Significant figures in multiplication:
the answer is only as precise as its weakest input

Digits that carry information, and digits that do not

Every measured number has a precision limit. A ruler graduated in millimetres lets you say a length is 12.4 cm, with the last digit estimated; it does not let you say 12.4000 cm. Significant figures are the digits that the measurement can actually vouch for: all the certain ones, plus one estimated digit.

When two measurements are multiplied, the uncertainty of each carries into the product. A calculator happily prints 9.42 from 7.85 × 1.2, but the 1.2 might really be anything from 1.15 to 1.25, so the third digit of the product is pure decoration. The rounding rule exists to stop you claiming precision that you do not have.

Counting the significant figures in each input

Rounding the product is easy once you have counted the figures correctly. The counting rules are short, but the places people go wrong are always the zeros.

• Non-zero digits always count: 4.56 has three.
• Zeros between non-zero digits count: 1.205 has four.
• Leading zeros never count; they only mark the decimal place: 0.0350 has three, the 3, 5 and the final 0.
• Trailing zeros count when there is a decimal point: 2.30 has three, but 2300 without a point is ambiguous and is usually read as two.
• Exact numbers, such as 12 items in a dozen or a conversion defined as 100 cm per metre, have unlimited figures and never limit the answer.

The rule for products and quotients

Notice what the rule counts: figures, not decimal places. That is the difference from addition and subtraction, where the answer is limited by the least precise decimal place. Mixing the two rules up is probably the most frequent error in introductory courses.

1. 1Count the significant figures in every input.
2. 2Multiply with all digits kept in the calculator.
3. 3Identify the smallest count among the inputs.
4. 4Round the product to that many significant figures, and do it only once.

Worked example: 7.85 × 1.2

A rectangular plate measures 7.85 cm by 1.2 cm. The first measurement has three significant figures, the second two, so the product is limited to two.

Now a case where the rounding changes the leading digits. Multiply 2.3 by 4.56. The raw product is 10.488. The factor 2.3 has two figures, so the answer is limited to two, and 10.488 rounds to 10, which is written 1.0 × 10¹ to show that both digits are significant. Writing just 10 would leave the reader unsure whether the zero is meaningful.

Rounding once, at the end

In a chain calculation, round only the final result. Rounding after every step stacks small errors on top of each other. Keep one or two extra guard digits in the intermediate results and drop them at the end.

Take 7.85 × 1.2 × 3.4. The three-figure and two-figure inputs make the answer two figures. Multiplying straight through gives 32.028, which rounds to 32. If you had rounded 9.42 to 9.4 first, then multiplied by 3.4, you would have 31.96, which also rounds to 32 here, but with other numbers the two routes can end in different last digits.

Exact constants and conversion factors do not limit the result, but measured constants do. Use pi and standard conversion values with more figures than the data so they never become the weak link.

Division and mixed calculations

Quotients follow the same rule as products. A mass of 23.47 g (four figures) in a volume of 8.5 mL (two figures) gives a density of 23.47 ÷ 8.5 = 2.761 g/mL, which is reported as 2.8 g/mL because the volume limits it to two figures.

Mixed expressions take the rules in the order the operations occur. Consider (3.45 + 2.1) × 1.20. The sum is limited by the least precise decimal place, which is the tenths of 2.1, so 5.55 becomes 5.6. That intermediate value has two significant figures, so multiplying by 1.20 gives 6.72, reported as 6.7. Keeping the unrounded 5.55 as a guard value would give 6.66, which also rounds to 6.7, a handy confirmation that the answer is stable.

Another place the rule shows up is unit conversion with measured factors. Converting 4.5 km to metres with the exact factor 1000 m/km keeps two figures, giving 4.5 × 10³ m, not 4500 m, since the bare number would hide how precise the original was.

When the rule is only a rough guide

Teachers and journals differ on how strictly they apply the rule, and some ask for one extra figure in intermediate working. Follow the convention of the course or report you are writing for, and say so when you present a rounded value so the reader knows how much weight the final digit deserves.

The fewest-figures rule is a quick approximation of how uncertainty propagates. When a number is close to a power of ten boundary, such as 9.5 versus 10.4, the relative precision is different even though the count of figures is the same. A measurement written 9.9 has a relative uncertainty of about 0.5%, while 1.0 has about 5%; the rule treats both as two figures.

For laboratory reports that need defensible uncertainties, propagate them explicitly: add the relative uncertainties of the factors for a product. Significant figures remain the right shorthand for everyday work, homework and quick estimates.

A percentage tool can help with the follow-on step of expressing a rounded result as a share of something else, but it does not apply significant-figure rules, so round the inputs and outputs yourself.