# Brush up your maths: Significant figures

## Rounding

When we carry out calculations, we often get the result as a number with many digits. We usually do not need them all.

Sometimes we report to the nearest whole number. For example, 53.2 to the nearest whole number is 53. This is rounding to no decimal places, if you like. We round up or down just as we did for any other number of decimal places. If the first figure after the decimal point is 0, 1, 2, 3, or 4 we round down, if the first figure after the decimal point is 5, 6, 7, 8, or 9 we round up. Hence 53.2 to the nearest whole number is 53. We round down because the first digit to be cut off is 2. On the other hand, 53.6 to the nearest whole number is 54. We round up because the first digit to be cut off is 6.

We round to other units than whole numbers. We might round £1,512,132 to £1.5 million, for example. The annual budget of a hospital might be given in millions of pounds, your annual salary in thousands. If your annual salary was £151,124 you might be happy to report that as £151 thousand, or even £150 thousand. (On £150 thousand per year, you might be quite happy, full stop, of course.)

When we round in this way, the rules for rounding described above are applied. For example, 1,598,121 would be 1.6 million, not 1.5 million.

### Exercise: Rounding

What is 23,432 to the nearest thousand?

What is 47.743 to the nearest whole number?

What is 2,317,995 to the nearest tenth of a million?

## Significant figures

How do we decide how many decimal places we use or to what unit we should round? One way is to look at significant figures. This has nothing to do with the statistical use of the word "significant". (The English language has more words than any other, but there are not enough for us to avoid words having multiple meanings.)

The first significant figure of a number is the first digit which is not zero. Hence the first significant figure of 20,499 is 2 and the first significant figure of 0.0020499 is 2.

The second significant figure of a number is the digit after the first significant figure. This is true even if the digit is zero. Hence the second significant figure of 20,499 is 0, as is the second significant figure of 0.0020499.

The third significant figure of a number is the digit after the second significant figure. This is true even if the digit is zero, and so on. Hence the third significant figure of 20,499 is 4 and the fourth is 9, as are the third and fourth significant figures of 0.0020499.

We round a number to three significant figures in the same way that we would round to three decimal places. We count from the first non-zero digit for three digits. We then round the last digit. We fill in any remaining places to the right of the decimal point with zeros. This is because we need them to hold the correct place value for the significant digits.

For example, 20,499 to three signifcant figures is 20,500. We round up because the first figure we cut off is 9. 0.0020499 to three significant figures is 0.00205. We do not put any extra zeros in to the right after the decimal point. This is because we do not need them to hold the correct place value for the significant digits.

If the last significant digit of a number is 0, we include this. For example, 0.0020499 to two significant figures is 0.0020. The first significant digit is 2, the second significant digit is 0. The next digit is 4, so we round down.

### Exercise: significant figures

Give the following numbers to three significant figures:
654.389
65.4389
654,389
56.7688
0.03542210
0.0041032
45.989