Significant figures cheat sheet

Everything you need to count sig figs and propagate them through a calculation, in one page.

Counting sig figs

Fix ambiguity with scientific notation. “1200” could be 2, 3, or 4 sig figs. 1.2×10³ is unambiguously 2; 1.200×10³ is unambiguously 4.

Arithmetic rules — two different rules

Multiplication and division — the answer keeps the LEAST number of sig figs among the factors:

$4.56 \times 1.4 = 6.384 \longrightarrow 6.4\ (\text{2 sig figs})$

Addition and subtraction — the answer keeps the LEAST number of decimal places among the terms:

$12.11 + 8.0 + 1.013 = 21.123 \longrightarrow 21.1\ (\text{1 decimal place})$

The two rules disagree in practice. 100. × 0.1 is 10. (1 sig fig), but 100.0 + 0.1 is 100.1 (1 decimal place). Which rule to apply depends on the operation, not the numbers.

Exact numbers

Numbers that come from definitions, counted objects, or integer coefficients in equations carry infinite sig figs and never limit the answer:

• “12 eggs” (counted) → exact.
• “1 dozen = 12” (definition) → exact.
• “2 in 2 H₂ + O₂ → 2 H₂O” (stoichiometric coefficient) → exact.
• “1 in = 2.54 cm” (defined conversion) → exact.
• Avogadro’s number 6.02214076×10²³ (defined since 2019 SI redefinition) → now exact.

Weights measured on a balance, concentrations read off a bottle, and volumes read off a flask are not exact — they carry the sig figs of the instrument reading.

Rounding

Two conventions are in use. Both are common; know which one your course expects.

• 5-up (round half away from zero) — the schoolbook rule. 2.5 → 3, 2.45 → 2.5. Simple and slightly biased upward across many rounds.
• 5-then-even (round half to even, “banker’s rounding”) — the modern IUPAC-style recommendation used in most scientific reporting. 2.5 → 2, 3.5 → 4, 2.45 → 2.4, 2.55 → 2.6. Unbiased across many rounds.

Programming note: Python’s built-in round() uses half-to-even; JavaScript’s Math.round() uses half-up. Spreadsheets vary.

Always round only once, at the very end. Rounding intermediate results propagates error. Carry at least one extra digit (“guard digit”) through the calculation and round the final answer to the correct precision.

Sig figs in logarithms — the pH trip-up

For a logarithm, sig figs live differently than for ordinary numbers:

The number of decimal places in the log equals the number of sig figs in the argument.

Because pH = -log[H⁺]:

The digits before the decimal in a pH are the “characteristic” (from the power of 10) and do not count as sig figs — they are placeholders. Only the digits after the decimal count. So pH = 3 implies only 1 sig fig in [H⁺], not 1 sig fig in the concentration’s leading digit.

The same rule applies to pKa, pOH, and any log₁₀-transformed quantity.

Worked examples

1. Multiplication. Compute molarity from mass, MW, and volume: M = m / (MW · V). Given m = 0.503 g (3 sf), MW = 58.44 g/mol (4 sf), V = 0.2500 L (4 sf).

$M = \frac{0.503}{58.44 \times 0.2500} = 0.034428... \longrightarrow 0.0344\ \mathrm{M}\ (\text{3 sf})$

Answer takes 3 sig figs — the mass was the weakest factor.

2. Subtraction — different rule. Titration burette reads V_final = 27.53 mL, V_initial = 0.24 mL.

$\Delta V = 27.53 - 0.24 = 27.29\ \mathrm{mL}\ (\text{2 decimal places})$

Both readings have 2 decimal places, so the difference has 2 decimal places (4 sf here — different from the ×/÷ rule).

3. pH. [H⁺] = 4.2×10⁻⁵ M (2 sf).

$\mathrm{pH} = -\log(4.2 \times 10^{-5}) = 4.376... \longrightarrow \mathrm{pH} = 4.38\ (\text{2 decimals})$

Two sig figs in [H⁺] → two decimals in pH.

Common student mistakes

1. Mixing up ×/÷ and +/− rules. Adding sig figs like they were decimal places (or vice versa) is the single most common error.
2. Rounding too early. 1/3 = 0.33 · 3 = 0.99, but the true answer is 1. Carry at least one guard digit through the calculation.
3. Miscounting leading zeros. 0.00420 has 3 sig figs — the leading zeros locate the decimal, the trailing zero (with a decimal point present) counts.
4. Applying sig-fig rules to exact numbers. A stoichiometric “2” doesn’t limit your answer to 1 sig fig — it’s exact.
5. Wrong pH precision. Reporting pH = 7.4123 from a meter that reads to 0.01 pretends to more precision than the instrument delivers.