Significant figures: the rules, with examples

Significant figures are the digits in a number that carry meaningful information about its precision. The rules are short: non-zero digits always count, zeros between them count, leading zeros never count, and trailing zeros count only when there is a decimal point. When you calculate, the answer should be no more precise than the least precise measurement you started with. The significant figures calculator counts them and tells you which rule applied, and this guide works through the rules with examples.

Why significant figures matter

A measurement is never exact. A ruler marked in millimetres can tell you a pencil is 12.3 cm long, but not 12.347 cm. Writing 12.3 says you know three digits; writing 12.300 would claim precision you do not have. Calculators make this worse, because dividing 10.0 by 3.0 gives 3.3333333, and copying all those digits suggests a precision the inputs never had. Significant figures are a simple way to keep answers honest.

The counting rules

• Non-zero digits always count. 4.56 has 3 significant figures.
• Zeros between non-zero digits count. 1002 has 4; 3.05 has 3.
• Leading zeros never count. They only place the decimal point: 0.0045 has 2, and so does 0.000045.
• Trailing zeros after a decimal point count. 2.50 has 3, and 100.0 has 4, because someone chose to write those zeros.
• Trailing zeros in a whole number without a decimal point are ambiguous. They are usually not counted, so 4560 is taken to have 3.

Put together: 0.004560 has 4 significant figures. Counting starts at the 4, and the final zero counts because it follows a decimal point.

Trailing zeros

The awkward case is a number like 4560 or 1500. If a crowd is estimated at 1500, the zeros are probably placeholders; if a scale reads 1500 g to the nearest gram, they are real. The number alone cannot say which. Scientific notation removes the doubt: 1.5 × 10³ has 2 significant figures, 1.500 × 10³ has 4. That is the main reason scientists write measurements this way; Scientific notation explained, with e-notation and engineering notation covers the notation, and the scientific notation converter switches forms. The calculator follows the usual convention and does not count trailing zeros in a whole number, telling you so when it applies.

Exact numbers are a separate case. Counted values like 12 eggs, and defined values like 100 cm in a metre, have unlimited significant figures and never limit the precision of an answer.

Multiplying and dividing

For multiplication and division, the answer keeps as many significant figures as the input with the fewest:

• 4.56 × 1.4 = 6.384. The inputs have 3 and 2 significant figures, so the answer is 6.4.
• 9.8 ÷ 3.14159 = 3.1194… The 9.8 has only 2, so the answer is 3.1, however many digits of π you used.

The logic is that a relative error carries through multiplication: if one input is only good to about 1 part in 100, the answer cannot be better.

Adding and subtracting

Addition and subtraction use a different rule: the answer keeps as many decimal places as the input with the fewest decimal places. Significant figures do not matter here, only the position of the last reliable digit.

• 12.11 + 0.3 = 12.41, which rounds to 12.4, because 0.3 is only known to one decimal place.
• 123.25 + 46.0 + 86.256 = 255.506, which rounds to 255.5.

Mixing the two rules up is the most common mistake. Note that adding can increase the number of significant figures, as in 9.8 + 0.5 = 10.3, while subtracting nearly equal numbers can wipe most of them out: 10.52 − 10.47 = 0.05, with only one significant figure left.

Rounding well

Two habits avoid most rounding errors. First, keep a couple of extra digits through every intermediate step and round only the final answer; rounding at each step lets errors build up. Second, know which tie-breaking rule you are using when the dropped digit is exactly 5. Rounding half up turns 2.45 into 2.5 at two significant figures, while rounding half to even, common in statistics and computing, gives 2.4. The rounding calculator shows the rules side by side. The sig figs tool rounds any number you type to between 1 and 6 significant figures, and the scientific calculator keeps full precision for the steps in between.