Transcript Significant Digits

Significant Digits
Copyrighted © by T. Darrel Westbrook
Significant Digits
There are two parts to dealing with uncertainty in
science. The first part is covered in this presentation,
Significant Digits, and the second part covers error
propagation in the presentation
Propagation of Uncertainty
It is recommended you view Significant Digits
presentation first.
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Significant Digits
What You Will Learn:
– About Accuracy and Precision
– About Types of Errors
– About Significant Digits (also called Significant
Figures)
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Significant Digits
Definition: Certainty – is perfect knowledge that has total
security from error.
Definition: Uncertainty – The lack of certainty. A state of
limited knowledge where it is impossible to
exactly describe existing state or future
outcome from more than one possible
outcome.
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Significant Digits
We begin our study of significant digits
(also called significant figures) with some
definitions.
Digit: one of the numbers 1, 2, 3, 4, 4, 5, 6, 7,
8, 9, and 0.
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Significant Digits
Accuracy: the degree of closeness of a measure to its
actual value.
Precision: is the degree of reproducibility of a measure or
the claimed / implied closeness of a measure.
“The results of calculations or a measurement can be
accurate but not precise, precise but not accurate,
neither, or both.
A measurement system or computational method is
called valid if it is both accurate and precise.” Wikipedia
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Significant Digits
We are trying to measure as precise as we can, and we do
this by making our measurements as consistently accurate
as we can.
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Significant Digits
Measurement Accuracy: describes how close a measured
value is to the true value of the quantity being
measured.
Measurement Precision: refers to the degree of exactness
with which a measurement is made and stated.
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Significant Digits
ERRORS
Method Error: using different methods to measure the
same variable. This can be reduced by
standardizing the method.
Parallax Error: the error of sight that occurs when not
reading a scale perpendicular to the scale.
Instrument Error: caused by the devices used to measure
experiments.
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Significant Digits
When we measure things, the accuracy of our
measurements are known within limits of our measuring
equipment, our skills of using the equipment, and the
number of measurements we take.
These all contribute to the uncertainty of our
measurements and their use in calculations.
We do not want to claim any more accuracy than we are
capable of producing.
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Significant Digits
Significant digits are all about measurement and/or
calculations.
1 2 3 4 5
cm
How accurate can you measure with the ruler below?
How accurate can you measure with the ruler now?
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Significant Digits
What is a significant digit ?
Significant Digits: those digits in a measurement and/or
calculations that are known with certainty plus the first
digit that is uncertain.
1 2 3 4 5
cm
Using the ruler, how many digits would be in the level of
certainty?
Where is the uncertain digit located?
So, how many significant digits total?
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Significant Digits
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Significant Digits
There are several rules associated with the use of
significant digits.
The first rule is,
all non-zero digits are ALWAYS significant.
22
22.3
3
32
2133
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Significant Digits
The bulk of the rules come from having to deal with ZEROS.
1. Leading zeros are NOT significant. 046, 0.046, 0.135
2. Zeros trapped between other digits (1, 2, 3, 4, 5, 6, 7, 8, 9) ARE
ALWAYS significant. 2033, 40019
3. Zeros after other numbers AND to the RIGHT of the decimal
ARE significant. 2.30, 0.0503
4. Zeros at the end of a number ARE significant ONLY if they are
to the RIGHT of the decimal point. 0.003000, 1.20, 1.230
How about the number 7900. How many significant digits?
Use scientific notation to avoid confusion.
7.900  103
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Significant Digits
Calculations
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Significant Digits
Multiplication, Division, and Trig Functions Rule:
the final answer can not be larger than the number in
the calculations with the least significant digits.
For example: (15.23) 36 (14.2) =
7785.576
7785.58
7785.6
7780
7700
7000
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Significant Digits
Addition and Subtraction Rule: the final answer
cannot be larger than the number in the calculations
with the least number of decimal places.
Note the difference:
Multiplication, Division, and Trig Functions Rule:
the final answer cannot be larger than the number in the
calculations with the least significant digits.
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Significant Digits
Addition and Subtraction Rule: the final answer cannot
be larger than the number in the calculations with the
least number of decimal places.
123.25
14.3
+ 16.87525
154.42525
154.4253
154.425
154.43
154.4
154
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Significant Digits
Addition and Subtraction Rule: the final answer cannot
be larger than the number in the calculations with the
least number of decimal places.
123.25
14.3
+ 16.87525
154.4
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Significant Digits
Rounding Rules
1. The last digit to be retained is increased by 1, if the
digit to the right is 6 or more.
2. The last digit to be retained is not changed, if the digit
to the right is 4 or less.
3. The last digit to be retained is followed by a 5 followed
by zeros then round to the nearest even number. This
minimizes error accumulation in long calculations.
Another technique to avoid error accumulation is to retain
an extra significant digit until the answer is obtained.
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Significant Digits
Rounding Rules
For 3 significant digits, so we keep 1.3 plus?
becomes 1.36, round up
1.35623
1.34499
becomes 1.34, round down
1.33500
becomes 1.34, round to nearest even
1.32500
becomes 1.32, round to nearest even
1.30501
becomes 1.31, round up
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Significant Digits
A carpet is to be installed in a room that measures 12.71
meters by 3.46 meters. Find the area of the room.
Multiply the length and width and you get 43.9766 m2.
How many numbers should you claim?
43.9 m2
Then round the results based on the 7 immediately to the
right of the 9
44.0 m2
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Significant Digits
What You Have Learned:
– About Accuracy and Precision
– About Types of Errors
– About Significant Digits
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Significant Digits
Study Guide
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Significant Digits
Accuracy: the degree of closeness of a measure.
Precision: is the degree of reproducibility of a measure.
Measurement Accuracy: describes how close a measured value is to the true value of the quantity being measured.
Measurement Precision: refers to the degree of exactness with which a measurement is made and stated.
Method Error: using different methods to measure the same variable. This can be reduced by standardizing the method.
Parallax Error: the error of sight that occurs when not reading a scale perpendicular to the scale.
Instrument Error: caused by the devices used to measure experiments.
Significant Digits: those digits in a measurement and/or calculations that are known with certainty plus the first digit that is uncertain.
Significant Digits Rules:
– all non-zero digits are ALWAYS significant.
Rules dealing with Zero
1. Leading zeros are NOT significant. 046, 0.046, 0.135
2. Zeros trapped between other digits (1, 2, 3, 4, 5, 6, 7, 8, 9) ARE ALWAYS significant. 2033, 40019
3. Zeros after other numbers AND to the RIGHT of the decimal ARE significant. 2.30, 0.0503
4. Zeros at the end of a number ARE significant ONLY if they are to the RIGHT of the decimal point. 0.003000, 1.20, 1.230
Multiplication, Division, and Trig Functions Rule : the final answer can not be larger than the number in the calculations with the
least significant digits.
Addition and Subtraction Rule : the final answer can not be larger than the number in the calculations with the least number of
decimal places.
Rounding Rules
1. The last digit to be retained is increased by 1, if the digit to the right is 6 or more
2. The last digit to be retained is not changed, if the digit to the right is 4 or less.
3. The last digit to be retained is followed by a 5 followed by zeros then round to the nearest even number. This minimizes error
accumulation in long calculations.
Another technique to avoid error accumulation is to retain an extra significant digit until the answer is obtained.
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Significant Digits
End of Line
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