1.3 The role of brackets

Sometimes, you need to do the addition or subtraction in a calculation before the multiplication, or to add two numbers together before raising to a power. The way to over-ride the standard rules of ‘multiplication before addition’ and ‘powers before multiplication’, etc., is to use brackets:

So, in the calculation , you should add the 3 and the 2 first (to give 5), then multiply by 4, i.e. Similarly, in the calculation , you should add the 3 and the 4 first before squaring. So

You can do calculations including brackets on a scientific calculator by making use of its brackets keys, usually labelled as ‘(’ and ‘)’. Try the calculation on your calculator now.

If you have a calculation which involves nested brackets, work out the innermost sets first. For example:

Note that using different symbols for the brackets, for example ( ), { } and [ ], makes the calculation clearer than using the same symbol throughout the calculation.

Strictly speaking, brackets are only needed to override the other rules of precedence, and they are not needed in calculations such as . In the absence of the brackets, you or your calculator would follow the rule and do the multiplication first in any case. However, brackets are often used in calculations for clarity, even when they are not strictly necessary. For example, the calculation is more understandable and ‘readable’ if it is written as , even though the brackets are not essential here. You are encouraged to write brackets in your calculations whenever they help you to express your working more clearly.