digits
added to your number have no physical meaning.
Any physically measured number (including physical
constants) will be written with the correct number of
significant figures unless otherwise noted.
Numerical constants, such as the 4 or the π in the equation

have an infinite number of
figures because they
4 significant
3
V sphere   r
are NOT measured.

3

13

Mathematics
Math in Physics

Using significant figures

When multiplying or dividing numbers, the calculated
number has the same number of significant figures as the
number with the least significant figures used in the
calculation.

6 . 2  5 . 367  10
6 .2
5 . 367  10
14

8

8

  3 .3  10

 1 . 2  10

8

9

Mathematics
Math in Physics

Using significant figures

When adding or subtracting numbers, the number of
significant figures in the calculated number must be such
that the decimal place of the result is not beyond the
least decimal number in the numbers used in the
calculation.



6.2  5.367  10

8



0.000000062  10
 5.367
15

5.367

8

 10 
8

 10

8

Mathematics
Math in Physics

Examples using significant figures

1 . 58  0 . 0030  4 . 7  10

3

0 .078  10  5 .688  10
8

1.28  10

16

8

 3 .26  10

9

7

 0 . 44

 3 . 39  10

9

Mathematics
Math in Physics

Why do we use order of magnitude?

Order of magnitude is used to make estimates.
For example: How many professors are there in the U.S.?
(These kinds of questions are named for Enrico Fermi,
who first proposed them.)
Order of magnitude is also used to check calculations.

17

Mathematics
Math in Physics

Order of magnitude defined
To determine the order of magnitude of a number, you
must put the number in scientific notation using one digit
before the decimal.
(e.g. 1345  1.345x103
0.00845090  8.45090x10-3)

If the decimal number is less than five, the order of
magnitude is then the exponent. If it is greater than or
equal to five, the order of magnitude is the exponent plus
one.

18

1345 has order of magnitude 3
0.00845090 has order of magnitude -2

Mathematics
Math in Physics

Estimation
Estimation is not the same as calculation. However, it will almost
always be within one exponent of the calculated answer.
An estimate is a fast way to check a number or choose between two
numbers given as an answer
Actual

1324  1 . 235  1 . 67
230  10

Estimate

1  10  1  1  1  10

 7 . 12  10

3

1  10

19

8

10

7

8

Mathematics
Math in Physics

An Example Fermi Problem
To within an order of magnitude how many bars of soap are
sold in the United States each year?
1.
2.
3.
4.

There are about 1x108 people in the U.S. (the actual number is
closer to 3x108).
Each person lives in a family of about 1 person (the average is
really closer to 3).
Each family uses about 1 bar of soap each week (a better number
would be 0.5).
There are about 100 weeks in a year (the number is actually 52).

1  10  1  1  100
8

 1  10

10

Compare this to the actual answer

3  10  3  0 .5  52  2 .34  10
8

20

10