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Significant Figures

(Adapted from Prof. Pasternack, JHU Chemistry)

A good tutorial on significant figures can be found

at http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/

The rules for significant figures can be summarized as follows:

1. To determine the number of significant figures:

o All nonzero digits are significant. (1.234 has 4 sig figs)

o Zeroes between nonzero digits are significant. (1.02 has 3 sig figs)

o Zeroes to the left of the first nonzero digits are not significant. (0.012 has 2

sig figs)

o Zeroes to the right of a decimal point in a number are significant. (0.100

has 3 sig figs)

o When a number ends in zeroes that are not to the right of a decimal point,

the zeroes are not necessarily significant. (120 may have 2 or 3 sig figs)

To avoid ambiguity, use scientific notation. (1.9 x 102 has 2 sig

figs and 1.90 x 102 has 3 sig figs)

2. Some numbers are exact because they are known with complete certainty. Exact

numbers have an infinite number of significant figures. For example: there are

exactly 60 seconds in 1 minute. 325 seconds = 5.42 minutes.

3. For addition or subtraction, round off the result to the leftmost decimal place. For

example: 40.123+20.34=60.46.

4. For multiplication or division, round off the result to the smallest number

of significant figures. For example: 1.23x2.0=2.5, not 2.46.

5. For logarithms, retain in the mantissa (the number to the right of the decimal

point in the logarithm) the same number of significant figures as there are in the

number whose logarithm you are taking. For example: log(12.8)=1.107. The

mantissa is .107 and has 3 sig figs because 12.8 has 3 sig figs)

6. For exponents, the number of sig figs is the same as in the mantissa. For example

101.23 = 17 or 1.7 x 101, which has 2 sig figs because there are 2 sig figs in the

mantissa (.23).

7. For multiple calculations, compute the number of significant digits to retain in the

same order as the operations: first logarithms and exponents, then multiplication

and division, and finally addition and subtraction.

o When parentheses are used, do the operations inside the parentheses first.

o To avoid round off errors, keep extra digits until the final step.

8. When determining the mean and standard deviation based on repeated

measurements:

o The mean cannot be more accurate than the original measurements. For

example, the maximum number of significant figures in the mean when

averaging measurements with 4 sig figs is 4.

o The standard deviation provides a measurement of experimental

uncertainty.

o Experimental uncertainty should almost always be rounded to

one significant figure. The only exception is when the uncertainty (if

written in scientific notation) has a leading digit of 1 when a second digit

should be kept.

For example if the average of 4 masses is 1.2345g and the standard

deviation is 0.323g, the uncertainty in the tenths place makes the

following digits meaningless so the uncertainty should be written as

+/- 0.3. The number of significant figures in the value of the mean is

determined using the rules of addition and subtraction. The value

should be written as (1.2 +/- 0.3)g.

The exception is when the uncertainty (if written in scientific

notation) has a leading digit of 1 when a second digit should

be kept. For example (1.234 +/- 0.172)g should be written as

(1.23 +/- 0.17)g.

In the (less common) event that the uncertainty is in a digit that

is not significant, report it as such.

For example the value of the average of 122 and 123 should

be reported as 122 +/- 0.5. (Note that the mean value of

122.5 is rounded to the even digit. If it were 123.5, it would

be rounded to 124.)

What are the five rules of significant figures? Rules For Determining If a Number Is Significant or Not All non-zero digits are considered significant. ... Zeros appearing between two non-zero digits (trapped zeros) are significant. ... Leading zeros (zeros before non-zero numbers) are not significant. ... Trailing zeros (zeros after non-zero numbers) in a number without a decimal are generally not significant (see below for more details). ... More items...

Author: Jeffrey J. Gray

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