Significant Figures and Uncertainty
What are significant figures?
Well... For any given measurement, some digits (or figures) are significant, while some are nonsignificant. Significant figures always indicate precision. Knowing about sig figs allows you to round your answer properly.
Here are the rules for determining whether or not digits are significant:
1. Digits from 1-9 are always significant.
Examples:
a) 5 843 has 4 significant figures.
b) 23 has 2 significant figures.
2. Zero(s) between two other significant digits are always significant.
Examples:
a) 6008 has 4 significant figures.
b) 20 564 302 has 8 significant figures.
3. One or more additional zeros to the right of both the decimal place and another significant digit are significant.
Examples:
a) 23.0 has 3 significant figures.
b) 860.00 has 5 significant figures.
4. Zeros used solely for spacing the decimal point (placeholders) are not significant.
i.e. zeros at the right of a large number are not significant.
zeros at the left of a small number (<1) are not significant.
Examples:
a) 0.0025 has 2 significant figures.
b) 6 500 has 2 significant figures.
c) 2.31 x 104 has 3 significant figures.
5. Numbers that are not measurements (for example, constants in a formula or values that have been counted) have an infinite number of significant figures. We basically ignore them when we are trying to decide how many significant figure we need to round to.
Example:
a) There are 41 bunnies in a cage.
b) The acceleration due to gravity near the surface of the Earth.
NOW FOR THE FUN PART! YAY!!
The rule for addition and subtraction with significant figures
When adding or subtracting, round the asnwer to the least number of decimal places.
Examples:
a) 32.3 + 51 = 83.3 -> 83
b) 6.235 - 2.54 = 3.695 -> 3.70
c) 452.99 + 0.120005 = 453.110005 -> 453.11
The rule for multiplication and division with significant figures
When multiplying or dividing with numbers, round the answer to the least number of sig figs.
i.e. round off your answer to match the same number of significant figures as your measurement with the least number of significant figures.
Examples:
a) 0.0025 x 3568 = 8.92 -> 8.9
b) 4525 ÷ 320 = 14.14 -> 14
Multi-Step Problems
When a problem involves both addition and multiplication, each rule has to be applied seperately.
Examples:
a) (4.56 + 12.678)(3.99)
= (17.24)(3.99)
= 68.8
b) 12.98 x 2.75 - 7.39
= 35.7 - 7.39
= 28.3
c) 600.3 - 45
3.987
= 555
3.987
= 139
d) 35.98 x 34.09 + 107.9 x 8.09
= 1227 + 873
= 2100
Uncertainty
Uncertainty is present when a quantity has been measured with an instrument. The uncertainty in the measurement is a result of the uncertainty of the instrument used, or of the skill of the person taking the measurement.
There are two ways to express uncertainty:
Absolute uncertainty: is expressed in the same units of measurement itself.
Relative uncertainty: is expressed as a percentage of the measurement.
Relative uncertainty = Absolute uncertainty x 100
Value of measurement
There are two ways of determining the uncertainty of a measurement:
1) Sometimes, the uncertainty is written on the instrument itself
Example: On a balance, it may say that the mass indicated has an uncertainty of 0.01g.
2) When the instrument does not indicate a specific uncertainty, the uncertainty is equal to "one half of the
smallest measurement" provided by the instrument.
Well... For any given measurement, some digits (or figures) are significant, while some are nonsignificant. Significant figures always indicate precision. Knowing about sig figs allows you to round your answer properly.
Here are the rules for determining whether or not digits are significant:
1. Digits from 1-9 are always significant.
Examples:
a) 5 843 has 4 significant figures.
b) 23 has 2 significant figures.
2. Zero(s) between two other significant digits are always significant.
Examples:
a) 6008 has 4 significant figures.
b) 20 564 302 has 8 significant figures.
3. One or more additional zeros to the right of both the decimal place and another significant digit are significant.
Examples:
a) 23.0 has 3 significant figures.
b) 860.00 has 5 significant figures.
4. Zeros used solely for spacing the decimal point (placeholders) are not significant.
i.e. zeros at the right of a large number are not significant.
zeros at the left of a small number (<1) are not significant.
Examples:
a) 0.0025 has 2 significant figures.
b) 6 500 has 2 significant figures.
c) 2.31 x 104 has 3 significant figures.
5. Numbers that are not measurements (for example, constants in a formula or values that have been counted) have an infinite number of significant figures. We basically ignore them when we are trying to decide how many significant figure we need to round to.
Example:
a) There are 41 bunnies in a cage.
b) The acceleration due to gravity near the surface of the Earth.
NOW FOR THE FUN PART! YAY!!
The rule for addition and subtraction with significant figures
When adding or subtracting, round the asnwer to the least number of decimal places.
Examples:
a) 32.3 + 51 = 83.3 -> 83
b) 6.235 - 2.54 = 3.695 -> 3.70
c) 452.99 + 0.120005 = 453.110005 -> 453.11
The rule for multiplication and division with significant figures
When multiplying or dividing with numbers, round the answer to the least number of sig figs.
i.e. round off your answer to match the same number of significant figures as your measurement with the least number of significant figures.
Examples:
a) 0.0025 x 3568 = 8.92 -> 8.9
b) 4525 ÷ 320 = 14.14 -> 14
Multi-Step Problems
When a problem involves both addition and multiplication, each rule has to be applied seperately.
Examples:
a) (4.56 + 12.678)(3.99)
= (17.24)(3.99)
= 68.8
b) 12.98 x 2.75 - 7.39
= 35.7 - 7.39
= 28.3
c) 600.3 - 45
3.987
= 555
3.987
= 139
d) 35.98 x 34.09 + 107.9 x 8.09
= 1227 + 873
= 2100
Uncertainty
Uncertainty is present when a quantity has been measured with an instrument. The uncertainty in the measurement is a result of the uncertainty of the instrument used, or of the skill of the person taking the measurement.
There are two ways to express uncertainty:
Absolute uncertainty: is expressed in the same units of measurement itself.
Relative uncertainty: is expressed as a percentage of the measurement.
Relative uncertainty = Absolute uncertainty x 100
Value of measurement
There are two ways of determining the uncertainty of a measurement:
1) Sometimes, the uncertainty is written on the instrument itself
Example: On a balance, it may say that the mass indicated has an uncertainty of 0.01g.
2) When the instrument does not indicate a specific uncertainty, the uncertainty is equal to "one half of the
smallest measurement" provided by the instrument.
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