Significant Figures in Chemistry: Reporting Exactly What You Measured

Why precision matters

Every measurement carries uncertainty. A balance reading of 25.0 g means the true mass lies somewhere near 25.0 g — the 2 and 5 are certain, and the final 0 is the chemist's best estimate of the next digit. Significant figures are the agreed language for reporting exactly that much information and no more.

Writing 25.0000 g when the balance only reads to 0.1 g would be dishonest precision. Writing 25 g when you actually measured to 0.1 g throws away information. Significant figures keep you exactly on the line.

Counting significant figures

A digit is significant if it tells you something real about the measurement:

• Nonzero digits are always significant: 3.14 has three.
• Zeros between nonzero digits are significant: 2005 has four.
• Leading zeros are never significant: 0.0045 has two — they only set the decimal place.
• Trailing zeros after a decimal point are significant: 45.00 has four.
• Trailing zeros without a decimal point are ambiguous: 4500 could be two, three, or four — write it as 4.500 × 10^3 to be clear.

Scientific notation resolves the ambiguity: in 2.50 × 10^3, every digit of the coefficient is significant, so the value has three significant figures.

Multiplication and division

For multiplication and division, the answer keeps the same count of significant figures as the input with the fewest.

• 2.50 g ÷ 25.0 mL = 0.100 g/mL — three and three give three.
• 12.0 m × 3.5 m = 42 m² — three and two give two.
• 0.0480 mol × 58.44 g/mol = 2.80 g — three and four give three.

The logic: a chain of measurements is only as strong as its weakest link. The least certain measurement dominates the uncertainty of everything computed from it.

Addition and subtraction

For addition and subtraction, the rule switches from digit count to decimal places: the answer keeps the fewest decimal places of any input.

• 12.5 g + 3.25 g = 15.8 g — one decimal place wins over two.
• 100.0 mL − 25.55 mL = 74.4 mL — one decimal place again.

Why the change? When you add, the absolute uncertainty matters — the tenths place of 12.5 is genuinely unknown, so it swamps the hundredths place of 3.25. Counting rules apply to scaling (multiplication); place rules apply to combining (addition).

Exact versus measured numbers

Not every number in a calculation is a measurement. Two kinds of numbers are exact and never limit significant figures:

• Counted quantities: 25 students, 3 beakers, 2 atoms of hydrogen in H2O.
• Definitions: 1000 mL in 1 L, 100 cm in 1 m, 60 s in 1 min.

So in the calculation (25.0 g / 2 beakers), the 2 does not drag the answer down to one significant figure — only the 25.0 g does. Measured conversion factors, like a molar mass of 58.44 g/mol, are not exact and do count.

Rounding rules

Round the final answer to the allowed significant figures with the standard convention:

• If the first dropped digit is 5 or greater, round the last kept digit up.
• If it is 4 or less, leave the last kept digit alone.

Examples: 3.456 rounded to three significant figures is 3.46; 3.444 becomes 3.44; 2.845 becomes 2.85. In multi-step calculations, carry at least one extra digit through the intermediate steps and round only once, at the very end.

Chaining calculations

Real lab calculations mix operations, so apply the rules in order:

Example: (25.0 mL − 3.5 mL) × 1.25 g/mL

1. Subtraction first: 25.0 − 3.5 = 21.5 mL — one decimal place, so 21.5.
2. Multiplication: 21.5 × 1.25 = 26.875 g.
3. Fewest significant figures is three (21.5 and 1.25 both have three), so round to 26.9 g.

Work the parentheses first, apply each operation's rule as you go, and do a single final rounding. Your calculator will happily show 26.875 — your job is to know which digits earned their place.

Common mistakes

Real-world applications

• Reporting titration volumes and calculated concentrations honestly

• Determining density from measured mass and volume

• Computing percent yield without overstating precision

• Publishing analytical results that other labs can reproduce

• Grading lab reports against professional measurement standards