Mastering Significant Figures: Precision, Calculations, and Rounding Rules

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.

Rules for Determining Significant Figures

A. All nonzero digits are significant. B. Zeros are also significant, with two important exceptions for quantities less than one:

• Any zeros preceding the decimal point.
• Any zeros following the decimal point but preceding the first nonzero digit. C. Zeros between nonzero digits are always significant. D. Terminal zeros preceding the decimal point in numbers greater than one are ambiguous.

For example, in the quantity 7500 m, it is unclear whether we mean 7500 m measured to the nearest meter or to the nearest 10 meters. If all the zeros are significant (indicating four significant figures), we can write 7500. m. To remove ambiguity, exponential notation should be used. The coefficient determines the number of significant figures, while the power of ten locates the decimal point.

Identifying Significant and Non-Significant Zeros

• Significant: Zeros between nonzero numbers (e.g., 4500.7 has five significant figures).
• Significant: Zeros at the end of a number and to the right of a decimal point (e.g., 7.500 × 10³ m has four significant figures).
• Not significant: Zeros used for cosmetic purposes or just to locate the decimal point (e.g., 0.00400 has three significant figures; the first two zeros are not significant).

Significant Figures in Numerical Calculations

Precision must neither be gained nor lost in calculations involving measured quantities. The number of significant figures in the result of a calculation is determined by the least precise measurement involved.

Multiplication and Division Rule:

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

For example, consider the calculation 14.79 cm × 12.11 cm × 5.05 cm. The least precisely known value is 5.05 cm (three significant figures). The calculator result may show more digits, but variations begin in the third digit, so we must round accordingly. The final volume should be expressed as 904 cm³, rounded to three significant figures.

Addition and Subtraction Rule:

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.

For instance:

9986.0 g

+   12.11 g

-------------

9998.1 g

Since 9986.0 g has one digit beyond the decimal point, the sum must also have one digit beyond the decimal point.

Exact Numbers and Significant Figures

There are two cases where numbers are considered exact and have an unlimited number of significant figures:

1.  By definition (e.g., 1 min = 60 s, 1 inch = 2.54 cm).

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

Rounding Off Numerical Results

To three significant figures, numbers should be rounded as follows:

• 15.453 → 15.5
• 14,775 → 1.48 × 10⁴

If rounding off a number, increase the final digit by one unit if the digit dropped is 5, 6, 7, 8, or 9. If the dropped digit is 0, 1, 2, 3, or 4, leave the final digit unchanged.

For example:

• 15.44 → 15.4
• 15.45 → 15.5

Best Practices for Using Significant Figures in Calculations

When using electronic calculators, store all intermediate results without rounding. Apply rounding only to the final answer to avoid unnecessary errors due to early rounding. This ensures accuracy while maintaining proper significant figure conventions.