Significant Figures: The Rules, With Worked Examples

Published 2026-03-31 · By the CalculatorHive editorial team

Significant figures are how you tell a reader the precision of a measurement without writing an error bar. Report a length as 2.5 cm and you are claiming to know it to about a millimetre; report it as 2.500 cm and you are claiming ten times better. Getting this wrong in a lab report costs marks; getting it wrong in engineering costs more. Here are the rules, the ambiguity nobody warns you about, and the arithmetic conventions that actually differ between operations.

What are the rules for counting significant figures?

Read the number left to right and apply four rules in order.

1. Every non-zero digit is significant. 4,782 has four.
2. Zeros between non-zero digits (captive zeros) are always significant. 4,502 has four; 30.06 has four.
3. Leading zeros are never significant. They only place the decimal point. 0.0034 has two; 0.000500 has three (the two trailing zeros count, the leading ones do not).
4. Trailing zeros are significant only if the number contains a decimal point. 38.00 has four. 3,800 has two — or possibly three or four; see the next section.

Why are trailing zeros in whole numbers ambiguous?

Write 4,500 and you have said nothing about whether you measured to the nearest unit, the nearest ten, or the nearest hundred. The two zeros might be measured digits or they might be placeholders holding the 4 and 5 in position. Plain decimal notation has no way to distinguish them.

Scientific notation solves this completely, because the placeholders move into the exponent and only measured digits remain in the mantissa:

• 4.5 × 10³ — two sig figs, measured to the nearest hundred.
• 4.50 × 10³ — three sig figs, measured to the nearest ten.
• 4.500 × 10³ — four sig figs, measured to the nearest unit.

This is the main reason scientific work uses the notation even for numbers that are not especially large or small. The alternatives — a trailing decimal point (4500.) or an overbar on the last significant digit — are older conventions that survive in textbooks but are easy to lose in typesetting. Convert with the scientific notation calculator if you'd rather not shift decimal points by hand.

How do significant figures work in multiplication versus addition?

This is the point most people get wrong, because they learn one rule and apply it everywhere. The two operations use different rules for a real physical reason.

Multiplication and division: match the fewest significant figures. Relative uncertainties add when you multiply, so the least precise factor — measured as a fraction of itself — governs the result.

Worked example. You weigh a sample at 4.237 g (four sig figs) and measure its volume in a graduated cylinder as 1.6 mL (two sig figs). Density = 4.237 ÷ 1.6 = 2.6481 g/mL. The volume carries only two sig figs, so the answer is 2.6 g/mL. Reporting 2.6481 would claim a precision the cylinder cannot deliver.

Addition and subtraction: match the fewest decimal places. Absolute uncertainties add here, so what matters is the rightmost place any input actually resolves.

Worked example. 12.11 + 18.0 + 1.013 = 31.123. The 18.0 term is known only to one decimal place, so the sum is 31.1 — regardless of the fact that two of the three inputs have four significant figures.

Look at row four. Both inputs have four significant figures, yet the difference has only one. That collapse is called subtractive cancellation, and it is the reason experiments that measure a small difference between two large numbers are so hard to do well — the precision of your instruments is mostly destroyed by the subtraction. Row five is the mirror image: a one-sig-fig input yields a five-sig-fig sum, which is perfectly correct under the decimal-places rule.

How should I round, and when?

Round once, at the very end. If a calculation has several steps, carry at least two extra digits through the intermediate results and round only the final answer. Rounding at each step introduces errors that compound, and in a long chain they can shift the last reported digit.

For the rounding itself there are two conventions:

• Round half up — the school rule. 2.45 → 2.5, 2.25 → 2.3. Simple, but it biases a large batch of results slightly upward, because ties always go the same way.
• Round half to even (banker's rounding, the IEEE 754 default and the ASTM/analytical-chemistry convention). When the part being dropped is exactly half, round to make the last kept digit even: 2.45 → 2.4, 2.35 → 2.4, 2.25 → 2.2. Ties split evenly between up and down, so the bias cancels over many values.

Both are defensible; what matters is stating which you used and applying it consistently. The rounding calculator supports several modes if you need to check a boundary case, and the significant figures calculator will count and round to a chosen number of sig figs in one step.

Two special cases worth knowing:

• Exact numbers have unlimited significant figures. Counted objects (12 flasks) and defined conversions (1 inch = 2.54 cm exactly, 1 minute = 60 seconds) never limit your result. Dividing a 3.72 g total by exactly 4 samples still leaves three sig figs.
• Logarithms keep sig figs in the mantissa. For a log, only the digits after the decimal point carry precision — the characteristic just records the power of ten. A pH of 4.35 has two significant figures, corresponding to a hydrogen ion concentration of 4.5 × 10⁻⁵ M.

Where significant figures meet percent error

Suppose a pendulum experiment gives 9.72 m/s² for gravitational acceleration against an accepted 9.81 m/s². Percent error is |9.72 − 9.81| ÷ 9.81 × 100 = 0.9174%. But look at the numerator: both values are known to two decimal places, so their difference, 0.09, carries only one significant figure. The division then inherits that limit, and the honest report is 0.9%, not 0.92% and certainly not 0.9174%. The percent error calculator gives the full-precision figure; you decide how many digits are defensible.

Common questions

Is zero ever a significant figure?

Frequently. A zero is significant whenever it is a measured digit rather than a placeholder: between non-zero digits (4,502), or trailing a decimal point (38.00, 0.0500). It is never significant when it is only positioning the decimal point, which is why leading zeros in 0.0034 never count no matter how many there are.

How many guard digits should I carry through a multi-step calculation?

Two beyond your intended final precision is the usual advice, and keeping the full calculator display is better still. If you are reporting three sig figs, work at five or more internally. The one thing never to do is retype a rounded intermediate value into the next step — that locks the rounding error in permanently.

What is the difference between significant figures and decimal places?

Decimal places count digits to the right of the decimal point only; significant figures count all measured digits wherever they sit. 0.00123 has five decimal places but three significant figures. 1234 has zero decimal places and four significant figures. Multiplication rules use sig figs; addition rules use decimal places — which is exactly why the two must not be confused.

Do significant figures apply to numbers I look up, like constants?

Yes, and they usually limit you less than your own measurements do. Published constants are given to many digits precisely so they don't become the weakest link — use enough of them that the constant is more precise than your data, then let your measurement set the final precision. If you truncate a constant to two digits, you have artificially made it the limiting factor.