Skip to content
ToolBoxGeniehome

Significant Figures Calculator

Calculators · Added

Significant figures are a property of how a number was written, not of its value — 1200, 1.2 × 10³ and 1200. are the same quantity with two, two and four figures. This counts them digit by digit and says why each one does or does not count, rounds correctly on the cases floating point gets wrong, and applies the right propagation rule to a calculation rather than the one people usually reach for.

What do you need

How it is written matters: 1200, 1200. and 1.2e3 are the same value with different precision.

Enter a number and press the button.

How to use the significant figures calculator

  1. 1Pick what you need: counting the figures in a number, rounding to a set number of them, or calculating with them.
  2. 2Type the number exactly as it was given to you — the trailing zeros and the decimal point are the information.
  3. 3For a calculation, put each measurement on its own line and choose one operation.
  4. 4Read the answer, and the explanation of which operand set the precision and why.

Examples

Counting, with the reason for each digit

Input
0.004560
Result
4 significant figures, 6 decimal places

The leading zeros only place the point; the trailing zero after the point is significant or it would not have been written.

The case a computer gets wrong

Input
1.005 to 2 decimal places
Result
1.01

Type (1.005).toFixed(2) into any browser console and it gives 1.00, because the double nearest 1.005 is slightly below it.

Two rules, two answers

Input
12.11 + 0.3 + 4.156, then 2.5 × 3.42
Result
16.6 (one decimal place) and 8.6 (two significant figures)

Addition keeps decimal places, multiplication keeps significant figures. Swapping the rules is the standard exam mistake.

About the significant figures calculator

The rules, and the one that is not a rule

Four of the five rules are unambiguous. Every non-zero digit is significant. Zeros between non-zero digits are significant. Leading zeros never are — they only place the decimal point, which is why 0.0045 has two figures and not four. Trailing zeros after a decimal point are significant, because nobody writes them by accident.

The fifth case is not a rule at all but a hole in the notation. Trailing zeros in a whole number with no decimal point could be either, and no convention settles it. Some teachers say assume they are not significant, some say assume they are, and some say the number is simply badly written. All three are defensible, and this tool takes the third position: it reports the range and shows you the scientific notation that would say what you meant.

Why this rounds on the digits rather than on a number

The obvious implementation of rounding is to parse the input into a floating point number and call the language's rounding function. It produces wrong answers on exactly the examples a page about precision is likely to be asked. In JavaScript, (1.005).toFixed(2) is "1.00", not "1.01", and (1.015).toFixed(2) is "1.01" — because the doubles nearest those decimals sit fractionally below them, so the correct rounding of the stored value is downward. The same trap catches Python's round and most spreadsheet functions.

So this parses the literal into a sign, a string of digits and an exponent, rounds by looking at the digit after the cut and carrying through any nines, and renders the string back. No floating point value exists anywhere in the path, and every textbook case comes out the way the textbook says it should.

The same reasoning is why the input is a string rather than a number. As soon as the digits become a value, the difference between 1200 and 1.2 × 10³ is gone, and with it the entire question the tool exists to answer.

Where the rules break down

Subtraction of two close numbers destroys precision in a way the rules describe but do not warn about. 1.00003 minus 1.00001 has six significant figures on each side and two in the answer. This is catastrophic cancellation, and it is the reason a measurement intended to be subtracted from another has to be far more precise than the difference needs to be. The tool flags it when the result collapses to a small fraction of the first operand.

Rounding also compounds. Applying the rules after every step of a long calculation gives a different answer from carrying full precision and rounding once at the end, and the difference grows with the number of steps. The usual advice — keep an extra digit or two through the working and round only the final answer — is right, and it is another reason a mixed expression has no single correct treatment.

Frequently asked questions

Why is 1200 ambiguous?
Because decimal notation has no way to say whether those trailing zeros were measured or are just holding the place. A ruler reading to the nearest hundred gives two significant figures; a precise instrument reporting exactly twelve hundred gives four. Nothing in the written form distinguishes them, which is why this tool reports a range rather than picking a side. Writing 1.2 × 10³ or 1.200 × 10³ removes the ambiguity entirely, and that is the reason scientific notation exists.
Why does multiplication use significant figures and addition use decimal places?
Because the two operations propagate different kinds of error. Multiplying combines relative uncertainties — a 1% error in one factor gives roughly a 1% error in the product — so the answer inherits the worst relative precision, which is what significant figures measure. Adding combines absolute uncertainties, so the answer is only as precise as the coarsest measurement's last decimal place. A sum of 12.11 and 0.3 cannot be known past the first decimal, however many figures the first number had.
Why does this not accept a mixed expression?
Because there is no single correct answer to give. A mixed expression needs the rule applied at each step in evaluation order, and textbooks genuinely disagree about whether intermediate results should be rounded as you go or carried at full precision and rounded once at the end. The two approaches give different answers. Offering one of them silently would be presenting a convention as the convention, so this handles one operation at a time and says so.
Are exact numbers counted?
No, and they should not be entered here. A count of 12 items, a conversion factor defined as exactly 2.54, or the 2 in a formula are exact — they have infinite significant figures and never limit the answer. Only measured quantities carry uncertainty. Putting an exact number into this calculator will make it the limiting operand and give you an answer rounded far more than it should be.
Are significant figures the same as an uncertainty calculation?
No. They are a rule of thumb for reporting, and a fairly crude one: they can be out by a factor of a few in either direction and they say nothing about which way the error lies. Where each measurement's uncertainty is actually known, propagate it properly with the standard partial-derivative formulas and quote the result with its uncertainty. Significant figures are what you use when you have the reading but not its error bar.

Related tools