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Significant Figures (Digits)

When reporting values that were the result of a measurement or calculated using measured values, it is important to have a way to indicated the certainty of the measurement. This is accomplished through the use of significant figures.  Significant figures are the digits in a value that are known with some degree of confidence.  As the number of significant figures increases, the more certain the measurement.  As precision of a measurement increases, so does the number of significant figures. Consider the weight measurements made using the following three instruments.  Notice that the number of significant digits increase as the measured value gets more precise and the range of uncertainty gets smaller. 

Instrument Measured Value Precision of Measurement Minimum Amount of Uncertainty in the Measurement Significant Figures of Measured Value
Postage Scale
    1g    +/- 0.5g
Two-pan balance
    0.01g    +/- 0.005g
Analytical balance
    0.001g    +/- 0.0005g

There are conventions that must be followed for expressing numbers so that their significant figures are properly indicated.  These conventions are:

All non zero digits are significant. 549 has three significant figures
1.892 has four significant figures
Zeros between non zero digits are significant. 4023 has four significant figures
50014 has five significant figures
Zeros to the left of the first non zero digit are not significant. 0.000034 has only two significant figures. (This is more easily seen if it is written as 3.4x10-5)
0.001111 has four significant figures.
Trailing zeros (the right most zeros) are significant when there is a decimal point in the number.  For this reason it is important to give consideration to when a decimal point is used and to keep the trailing zeros to indicate the actual number of significant figures.  400. has three significant figures
2.00 has three significant figures
0.050 has two significant figures
Trailing zeros are not significant in numbers without decimal points.
470,000 has two significant figures
400 or 4x102 indicates only one significant figure. (To indicate that the trailing zeros are significant a decimal point must be added. 400. has three significant digits and is written as 4.00x102 in scientific notation.)

Exact numbers have an infinite number of significant digits but they are generally not reported.

Defined numbers also have an infinite number of significant digits.

If you count 2 pencils, then the number of pencils is 2.000...

The number of centimeters per inch (2.54) has an infinite number of significant digits, as does the speed of light (299792458 m/s).

Maintaining Significant Digits in Calculations
Once the number of significant figures various values have been determined, the issue then becomes dealing with significant figures when these values are used in calculations. When combining values with different degrees of precision, the precision of the final answer can be no greater than the least precise measurement. However, it is a good idea to keep one more digit than is significant during the calculation to reduce rounding errors. In the end, however, the answer must be expressed with the proper number of significant figures.

Addition and Subtraction
When adding and subtracting, round the final result to have the same precision (same number of decimal places) as the least precise initial value, regardless of the significant figures of any one term.  For example,

98.112 + 2.3 = 100.412 but this value must be rounded to 100.4 (the precision of the least precise term).

Multiplication, Division, and Roots
When multiplying, dividing, or taking roots, the result should have the same number of significant figures as the least precise number in the calculation.  For example,

(3.69) (2.3059) = 8.5088, which should be rounded to 8.51 (three significant figures like 3.69). 

Logarithms and Antilogithms
When calculating the logarithm of a number, 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 is being found.  For example,

log(3.000x104) = 4.477121, which should be rounded to 4.4771

log(3x104) = 4.477121, but this value should be rounded to 4.5

When calculating the antilogarithm of a number, the resulting value should have the same number of significant figures as the mantissa in the logarithm.  For example,

antilog(0.301) = 1.9998, which should be rounded to 2.00,

antilog(0.30) = 1.9998, which should be rounded to 2.0

Multiple Mathematical Operations
If a calculation involves a combination of mathematical operations, perform the calculation using more figures than will be significant to arrive at a value.  Then, go back and look at the individual steps of the calculation and determine how many significant figures would carry through to the final result based on the above conventions. For example,

X = ((5.254+0.0016)/34.6) - 2.231*10-3

Calculate the value of X using more digits than will be significant.  In this case X = 0.1496649538

Then, go back and look at each piece of the equation to determine the significant figures.

5.254 + 0.0016 = 5.256 (since the sum is limited to the thousandths place by 5.254)

5.256 / 34.6 = 0.152 (since the quotient is limited to 3 significant figures by 34.6)

0.152 - 0.002231 = 0.149 (since the difference is limited to the thousandths place by 0.152)

The value initially obtained for X (0.1496649538) should be rounded to have 3 significant digits

Therefore, the final answer is 0.150 or 1.50x10-1

The Rules of Rounding
When a value contains too many significant figures, it must be rounded off.  There are two methods that are commonly used to minimize the error introduced into a value do to rounding.

Method 1: This method involves underestimating the value when rounding the five digits 0, 1, 2, 3, and 4, and overestimating the value when rounding the five digits 5, 6, 7, 8, and 9.  With this approach, if the value of the digit to the right of the last significant figure is smaller than 5, drop this digit and leave the remaining number unchanged. Thus, 2.794 becomes 2.79.  If the value of the digit to the right of the last significant digit is 5 or larger, drop this digit and add 1 to the preceding digit. Thus, 2.795 becomes 2.80.

Method 2:  This method takes into account that zero doesn't really require rounding and when rounding 5, this value is exactly centered between the underestimated value if it is rounded down and the overestimated value if it is rounded up.  Therefore, five should be rounded up half of the time and down half of the time.  Since it would difficult to keep track of this when performing numerous measurements or calculations, 5 is rounded down when the preceding significant digit is even and 5 is rounded up when the preceding significant digit is odd.  Values less than 5 are rounded down and values greater than 5 are rounded up.  For example, 2.785 would be rounded down to 2.78 and 2.775 would be rounded up to 2.78.