Significant Figures in Chemistry Calculations

Measured numbers versus exact numbers

A measured value is uncertain because every instrument has a limit. When you read a buret graduated in 0.1 mL steps, you record the certain marks and then estimate one more digit between them, so a reading such as 18.36 mL has four significant figures: three certain digits and one estimated digit. Writing 18.360 mL would claim a precision the buret cannot deliver.

Exact numbers carry no measurement uncertainty. They come from counting (12 test tubes in a rack), from definitions (1 inch is exactly 2.54 cm, and 1 L is exactly 1000 mL), and from coefficients and subscripts in chemical formulas and equations. Treat exact numbers as having unlimited significant figures, so they never limit the precision of a calculated result.

Counting digits, including the troublesome zeros

Nonzero digits are always significant. Zeros are significant or not depending on whether they report measured information or only mark the position of the decimal point. The rules below cover every case you will meet in general chemistry.

Trailing zeros in a whole number without a decimal point are the one truly ambiguous case. A mass written as 1200 g might be known to two, three, or four significant figures. Scientific notation removes the ambiguity: 1.2 × 10³ g has two, 1.20 × 10³ g has three, and 1.200 × 10³ g has four.

• Zeros between nonzero digits are significant: 30.506 g has five.
• Leading zeros are never significant; they only place the decimal point: 0.004070 m has four (4, 0, 7, 0).
• Trailing zeros to the right of a decimal point are significant: 2.500 mL has four.
• Trailing zeros in a whole number with no decimal point are ambiguous; use scientific notation to state your intent.

Adding and subtracting: line up the decimal places

When quantities are added or subtracted, the uncertainty that matters is the absolute uncertainty in the last decimal place, not the count of significant figures. Round the result to the same number of decimal places as the measurement with the fewest decimal places.

Worked example 1. You combine three portions of a solid weighed on different balances: 12.6 g, 0.237 g, and 105.42 g. The calculator shows 118.257 g. The first mass is known only to the tenths place, so the sum is known only to the tenths place: 118.3 g. Notice that the answer has four significant figures even though one input had only three; in addition, digit counts are not what limit the result.

Subtraction can destroy significant figures. Suppose a flask containing a liquid weighs 58.36 g on a top-loading balance and the empty flask weighed 57.912 g on an analytical balance. The difference is 0.448 g on the calculator, but the first mass is known only to the hundredths place, so the liquid's mass is 0.45 g. Two significant figures survive from inputs that had four and five. When a small difference between two large numbers matters, weigh both on the more precise balance.

Multiplying and dividing: count significant figures

For products and quotients, the relative uncertainty is what carries through. Round the result to the same number of significant figures as the input with the fewest significant figures.

Worked example 2. A rectangular block measures 4.20 cm by 2.15 cm by 1.8 cm. The volume is 4.20 cm × 2.15 cm × 1.8 cm = 16.254 cm³ on the calculator. The height has only two significant figures, so the volume is reported as 16 cm³. Measuring the height more carefully, say as 1.80 cm with a caliper, would be the only way to justify a third digit.

Logarithms follow a related rule that appears in pH work: the number of decimal places in the logarithm equals the number of significant figures in the original value. A hydrogen-ion concentration of 3.4 × 10⁻⁴ mol/L has two significant figures, so its pH is reported to two decimal places, 3.47.

Rounding once, at the end of a multi-step calculation

In a calculation with several steps, keep at least one or two extra digits (guard digits) in every intermediate value and round only the final answer. Track how many significant figures each step justifies, but do not discard digits until the end. Rounding at every step adds a small error each time, and those errors can shift the last reported digit.

Worked example 3. You dissolve 1.37 g of sodium chloride in water and dilute it to exactly 50.00 mL in a volumetric flask. Using IUPAC standard atomic weights rounded to two decimal places (Na 22.99, Cl 35.45), the molar mass of NaCl is 58.44 g/mol. The amount is 1.37 g ÷ 58.44 g/mol = 0.023443 mol, kept unrounded. The volume is 0.05000 L, so the concentration is 0.023443 mol ÷ 0.05000 L = 0.46886 mol/L, which rounds to 0.469 mol/L because the mass has three significant figures.

If you had rounded the amount to 0.0234 mol first, you would get 0.0234 mol ÷ 0.05000 L = 0.468 mol/L. The difference is small, but it is a real error introduced by the arithmetic rather than by the measurements, and in longer chains it can grow.

When to round half-way digits up or to the even digit

If the first digit you drop is less than 5, leave the retained digit alone; if it is greater than 5, or is 5 followed by any nonzero digits, round up. So 0.064371 rounds to 0.0644 and 7.2849 rounds to 7.28 at three significant figures.

The only disputed case is a dropped digit of exactly 5 with nothing after it. OpenStax Chemistry 2e rounds so that the retained digit is even (6.125 becomes 6.12, while 6.135 becomes 6.14), which avoids a slight upward bias over many roundings. Many instructors simply round 5 up. Check which convention your course uses and apply it consistently; it affects only this one case.

Significant-figure errors to look for in your own work

Most significant-figure errors come from a handful of predictable slips. Before you box an answer, read the inputs again and ask which one limits the result and by which rule.

A quick habit makes this routine: next to each answer, note the limiting input and the rule you applied, decimal places or digit count. If the answer shows more digits than that input justifies, round it; if it shows fewer, look for a trailing zero the calculator dropped.

• Letting an exact number limit the answer. A factor of 1000 mL/L or a coefficient of 2 never reduces significant figures.
• Using the multiplication rule for a sum. Sums and differences are limited by decimal places, not by digit counts.
• Rounding every intermediate value. Keep guard digits and round once at the end.
• Dropping meaningful trailing zeros. A calculator shows 2.5 when the measurement was 2.50 mL; write the zero back.
• Counting leading zeros. 0.0081 g has two significant figures, not four.
• Reporting a whole number with ambiguous zeros. Write 3.40 × 10² g rather than 340 g when three digits are meaningful.