Significant figures and round-off
How to count significant figures for numbers larger and smaller than 1 and in scientific notation, and where a machine's round-off errors come from.
Counting significant figures
The same number admits several representations. Significant figures are those that carry information. There is no single definition, but there are usual rules:
- Numbers larger than 1: count the figures shown, excluding trailing zeros in the decimal part.
- Numbers smaller than 1: count the figures, excluding leading zeros before the first nonzero digit and trailing decimal zeros.
- Scientific notation x·10^z: the significant figures are those of the mantissa x.
| Number | Significant figures | Feature |
|---|---|---|
| 232.870000 | 5 | Decimal > 1 |
| 0.001100 | 2 | Decimal < 1 |
| 3·10^8 | 2 | Notación científica |
Round-off and machine precision
Round-off errors come from finite representation capacity. A computer stores each number with a limited number of figures; beyond that capacity, it rounds.