2.2 Calculations with significant figures

Remember the key rule! With calculations involving multiple measurements, your answer should be reported in such a way that it reflects the reliability of the least precise measurement. This means that the number of significant figures in your answer will be limited to the measurement with the lowest number of significant figures.

Multiplication / Division

If you multiply two numbers that have different numbers of significant figures, then the answer should have the same number of significant figures as the ‘weaker’ number.

For example, if you multiply 15.20 (4 significant figures) with 1.25 (3 significant figures) the answer would be limited to 19.0 (3 significant figures).

When you multiply two numbers, you more or less multiply the uncertainties. Therefore, it is the percentage by which you are uncertain that is important – the uncertainty in the number divided by the number itself. This is given roughly by the number of digits, regardless of their placement in terms of powers of ten. Hence the number of digits is what is important.

Addition / Subtraction

When adding and subtracting numbers, the rules of significant figures require that the number of places after the decimal point in the answer is less than or equal to the number of decimal places in every term in the sum. That is, limit the reported answer to the right-most column that all numbers have significant figures in common.

For example, if you were to add 1,992.123 and 34.12, note that the first number stops its significant figures in the thousandth’s column, while the second number stops its significant figures in the hundredth’s column. We therefore limit our answer to the hundredth’s column.

1,992.123

+ 34.12

––––––––––

2,026.243

2,026.24

This can be tricky to understand as it is not unusual for a sum to have more significant figures than the measurements added. What is important here is to limit the answer to the right most column where all of numbers have a significant figure in common.

If some of the numbers have no digits after the decimal point, use the same basic rule.

For example:

1,200

+ 85.88

––––––––––

1,285.88

1,300

In this case, the answer has to be limited to the hundreds column as 1,200 only has two significant figures. Therefore, the answer is rounded to 1,300.

There is a difference in how addition/subtraction is handled compared to multiplication/division because when you add two or more numbers you add their uncertainties. If one of the numbers is smaller than the uncertainty of the other, it does not make much difference to the value (and hence, uncertainty) of the final result. Therefore, it is the location of the digits, not the number of digits that is important.