In chemistry and all experimental sciences, every measurement carries some degree of uncertainty because no measuring instrument is perfectly precise. Significant figures (sig figs) are the reliable digits in a measured number that communicate the precision of the device used. Following these rules ensures that calculated results honestly reflect the limitations of the original data and prevents reporting an answer that suggests greater accuracy than the equipment can provide.
Rules for Identifying Significant Figures
Determining the number of significant figures begins with the rule that all non-zero digits are always counted as significant (e.g., 437.2 grams has four sig figs). Zeroes follow specific rules based on their location. Captive zeros, found between two non-zero digits, are always significant, meaning 4005 kilograms contains four significant figures.
Leading zeros, which appear before the first non-zero digit, are never significant as they only locate the decimal point (e.g., 0.0052 grams has two sig figs). Trailing zeros are only significant if the number contains a decimal point; 200 liters is ambiguous, but 200.0 liters clearly contains four sig figs. Scientific notation is the preferred method to avoid this ambiguity.
Applying Significant Figures in Multiplication and Division
When multiplying or dividing measured quantities, the final answer must be limited to the same total number of significant figures as the input measurement that has the fewest significant figures. This ensures the calculated result does not imply greater precision than the least-precise initial measurement.
For example, dividing \(12.55\) grams (four sig figs) by \(3.0\) milliliters (two sig figs) yields a raw answer of \(4.18333…\) grams per milliliter. Since \(3.0\) milliliters is the limiting measurement, the final result must be rounded to two significant figures, yielding \(4.2\) grams per milliliter.
Applying Significant Figures in Addition and Subtraction
The rule for addition and subtraction focuses on the precision of the place value, differing from the rules for multiplication and division. When adding or subtracting measurements, the result must be rounded to the same number of decimal places as the measurement with the fewest decimal places. This ensures the answer is not more precise than the measurement known with the least certainty at the decimal level.
For instance, adding \(35.7\) milliliters (known to the tenths place) to \(6.342\) milliliters (known to the thousandths place) yields a raw sum of \(42.042\) milliliters. Since \(35.7\) milliliters is the limiting measurement, the final answer must be rounded to the tenths place. The correctly reported result is \(42.0\) milliliters.
Handling Exact Numbers and Defined Constants
Exact numbers are a significant exception to the standard rules because they have no uncertainty associated with them. These numbers are considered to have an infinite number of significant figures and therefore never limit the precision of a calculation. Exact numbers arise from counting discrete objects or from defined constants.
Examples include counting exactly 12 test tubes or defined conversion factors, such as exactly 100 centimeters in 1 meter. When an exact number is used alongside a measured value, only the measured value’s significant figures determine the final precision of the result.