Significant figures calculator

What are significant figures?

Significant figures are the digits in a number that carry real information about how precisely it was measured. If a ruler reads 12.4 centimetres, the 1, the 2 and the 4 are all telling you something and the number has three significant figures. Writing 12.400 would be a stronger claim: it says you know the value to a thousandth of a centimetre, which that ruler cannot support.

The rules are short. Every non-zero digit counts. Zeros between non-zero digits count. Leading zeros never count, because they only place the decimal point, so 0.0012 has two significant figures and not four. Trailing zeros after a decimal point count, which is why 1.200 has four and 1.2 has two even though the two numbers are equal.

The one case where the rules run out is trailing zeros in a whole number, and that is the case people actually look up. This calculator handles that case by refusing to guess.

How to use it

1. Pick what you want to do. Count them tells you how many significant figures a number has and how many decimal places. Round to gives you the number rounded to between one and six of them, in plain and scientific notation. Calculate takes two numbers and an operation.
2. Type the number as it is actually written. The notation matters, so 1200, 1200. and 1.2e3 are three different claims about precision and the tool treats them as such. Spaces and thousands separators are ignored, and negatives and scientific notation are both accepted.
3. Read the rule as well as the answer. In calculate mode the panel names which rule the operation follows and how many figures or places it kept, because that is where most mistakes happen. It also shows the unrounded result, so you can see what was thrown away.

Why 1200 has no single answer

Write 1200 and you have said nothing about whether those two zeros were measured or are simply holding the decimal point in place. If you counted 1200 people one by one, all four digits are significant. If you rounded 1183 to the nearest hundred, only two are. The string on the page is identical in both cases, so there is no rule that recovers the intent, and a calculator that answers this with a single number has quietly made a decision that was yours to make.

This tool reports a range instead: 1200 has between two and four significant figures. It then shows you the three ways to write it that remove the doubt, because the fix is notation rather than convention. Scientific notation is the cleanest, since the mantissa contains exactly the digits you are claiming: 1.2 × 10³ is two figures, 1.20 × 10³ is three, 1.200 × 10³ is four. A trailing decimal point, as in 1200., is the older shorthand for all four, and some style guides use an overbar on the last significant zero.

Note where the ambiguity does not arise. A decimal point anywhere in the number pins every digit down, so 120.0 is unambiguously four and 0.00120 is unambiguously three. An integer that does not end in zero is never ambiguous either. It is specifically the trailing zeros of a bare whole number that cannot be read, which is why every power of ten is the worst case: 1000000 could be anything from one figure to seven.

The rule depends on the operation, and most tools get it wrong

Multiplication and division keep the fewest significant figures of the inputs. Addition and subtraction keep the fewest decimal places. These are different rules measuring different things, and swapping them is the most common defect in calculators of this kind.

The reason they differ is worth a sentence, because it makes the rule memorable rather than arbitrary. When you multiply, the relative uncertainties combine, and significant figures are a measure of relative precision. When you add, the absolute uncertainties combine, and decimal places are a measure of absolute precision. Adding 1000.0 and 0.25 gives 1000.3, because the first number is only known to a tenth; applying the significant-figures rule instead would round a perfectly good answer down to 1000.

We measured how much this matters. Over 20,000 random pairs written with explicit decimal points, applying the multiplication rule to a sum changes the answer 57.1% of the time. It is not an edge case, it is most of them.

One further habit is worth more than either rule: round once, at the end. Carry the full precision through a multi-step calculation and round only the final answer. Rounding after every step changed the result in 28.4% of three-operand chains we tested, with a median gap of 6.7%. The largest gaps are all at one significant figure, where the only available answers either side are 1 × 10ᵏ and 2 × 10ᵏ, so a single step of difference is necessarily a doubling.

Honest limits

Significant figures are a convention for tracking precision, not a measurement of it. They are a rule of thumb taught because they are quick and because they usually give the right order of magnitude for the uncertainty. Real error analysis propagates the uncertainties themselves, and where the answer actually matters, that is what you should be doing instead.

The rules also have widely taught exceptions this tool does not try to guess at. Exact counts and defined constants have unlimited significant figures: if there are 12 items, the 12 does not limit anything. So do conversion factors that are exact by definition, such as 2.54 centimetres to the inch. If one of your inputs is exact, the limiting figure count comes from the other one alone, and you will need to tell the tool that by entering enough digits.

Rounding here is half away from zero, so 1250 to two significant figures gives 1300. That matches the ROUND function in every spreadsheet, which is what most people are checking against. Some scientific conventions round half to even instead, which would give 1200, and neither is more correct than the other; they simply differ, and the difference only shows up on an exact half.

Finally, division is carried to thirty digits before rounding, so a non-terminating quotient is shown truncated rather than exact and the panel says so. Every other operation here is exact: the numbers are held as an integer digit string and a power of ten, and nothing passes through binary floating point on the way.

Why is it free?

Because it is arithmetic on digit strings and there is nothing to pay for. No server sees your numbers: everything runs in your browser, and nothing you type is uploaded, logged or stored anywhere.

So there is no account, no sign-up and nothing held back. The engine and the 74 checks that verify it sit in the repository beside the page, including a differential test against Python's own decimal module and the two sweeps behind the figures quoted above.