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Scales in space and time
Scales in space and time

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1.3 Precision and magnitude

When working with quantities at any scale, and when combining numbers through addition, multiplication, etc., it is important to report the answers with an appropriate level of precision. To do this you need to consider the number of significant figures, which is the number of digits that carry a ‘meaning’ for the measurement.

The cardinal rule here is shown below:

Key point

An answer in a calculation cannot be more precise than the least precise number used in that calculation.

For example, 361 has three significant figures and 31.01 has four significant figures. When reporting an answer to a calculation, you should use the number of significant figures from the least precise value (or value with the smallest number of significant figures) in the calculation.

If you multiply 361 by 31.01, the answer provided by a calculator is 11 194.61. This value has seven significant figures, but the least precise value in the calculation (361) has just three significant figures. The answer should therefore be quoted to the same level of precision, which is 11 200. In this example, the part of the number that affects the third significant figure, namely 194.61, is rounded to 200 rather than 100. (More information on rounding is provided in Box 1.)

Box 1 A reminder about rounding

When rounding values, look only at the digit immediately to the right of where you want to stop the number of significant figures. If it is less than 5, round it down, if it is equal to or greater than 5, round it up.

For example, if rounding 11 194.61 to three significant figures, stop after the third significant figure. Therefore, look at the digit to the right of this third significant figure, and see it is a 9. As 9 is greater than or equal to 5, round up by increasing the third significant figure by 1. In effect, 11 194.61 is nearer to 11 200 than it is to 11 100.

When giving an answer to a calculation you should also quote the number of significant figures used. Note that you can do this by stating ‘to x significant figures’ in your final answer, or abbreviate it as ‘to x sig figs’ or even ‘to x s.f. (where x is the number of digits in question).

Precision and exact integers

There are some special cases where numbers do not affect the number of significant figures, such as exact integers. Sometimes this is encountered this when applying a formula.

For example, the perimeter of a square = 4 × edge length, because, by definition, a square has four sides of equal length. Here the integer 4 is exactly four, and not more or less. A square has exactly four sides so this number is a multiplier and not an amount measured.

The edge length, in contrast, is a measure, and therefore determines how precisely the perimeter can be reported. If the side of a square was measured be 2.62 cm, which has three significant figures, then:

multiline equation line 1 perimeter equation left hand side equals right hand side four multiplication 2.62 cm line 2 equals 10.48 cm line 3 equals 10.5 cm to three sig fig

Note, again, that the answer given by the calculator has been provided as well as the answer rounded to the appropriate number of significant figures to make work flow clear. This approach is good practice when it is appropriate to show working in calculations.