Significant Figures

Consider these measurements made on a low-precision balance: 10.4, 10.2, and 10.3 g. The reported result is best expressed as their average, that is, 10.3 g. A scientist would interpret these results to mean that the first two digits-10- are known with certainty and the last digit-3-is uncertain because it was estimated. That is, the mass is known only to the nearest 0.1 g, a fact that we could also express by writing 10.3 ± 0.1 g. To a scientist, the measurement 10.3 g is said to have three significant figures. If this mass is reported in kilograms rather than in grams, 10.3 g = 0.0103 kg, the measurement is still expressed to three significant figures even though more than three digits are shown. When measured on an analytical balance, the corresponding reported value might be 10.3107 g-a value with six significant figures. The number of significant figures in a measured quantity gives an indication of the capabilities of the measuring device and the precision of the measurements. We will frequently need to determine the number of significant figures in a numerical quantity. The rules for doing this, outlined in figure below, are as follows:

A. All nonzero digits are significant

B. Zeros are also significant, but with two important exceptions for quantities less than one.

C. Any zeros (1) preceding the decimal point, or (2) following the decimal point and preceding the first nonzero digit, are not significant.

D. The case of terminal zeros that precede the decimal point in quantities greater than one is ambiguous.

The quantity 7500 m is an example of an ambiguous case.

Do we mean 7500 m, measured to the nearest meter? Nearest 10 meters? If all the zeros are significant if the value has four significant figures we can write 7500. m. That is, by writing a decimal point that is not otherwise needed, we show that all zeros preceding the decimal point are significant. This technique does not help if only one of the zeros, or if neither zero, is significant.

The best approach here is to use exponential notation. The coefficient establishes the number of significant figures, and the power of ten locates the decimal point.

Significant Figures in Numerical Calculations

Precision must neither be gained nor be lost in calculations involving measured quantities. There are several methods for determining how precisely to express the result of a calculation, but it is usually sufficient just to observe some simple rules involving significant figures.

The result of multiplication or division may contain only as many significant figures as the least precisely known quantity in the calculation.

In the following chain multiplication to determine the volume of a rectangular block of wood, we should round off the result to three significant figures. Figure below may help you to understand this.

In adding and subtracting numbers, the applicable rule is as follows.

“THE RESULT OF ADDITION OR SUBTRACTION MUST BE EXPRESSED WITH THE SAME NUMBER OF DIGITS BEYOND THE DECIMAL POINT AS THE QUANTITY CARRYING THE SMALLEST NUMBER OF SUCH DIGITS. CONSIDER THE FOLLOWING SUM OF MASSES.”

The sum has the same uncertainty, ± 0.1 g, as does the term with the smallest number of digits beyond the decimal point, 9986.0 g. Note that this calculation is not limited by significant figures. In fact, the sum has more significant figures (six) than do any of the terms in the addition.

There are two situations when a quantity appearing in a calculation may be exact, that is, not subject to errors in measurement. This may occur

1.    By definition (such as 1 min = 60 s, or 1 inch = 2.54 cm)

2.    As a result of counting (such as six faces on a cube, or two hydrogen atoms in a water molecule)

Exact numbers can be considered to have an unlimited number of significant figures.

Rounding Off Numerical Results

To three significant figures, we should express 15.453 as 15.5 and 14,775 as 1.48 x 10⁴. If we need to drop just one digit, that is to round off a number, the rule that we will follow is to increase the final digit by one unit if the digit dropped is 5, 6, 7, 8, or 9 and to leave the final digit unchanged if the digit dropped is 0, 1, 2, 3, or 4.* To three significant figures, 15.44 rounds off to 15.4, and 15.45 rounds off to 15.5.

Applying Significant Figure Rules: Multiplication/Division

Applying Significant Figure Rules: Addition/Subtraction

In working through the preceding examples, you likely used an electronic calculator. What’s nice about using electronic calculators is that we don’t have to write down intermediate results. In general, disregard occasional situations where intermediate rounding may be justified and store all intermediate results in your electronic calculator without regard to significant figures. Then, round off to the correct number of significant figures only in the final answer.