### 1.11: Uncertainty in Measurement: Significant Figures

All the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits. Note that zero may be a measured value; for example, if a scale that shows weight to the nearest pound reads “140,” then the 1 (hundreds), 4 (tens), and 0 (ones) are all significant (measured) values.

A measurement result is properly reported when its significant digits accurately represent the certainty of the measurement process. Below are a set of rules to determine the number of significant figures in a measurement:

- All
**nonzero**digits are significant. Starting with the first nonzero digit on the left, count this digit and all remaining digits to the right. This is the number of significant figures in the measurement. For example, 843 has three significant digits, 843.12 has 5 significant digits. **Captive zeros**, which are zeros between two nonzero digits, are significant. For example, 8__0__8.1__0__1 has two captive zeros and 6 significant figures.**Leading zeros**are zeros to the left of the first nonzero digit. Leading digits are never significant; they merely represent the position of the decimal point. For example, the leading zeros in 0.008081 are not significant. This number can be expressed using exponential notation as 8.081 × 10^{−3}, then the number 8.081 contains all the significant figures, and 10^{−3}locates the decimal point.- The significance of
**trailing zeros**, which are zeros at the end of a number, depends on their position. Trailing zeros before (but after a nonzero digit) and after the decimal point are significant. However, for numbers that do not have decimal points, trailing zeros may or may not be significant. This ambiguity can be resolved with the use of exponential notation. For example, the measurement 1300 can be written as 1.3 × 10^{3}(two significant figures), 1.30 × 10^{3}(three significant figures, if the tens place was measured), or 1.300 × 10^{3}(four significant figures, if the ones place was also measured).

#### Significant Figures in Calculations

Uncertainty in measurements can be avoided by reporting the results of calculation with the correct number of significant figures. This can be determined by the following the rules for rounding numbers:

- When adding or subtracting numbers, round the result to the same number of decimal places as the number with the least number of decimal places.
- When multiplying or dividing numbers, round the result to the same number of digits as the number with the least number of significant figures.
- If the digit to be dropped (the one immediately to the right of the digit to be retained) is less than 5, “round down” and leave the retained digit unchanged.
- If the digit to be dropped (the one immediately to the right of the digit to be retained) is 5 or greater, “round up” and increase the retained digit by 1. Alternative rounding methods may also be used if the dropped digit is 5. The retained digit is rounded up or down, whichever yields an even value.

An important note is that rounding of significant figures should preferably be done at the end of a multistep calculation to avoid the accumulation of errors at each step due to rounding. Thus, significant figures and rounding, facilitate correct representation of the certainty of the measured values reported.

*This text is adapted from Openstax, Chemistry 2e, Section 1.5: Measurement Uncertainty, Accuracy, and Precision.*