## Significant Figures

The significant figures (also called significant digits or, informally, 'sig figs') of a number are those digits that carry meaning contributing to its precision. This includes all digits except:

- leading and trailing zeros which are merely placeholders to indicate the scale of the number.
- spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports.

## Identifying significant figures

The rules for identifying significant figures when writing or interpreting numbers are as follows:

- All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).
- Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2.
- Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.
- Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures since it has three trailing zeros. This convention clarifies the precision of such numbers; for example, if a measurement precise to four decimal places (0.0001) is given as 12.23 then it might be understood that only two decimal places of precision are available. Stating the result as 12.2300 makes clear that it is precise to four decimal places (in this case, six significant figures).
- The number 0 has one significant figure.
- The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is precise to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:

- A bar may be placed over the last significant figure; any trailing zeros following this are insignificant. For example, 1300 has three significant figures (and hence indicates that the number is precise to the nearest ten).

- The last significant figure of a number may be underlined; for example, "2000" has two significant figures.

- A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant.

- In the combination of a number and a unit of measurement the ambiguity can be avoided by choosing a suitable unit prefix. For example, the number of significant figures in a mass specified as 1300 g is ambiguous, while in a mass of 13 hg or 1.3 kg it is not.

## Sample Exercise

Give the following numbers to three significant figures:

654.389

65.4389

654,389

56.7688

0.03542210

0.0041032

45.989

Check your answers here.

654.389

65.4389

654,389

56.7688

0.03542210

0.0041032

45.989

Check your answers here.