Significant Figures: Rules and Calculations
Significant figures are the digits in a measurement that are known reliably plus one estimated digit, and they show how precise the measurement is. In brief: 0.00340 has three significant figures, 1,200 has two (unless a decimal point or scientific notation shows more), and in a calculation the answer must not be more precise than the least precise measurement used.
Every measurement has limits, and the number of digits you write should reflect them. Reporting 2.54387 g from a balance that reads to 0.01 g would claim precision that does not exist. This article gives the rules for counting significant figures, the rules for rounding, the rules for addition and multiplication, a table of examples, worked calculations and a short history of the idea.
Why do significant figures matter?
A ruler marked in millimeters lets you read a length such as 12.4 cm, where the 4 is an estimate between two marks. Writing 12.40 cm or 12.4000 cm would suggest that you measured with a much finer instrument. In a calculation, the uncertainty of each measurement carries over to the answer. The rules of significant figures give a simple, approximate way to keep track of this.
What are the rules for counting significant figures?
- Nonzero digits are always significant. 237 has three.
- Zeros between nonzero digits are significant. 2,005 has four.
- Leading zeros are not significant. They only show the position of the decimal point. 0.0042 has two.
- Trailing zeros are significant if there is a decimal point. 3.50 has three, and 120. (with a point) has three.
- Trailing zeros in a whole number without a decimal point are ambiguous. 1,200 has at least two. Use scientific notation to be clear: 1.2 × 103 has two, and 1.200 × 103 has four.
Counting examples
| Number | Significant figures | Reason |
|---|---|---|
| 4.56 | 3 | All nonzero |
| 100.05 | 5 | Zeros between nonzero digits count |
| 0.00340 | 3 | Leading zeros do not count, trailing zero after the point does |
| 0.0500 | 3 | 5, 0, 0 |
| 1,200 | 2 (ambiguous) | Trailing zeros without a decimal point |
| 1,200. | 4 | Decimal point shows trailing zeros are measured |
| 6.0 × 102 | 2 | Coefficient 6.0 |
| 2.500 × 10−4 | 4 | Coefficient 2.500 |
How do you round correctly?
Look at the first digit to be dropped. If it is less than 5, leave the last kept digit unchanged. If it is 5 or more, increase the last kept digit by one. For example, 3.4649 rounded to three significant figures is 3.46, and 3.465 is 3.47 by the common round-half-up convention. (Some scientific fields use round-half-to-even to avoid bias, so follow the rule your instructor or standard uses.) Only round at the end of a calculation, and keep one or two extra digits in the intermediate steps.
How do you use significant figures in calculations?
| Operation | Rule | Example |
|---|---|---|
| Multiplication and division | Answer has as many significant figures as the factor with the fewest | 12.5 × 3.2 = 40 (2 s.f.) |
| Addition and subtraction | Answer has as many decimal places as the number with the fewest decimal places | 1.234 + 12.5 = 13.7 |
| Logarithms | Decimal places in the log equal the significant figures in the original number | pH from 2.5 × 10−4 mol/L gives 3.60 |
| Exponents and roots | Same number of significant figures as the base | √2.5 = 1.6 |
Worked examples
- Density. A sample has a mass of 25.37 g and a volume of 10.0 mL. 25.37 ÷ 10.0 = 2.537. The volume has three significant figures, so the density is 2.54 g/mL.
- Addition. 45.2 g + 3.456 g + 0.12 g = 48.776 g. The least precise number is 45.2, with one decimal place, so the sum is 48.8 g.
- Subtraction. 100.0 mL − 99.65 mL = 0.35 mL. The first has one decimal place, the second has two, so the result should be given to one decimal place: 0.4 mL. Subtracting nearly equal numbers loses significant figures.
- Mixed operations. (4.52 × 3.1) + 2.35. First, 4.52 × 3.1 = 14.012, and with two significant figures this is 14 (the uncertain digit is in the units place). Adding 2.35 gives 16.35, which is rounded to the units place: 16. A good habit is to carry 14.0 and 16.35 until the end and then round to 16.
- Moles. 5.00 g of NaCl ÷ 58.44 g/mol = 0.08556 mol. The mass has three significant figures, so the answer is 0.0856 mol.
- Scientific notation. (2.50 × 103) × (4.0 × 10−2) = 10.0 × 101 = 1.0 × 102, with two significant figures.
What about exact numbers?
Exact numbers have no uncertainty and are treated as having unlimited significant figures. They include counted objects (12 eggs), defined conversion factors (1 m = 100 cm, 1 inch = 2.54 cm) and numbers in formulas (the 2 in 2πr). They never limit the number of significant figures in the answer. For example, the mass of 12 identical coins of 5.02 g each is 12 × 5.02 = 60.2 g, with three significant figures.
Where does the idea come from?
The need to express the limits of measurement became clear as science became more quantitative in the 18th and 19th centuries. Carl Friedrich Gauss and others developed the theory of errors in the early 1800s. The convention of significant figures is a simplified version of this more rigorous treatment, and it persisted because it is easy to apply by hand. More detailed methods that use uncertainties and statistics are now used for advanced work.
Which mistakes are common?
Mistake 1: Copying all digits from the calculator
A display of 2.5366666 does not make the answer more precise. Round it correctly.
Mistake 2: Mixing up the rules for adding and multiplying
For addition, count decimal places. For multiplication, count significant figures.
Mistake 3: Dropping trailing zeros
If a measurement is 2.50 g, the zero is significant and must be kept.
Mistake 4: Rounding in the middle of a calculation
This builds up rounding error. Keep extra digits and round only the final answer.