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A calculator will happily tell you that 5.2 g divided by 3.7 mL is 1.405405405 g/mL. It won’t tell you that most of those digits are meaningless. Your ruler, balance or pipette only measured to a certain precision, and an answer can’t be more precise than the measurements that went into it.
Significant figures are how chemists keep answers honest.
What makes a digit significant
A significant figure is a digit that carries real information about the measurement’s precision. The rules for counting them:
- All non-zero digits are significant. 4.56 has 3.
- Zeros between non-zero digits are significant. 1005 has 4; 2.08 has 3.
- Leading zeros are never significant. They just place the decimal point. 0.0045 has 2.
- Trailing zeros after a decimal point are significant. 2.50 has 3; 3.000 has 4. Writing that final zero is a claim: “I measured this to the hundredths place and it was zero.”
- Trailing zeros in a whole number without a decimal point are ambiguous. Does 1500 have 2, 3 or 4? You can’t tell. Scientific notation removes the doubt: 1.5 × 10³ (2), 1.50 × 10³ (3), 1.500 × 10³ (4).
Practice
| Number | Significant figures |
|---|---|
| 72.4 | 3 |
| 0.00310 | 3 |
| 6.022 × 10²³ | 4 |
| 100.0 | 4 |
| 5,000 | ambiguous (1 to 4) |
| 0.120 | 3 |
Multiplying and dividing
The answer gets the same number of significant figures as the input with the fewest.
Density of a sample with mass 5.2 g and volume 3.70 mL: 5.2 ÷ 3.70 = 1.405… → the mass has only 2 sig figs → 1.4 g/mL
Moles in 12.45 g of water: 12.45 g ÷ 18.015 g/mol = 0.691 09… → 4 sig figs → 0.6911 mol
Adding and subtracting
Here it’s not the number of significant figures that matters, but the decimal places. The answer is rounded to the fewest decimal places among the inputs.
Total mass: 25.1 g + 3.456 g + 0.02 g = 28.576 → 25.1 has only one decimal place → 28.6 g
The logic: 25.1 is only known to the nearest tenth, so the total can’t be known to the nearest thousandth, however precise the other numbers are.
Logarithms and pH
Logs follow a special rule: the number of decimal places in a logarithm equals the number of significant figures in the original number.
[H⁺] = 3.2 × 10⁻⁴ M (2 sig figs) → pH = 3.49 (2 decimal places)
The digit before the decimal point in a pH (the “3”) only reflects the power of ten, not the precision, which is why it doesn’t count. So pH 7.00 corresponds to [H⁺] known to 3 sig figs, and pH 7.0 to only 2. More on this in the pH scale explained.
Exact numbers have unlimited precision
Some numbers aren’t measurements, so they never limit your significant figures:
- Counted quantities: 3 atoms of oxygen in O₃, 12 eggs.
- Defined conversions: 1000 mL in a litre, 100 cm in a metre, 273.15 in the Celsius-to-kelvin conversion.
- Coefficients in a balanced equation.
- Defined constants: since 2019, the Avogadro constant (6.022 140 76 × 10²³) and several others are exact by definition.
Multi-step calculations
Don’t round at every step. Rounding errors accumulate. Instead:
- Keep one or two extra digits through intermediate steps (or keep everything in the calculator).
- Track how many significant figures the final answer is allowed.
- Round only once, at the end.
Rounding rules
- If the first dropped digit is less than 5, round down: 2.643 → 2.64
- If it’s greater than 5, or 5 followed by non-zero digits, round up: 2.6461 → 2.65
- If it’s exactly 5 with nothing after it, conventions differ: most introductory courses round up, while some scientific contexts round to the nearest even digit to avoid bias. Follow your course’s convention.
Why this matters beyond homework
In a real lab, significant figures communicate how good a measurement is. A drug dose written as “5 mg” and one written as “5.00 mg” imply very different measurement precision. Reporting 1.405405405 g/mL when your balance only reads to 0.1 g claims a precision you don’t have — and anyone reading your result will trust it more than they should.
Quick answers
Do atomic masses limit significant figures? Rarely. Atomic weights on the periodic table usually have four or more significant figures, more than most lab measurements.
What does a decimal point after a whole number mean? Some textbooks write “100.” with a trailing decimal point to show that all three digits are significant. It works, but it’s easy to miss — scientific notation (1.00 × 10²) is clearer.
Are sig figs the same as decimal places? No. 0.0045 has four decimal places but only two significant figures.
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