1. Significant figures

Numbers which are the result of an observation or measurement (measures) are approximations, and ARE SUBJECT TO ERROR. The precision of the measurement should be reflected by writing down only those figures in which the observer has confidence.

The confidence of the observer is reflected in the NUMBER OF SIGNIFICANT FIGURES used to write down the number.

For example, the quantity 1.23 metres is expressed here to 3 significant figures, implying that there is some uncertainty in the last (rightmost) digit. We know that the number is in the range 1.225 to 1.234 (all numbers in the range 1.225 to 1.234 would be rounded off to 1.23).

Integers (whole numbers) are considered to have an infinite number of significant figures.

For decimal fractions, leading zeros (zeros to the left of the first non-zero digit) are not considered to be significant figures, but trailing zeros (to the right of the last non-zero digit) are:

0.0123 has 3 significant figures
0.4560 has 4 significant figures.

Numbers which are written as whole numbers with trailing zeros (i.e., 1000, 102300) present a problem. The number of significant figures should be seen in context, but generally, the trailing zeros of such numbers are not considered to be significant figures.

For example, 1000 has 1 significant figure, unless it is clear that it is an exact number (resulting from an accurate count, for example).

If you want to express precision to four significant figures of a number subject to an error, use exponential notation:

1.000 x 103 has 4 significant figures.
2 x 10-2 has 1 significant figure.

2. Rounding off to a given number of significant figures

After adding or subtracting numbers, you should round off your answer in such a way as to retain digits as far as the first column in which there is uncertainty, as shown in the example on the left: in the first calculation, there is uncertainty about the digits in the third decimal place - the answer (69.383) can therefore be expressed to five significant figures, that is, in this case, three figures to the right of the decimal point. In the calculation on the right, there is uncertainty in the first decimal place, and so the answer must be expressed as 63.2 and not 63.220.

The result of a multiplication or division should be rounded off to as many significant figures as the least exact factor. Refer to the box on the right: the result of the multiplication must be expressed to three significant figures, because the least precise factor (21.2) only has three significant figures. The same principle applies for divisions. In the example, the divisor (0.056) only has two significant figures, and so the result must be expressed to the same degree of precision.

3. Additional questions