Chemistry · General chemistry I · Concept
Significant figures and rounding
Significant figures, or sig figs, are the digits in a measurement that carry meaning: every certain digit plus one estimated digit. When you multiply or divide, round the answer to the fewest significant figures in the data; when you add or subtract, round to the fewest decimal places.
What the digits tell you
A measurement is written with every digit you are sure of plus one estimated digit. 25.0 mL means the volume is known to about a tenth of a milliliter, while 25 mL is known only to the nearest milliliter. The extra zero is information, not decoration.
Counting significant figures
Nonzero digits always count. Zeros count when they sit between nonzero digits, or when they trail a number that has a decimal point. Leading zeros never count, because they only place the decimal point.
| Number | Significant figures | Why |
|---|---|---|
| 0.004050 | 4 | Leading zeros do not count; the final zero does |
| 1.050 | 4 | Zeros between digits and after a decimal point count |
| 2300 | 2, 3 or 4 | Trailing zeros without a decimal point are ambiguous |
| 2.30 × 10³ | 3 | Scientific notation removes the ambiguity |
| 12 eggs in a dozen | Unlimited | Exact numbers come from counting or from definitions |
Multiplying and dividing
The result has as many significant figures as the measurement with the fewest. 2.50 cm × 3.1 cm is 7.75 cm² on a calculator, but 3.1 has only two significant figures, so the area is reported as 7.8 cm².
Adding and subtracting
The result keeps as many decimal places as the measurement with the fewest. 12.52 g + 349.0 g + 8.24 g is 369.76 g on a calculator, but 349.0 has one decimal place, so the sum is 369.8 g. Subtraction can lose significant figures: 25.37 mL − 25.29 mL = 0.08 mL has only one.
Round once, at the end
In a calculation with several steps, keep at least one extra digit in the intermediate results and round only the final answer. When a calculation mixes steps, apply each rule to its own step: the decimal-place rule to a sum, then the significant-figure rule to a product.
Exact numbers never limit
Counted quantities and defined conversion factors, such as 1 in = 2.54 cm or 1 L = 1000 mL, are exact: they have unlimited significant figures and never limit a result. Three samples of 2.035 g each weigh 3 × 2.035 g = 6.105 g, still four significant figures.
Common mistakes
- Counting leading zeros: 0.0045 has two significant figures, not four.
- Using the significant-figure rule for a sum: 349.0 g + 8.24 g is 357.2 g by the decimal-place rule, not 357 g.
- Rounding intermediate results, which can change the last digit of the answer.
- Letting an exact number, such as a count or a defined conversion factor, limit the answer.
Key terms
- Significant figures
- The digits in a measurement that are known reliably, plus one final estimated digit. Exact counts and defined conversions, such as 1 m = 100 cm, never limit the significant figures of an answer.
- Precision
- How close repeated measurements are to each other. Precise results can still all be wrong, as when a scale is miscalibrated.
- Accuracy
- How close a measurement is to the true or accepted value. Measurements can be tightly grouped (precise) and still be inaccurate.
- Exact number
- A number with no uncertainty: a count, a defined relationship such as 1 kg = 1000 g, or a coefficient in a balanced equation. Exact numbers do not limit the significant figures of a result.
- Scientific notation
- Writing a number as a × 10ⁿ, with 1 ≤ |a| < 10 and n a whole number, such as 3.2 × 10⁻⁴. On a calculator, 1e-6 means 1 × 10⁻⁶, not Euler’s number e times −6.
- Decimal places
- The number of digits displayed after the decimal point. This differs from significant digits, which start at the first nonzero digit.
Work through an example
An empty flask has a mass of 45.216 g. Filled with 25.0 mL of a liquid, it has a mass of 72.95 g. What is the density of the liquid, to the correct number of significant figures?
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