9 Significant figures

You were introduced, in Section 4.1, to the idea of expressing answers to a specified number of decimal places. The more precisely you know a number, the more digits it seems reasonable to include; remember that writing down all the digits given on a calculator display cannot usually be justified. Quoting answers to a specified number of decimal places can be helpful, but it can be difficult to specify precisely how many digits are required. For example, suppose you have been asked to specify a distance of 34 178.921 metres to two decimal places. In metres, the correct answer would be 34 178.92 m, but in kilometres the correct answer would be 34.18 km, and if you were to choose to use scientific notation, the correct answer would be m.

It is frequently more reliable to quote answers to a specified number of significant figures where, in straightforward cases, the number of significant figures is found simply from counting the number of digits. So a temperature of 16.472 3 °C could be quoted to five significant figures as 16.472 °C, to four significant figures as 16.47 °C, to three significant figures as 16.5 °C and to two significant figures as 16 °C. The number of significant figures displayed reflects the certainty with which the value is known; in general the last digit will be somewhat uncertain, but you can be confident of the other digits.

Specifying the number of significant figures when zeros are involved can be a bit more tricky, as the following examples indicate:

• 0.082 m: here there are only two significant figures because initial zeros do not count. These initial zeros tell you only about the size of the number, and not about the precision to which it is known. The first significant digit in this value is the 8.
• 50.6 m: there are three significant figures here, since the zero in the middle of a number counts as a significant figure in the same way as the other digits.
• 79.0 m: there are three significant figures here too; a zero that is at the end of a number and after the decimal point has the same significance as any other digit; if this value was only known to two significant figures then it would have been quoted as 79 m.
• 900 m: this is the really tricky one! It could be that the value is known to three significant figures, that is, only the final zero is uncertain. But it might be that the distance has been measured only to the nearest 100 m (i.e. it lies between 850 m and 950 m). One way round this ambiguity is to state clearly the number of figures that are significant; for example to quote ‘900 m to one significant figure’. Alternatively, we can use scientific notation to resolve the ambiguity. Thus 9.00 10² m, 9.0 10² m and 9 10² m are all 900 m, but expressed to three significant figures, two significant figures and one significant figure respectively.