Significant Figures and Scientific Notation
Significant figures are the digits in a number that carry meaningful information about its precision, and scientific notation is a way of writing a number as a × 10n, where a is between 1 and 10 and n is an integer. In brief: all nonzero digits are significant, zeros between nonzero digits are significant, leading zeros are not, and trailing zeros count only after a decimal point (or when marked). When multiplying or dividing, the answer has as many significant figures as the least precise factor; when adding or subtracting, it has as many decimal places as the least precise term.
Physicists work with numbers from the size of an atom to the size of a galaxy, and every measured number has a limited precision. Significant figures show that precision, and scientific notation makes very large and very small numbers easy to read. This guide gives the rules for counting and rounding, shows how to use scientific notation, compares examples in a table, works through calculations and corrects common mistakes.
What are significant figures?
Significant figures (sig figs) are the digits that were actually measured, plus one estimated digit. The rules for counting them are:
- All nonzero digits are significant: 3.84 has three.
- Zeros between nonzero digits are significant: 4,005 has four.
- Leading zeros (before the first nonzero digit) are not significant, they only show the position of the decimal point: 0.0072 has two.
- Trailing zeros after the decimal point are significant: 2.50 has three, and 0.0300 has three.
- Trailing zeros in a whole number without a decimal point are ambiguous: 1,500 may have two, three or four. Scientific notation removes the doubt: 1.5 × 103 has two, and 1.500 × 103 has four.
- Exact numbers (such as 12 eggs, or the definition 100 cm = 1 m) have unlimited significant figures.
Rounding: look at the first digit to be dropped. If it is 5 or more, round up; if less than 5, leave the last kept digit unchanged. Round only at the end of the calculation, and keep an extra digit in intermediate steps.
How do significant figures work in calculations?
- Multiplication and division: the result has the same number of significant figures as the factor with the fewest. 4.56 × 1.4 = 6.384, which is rounded to 6.4 (two significant figures).
- Addition and subtraction: the result has the same number of decimal places as the term with the fewest decimal places. 12.11 + 18.0 + 1.013 = 31.123, which is rounded to 31.1 (one decimal place).
- The precision of the final answer cannot be greater than that of the data. A calculator gives many digits, but most of them are meaningless.
What is scientific notation?
In scientific notation, a number is written as a × 10n with 1 ≤ a < 10. To convert, move the decimal point until there is one nonzero digit before it, and count the places moved. Moving left gives a positive exponent, and moving right gives a negative exponent.
- The speed of light: 300,000,000 m/s = 3.00 × 108 m/s (three significant figures).
- The diameter of a hydrogen atom: 0.000 000 000 1 m = 1 × 10−10 m.
- The rules for the exponents: 10m × 10n = 10m+n, and 10m ÷ 10n = 10m−n.
- The number of digits in a is the number of significant figures, so it shows the precision directly.
Calculators show scientific notation as, for example, 3E8 or 3 × 10^8. Be careful to write it in full form in a solution.
Examples compared
| Number | Significant figures | Scientific notation |
|---|---|---|
| 0.000 45 | 2 | 4.5 × 10−4 |
| 0.0300 | 3 | 3.00 × 10−2 |
| 4,005 | 4 | 4.005 × 103 |
| 250.0 | 4 | 2.500 × 102 |
| 6,000,000 | 1 to 7 (unclear) | 6 × 106 (1 sig. fig.) |
Worked example: calculations with both
- Area: a plate is 12.4 cm by 3.1 cm. 12.4 × 3.1 = 38.44 cm2. The factor 3.1 has two significant figures, so the answer is 38 cm2.
- Total mass: 125.4 g + 8.26 g + 0.7 g = 134.36 g. The least precise term, 0.7 g, has one decimal place, so the answer is 134.4 g, rounded to one decimal place.
- Scientific notation in a product: (3.0 × 108 m/s) × (2.0 × 10−3 s) = (3.0 × 2.0) × 108 + (−3) = 6.0 × 105 m.
- Division: (9.6 × 106) ÷ (3.2 × 102) = (9.6 ÷ 3.2) × 106 − 2 = 3.0 × 104.
- Normalizing: 45 × 103 is not in standard form. Move one place: 4.5 × 104.
How did these notations develop?
Archimedes, in The Sand Reckoner (about 250 BCE), invented a system for naming very large numbers to estimate the number of grains of sand in the universe. Exponent notation in its modern form was introduced by René Descartes in the 1600s. The use of significant figures became standard in the experimental sciences and engineering during the nineteenth century, as measurements and calculation tables became more precise, and today calculators and spreadsheets present results in scientific notation.
Which mistakes are common?
Mistake 1: Copying all digits from the calculator
The answer should be rounded to the precision of the data.
Mistake 2: Counting leading zeros as significant
0.0072 has two significant figures, not four.
Mistake 3: Dropping trailing zeros after the decimal point
2.50 shows a precision of three digits, and 2.5 only two. They are not the same measurement.
Mistake 4: Mixing up the rules for addition and multiplication
Decimal places are used for sums and differences, and significant figures for products and quotients.