Significant Figures & Rounding — Maths for General Chemistry

Significant figures keep you honest

They encode how precisely something was actually measured

A measured value implies its own uncertainty. 2.5 g means somewhere between 2.45 and 2.55 g; writing 2.500 g claims far more precision. The significant figures (sig figs) are the digits that carry real information about a measurement — not the placeholder zeros that merely locate the decimal point.

The rules come down to three cases. Non-zero digits always count. Zeros between non-zero digits (captive zeros, e.g. the 0 in 4.05) always count. Trailing zeros count only if a decimal point is explicitly shown — so 1200 has ambiguous precision, but 1200. and 1.200 × 10³ each unambiguously signal four sig figs.

When combining measurements the key rule is: you cannot manufacture precision. For multiplication and division, the result takes the same number of sig figs as the least precise factor. For addition and subtraction, it takes the same number of decimal places as the measurement with the fewest. Carry full precision through intermediate steps and round only at the final answer.

Scientific notation is the cleanest way to express any number with explicit precision: $1.20 \times 1 0^{3}$ is unambiguously 3 sig figs, while $1.200 \times 1 0^{3}$ is 4.

A special case often overlooked is logarithms. When you calculate a pH or a pK, only the digits after the decimal point (the mantissa) are significant — the integer part (the characteristic) merely encodes the power of ten of the original concentration. So if $[H^{+}] = 2.5 \times 1 0^{- 4} M$ (2 sf), then $pH = - lo g (2.5 \times 1 0^{- 4}) = 3.60$ — the answer has two decimal places (not three sig figs). Conversely, taking an antilog: $[H^{+}] = 1 0^{- 3.60}$ gives a result with 2 sf.

Exact numbers — counting numbers (12 eggs, 2 in a balanced equation) and defined constants (1 inch = 2.54 cm exactly; the Avogadro constant since 2019) — have infinite sig figs and never limit the precision of a result. Only measured quantities do.

In a multi-step calculation always carry at least one or two extra guard digits through every intermediate step, and round only the final answer. Rounding at each step accumulates errors that can shift the last reported digit — the classic source of discrepancies between student answers on long calculations.

Common pitfalls

Rounding intermediate steps. Keep all digits through a multi-step calculation and round once at the end. Premature rounding can shift the last sig fig of the final answer.
Miscounting leading zeros. In 0.00408 the three zeros are placeholders — only 4, 0, 8 are significant (3 sf). Leading zeros never count.
Misreading trailing zeros in an integer. The number 1200 has ambiguous precision (2, 3, or 4 sf). Write $1.200 \times 1 0^{3}$ to make four sig figs explicit.
Applying the × rule to pH. pH = 3.60 has 2 decimal places (2 sf in the concentration), not 3 sig figs. The integer before the decimal is not significant — it tells you the order of magnitude only.

Going deeper

Sig-fig rules are a shorthand for formal uncertainty propagation. When multiplying two quantities each with a relative uncertainty of 1 %, the product has roughly $(0.01)^{2} + (0.01)^{2} \approx 1.4 %$ relative uncertainty — slightly worse than each factor. The sig-fig rule (keep the fewest) approximates this well for up to a few factors. For sums and differences, absolute rather than relative uncertainties add in quadrature, which is why the decimal-place rule applies there instead. When uncertainties are small (< ~1 %) the two approaches give the same number of digits.

The three sig-fig rules at a glance

Always count

Non-zero digits, and any zeros between them (captive zeros like the 0 in 405).

Never count

Leading zeros — they only locate the decimal point (0.0034 has 2 sig figs).

It depends

Trailing zeros count only if a decimal point is shown: 100.0 has 4, but 100 has only 1 (or ambiguously 3).

Rounding operations

\times, \div

: keep fewest sig figs.

+, -

: keep fewest decimal places.

In the lab

A balance reading 12.04 g and a pipette delivering 25.00 mL each carry their own precision. Carry the digits through the calculation, then round once at the end. Try the counter and rounding calculator to build the habit.

Sig-fig counter

Select a number — significant digits are highlighted in orange, non-significant ones are greyed out.

Pick a number — significant digits are highlighted

0.00408

3 significant figures

Leading zeros just place the decimal — only 4, 0, 8 are significant.

Worked example 1Reading precision from a burette

A burette reading is recorded as 23.45 mL. How many significant figures does this have, and what does that tell you about the burette's precision?

Worked example 2Reporting a mass measurement

An analytical balance gives 0.10200 g. How many significant figures is this? What is the implied uncertainty?

Rounding calculator

Choose two measured values and an operation — see the sig-fig-correct result and the rule that governs it.

Pick two numbers and an operation — see the sig-fig-correct answer

First number

Second number

calculator says0.0785

rulefewest sig figs = 2

reported answer0.079

× and ÷ keep the fewest significant figures; + and − keep the fewest decimal places. Your answer can never be more precise than your least precise measurement.

Worked example 3Reporting a titration result

In a titration, a 25.00 mL pipette delivers NaOH solution. The NaOH concentration is $0.1024 mol dm^{- 3}$ (4 sf). Calculate the moles of NaOH delivered, and report to the correct number of sig figs.

Worked examples

Multi-step calculations — practice rounding at the right stage.

Worked example 4Molar mass calculation with rounding

Calculate the molar mass of glucose $C_{6} H_{12} O_{6}$ and report to the correct number of sig figs. (Atomic masses: C = 12.011, H = 1.008, O = 15.999 g mol⁻¹.)

Worked example 5Percentage yield — mixed operations

Theoretical yield of aspirin is $3.68 g$ (3 sf). You isolate $3.1 g$ (2 sf). Calculate the percentage yield to the correct number of sig figs.

Worked example 6Beer–Lambert law — absorbance and concentration

The molar absorption coefficient of a dye is $ε = 2.35 \times 1 0^{4} L mol^{- 1} cm^{- 1}$ (3 sf). The path length is $l = 1.00 cm$ (3 sf). If the absorbance is $A = 0.470$ (3 sf), find the concentration. ( $A = εl c$ )

Worked example 7pH and the log sig-fig rule

A solution has $[H^{+}] = 4.7 \times 1 0^{- 3} mol dm^{- 3}$ (2 sf). Calculate the pH and state the correct number of decimal places. Then find $[H^{+}]$ from pH = 11.32 (4 sig figs) and state its precision.

Worked example 8Multi-step calculation — rounding only at the end

Calculate the number of molecules in a $0.250 g$ sample of water (M = 18.015 g mol⁻¹, $N_{A} = 6.022 \times 1 0^{23} mol^{- 1}$ ), rounding only the final answer.

ChallengePropagation through a three-step Hess's law calculation

From the following measured enthalpies (with stated precisions), calculate $Δ_{f} H^{\circ} (CO_{2})$ and report to the correct sig figs. Given:
① C(s) + O₂(g) → CO₂(g), $Δ H_{1} = - 393.5 kJ mol^{- 1}$ (4 sf)
② CO(g) + ½O₂(g) → CO₂(g), $Δ H_{2} = - 283.0 kJ mol^{- 1}$ (4 sf)
③ C(s) + ½O₂(g) → CO(g), $Δ H_{3} = - 110.5 kJ mol^{- 1}$ (4 sf)
Check consistency: ΔH₁ = ΔH₂ + ΔH₃.

Check yourself

Four quick questions on counting sig figs and applying the rounding rules.

Question 1 of 4 · Score 0

How many significant figures are in 0.00408?

Choose an answer.