Significant Figures Calculator

Significant figures are the digits in a measurement that carry information. A scale that reads 0.0040500 kg is telling you five things: that the first non-zero digit is a 4 in the thousandths place, and that the four digits from there to the end are known. The zeros in front are not measurements at all; they only place the decimal point. Counting significant figures is counting the digits that were measured, and the rules are a way of telling those apart from the ones that only hold a place.

There are four rules, and they all concern zeros, because non-zero digits are always significant. Leading zeros never count: 0.0045 has two significant figures, the same as 4.5 mm written in a smaller unit. Captive zeros, between two non-zero digits, always count: 405 has three, because the middle 0 is a measured tens digit. Trailing zeros after a decimal point always count: 2.50 has three, and writing that final zero is a claim that the hundredths digit was measured and found to be zero. And trailing zeros in a whole number with no decimal point are ambiguous.

That last case is the one every student meets and no rule can settle. 1500 might be a count accurate to the unit, or a measurement to the nearest hundred, and the digits alone do not say which. The usual convention is not to count those zeros, so 1500 has two significant figures, but that is a convention for reading other people's numbers, not a fact about them. The fixes are in the writing. A trailing decimal point, 1500., says all four digits are significant. Scientific notation removes the ambiguity entirely: 1.5 × 10^3 has two, 1.50 × 10^3 has three and 1.500 × 10^3 has four. That precision-carrying role, more than the convenience for very large or small numbers, is why measured values are so often written in scientific notation.

Rounding to a number of significant figures is ordinary rounding with a step set by the number itself. Three significant figures of 0.0040500 means cutting at the hundred-thousandths place, giving 0.00405; three of 40,560 means cutting at the hundreds, giving 40,600. That has the same trailing-zero ambiguity, so it is better written as 4.06 × 10^4. A carry can shift everything: 99.96 to three figures is 100., and the decimal point is the only thing that shows the result has three figures rather than one.

Arithmetic has two rules, and they are different because they protect different things. When you multiply or divide, the result keeps as many significant figures as the input with the fewest. 4.52 × 2.1 is 9.492 by arithmetic, but 2.1 is only known to two figures, so the answer is reported as 9.5. When you add or subtract, what matters is the last decimal place, not the count: 4.52 + 2.1 is 6.62, but 2.1 is only known to the tenths, so the sum is 6.6. Mixing the rules up is the most common error. Adding 1,500 to 12.3 to get 1,512.3 claims a precision the 1,500 never had; rounded to its last significant place, the hundreds, the sum is 1,500.

Two cautions. Exact numbers do not limit a result: a count of 12 items, the 2 in the formula for a circumference, or the defined 2.54 cm in an inch all have unlimited significant figures. And in a multi-step calculation, carry every digit through the intermediate steps and round once at the end. Rounding as you go accumulates error, and the final digit can come out wrong.

Significant figures are a rule of thumb for uncertainty, not a substitute for it. A physics or chemistry lab that states an uncertainty explicitly (4.52 ± 0.03) is using the more precise tool; the counting rules are what you use when that uncertainty is only implied by how the number is written.