Rules for Significant Figures

Significant figures are all digits in a measured quantity, including the last uncertain digit, that reflect the precision of the measuring instrument. They represent how precisely a measurement was taken. The certainty of any measurement depends on both the number of digits recorded and the precision of the instrument used. More precise instruments yield measurements with more significant figures.

Start from the first non-zero digit on the left and count all digits through the last digit written on the right. Non-zero digits and captive zeros between non-zero digits are always significant. Leading zeros never count as they only locate the decimal point. For example, 0.00208 has three significant figures, while 26.25 has four.

Leading zeros serve only to locate the decimal point and do not represent measured precision. They are placeholders that indicate the magnitude of the number. For instance, 0.00208 and 2.08 × 10−3 represent the same value with identical significant figures. The zeros in front simply position the decimal without adding measurement information.

Trailing zeros in decimal-formatted numbers are always significant. However, trailing zeros in whole numbers without decimal points create ambiguity. For example, 2200 has two significant figures, but 2200.0 has five. Using exponential notation like 2.2 × 103 clarifies that only two figures are significant, while 2.20 × 103 indicates three.

More precise measuring tools produce measurements with more significant figures. An ordinary ruler measures to the nearest millimeter, while a caliper measures to 0.01 mm, making it more precise. The last digit in any measurement is estimated by the person taking it, and this uncertainty is reflected in the count of significant figures recorded.

The last digit written in a measurement is the first digit with some uncertainty because it has been estimated by the person making the measurement. For example, when measuring a stick's length between 13.2 cm and 13.3 cm with a ruler, the observer must estimate the final decimal place. This estimation reflects the limit of the measuring instrument's precision.

Exponential notation eliminates ambiguity about trailing zeros in whole numbers. The number 2200 could have two or three significant figures, but writing it as 2.2 × 103 clearly shows two significant figures, while 2.20 × 103 shows three. This notation makes the precision of the measurement explicit and unambiguous.