# 3.2 Square roots of negative numbers

So, far you have only been looking at the square roots of positive numbers, but what about negative numbers? Let’s see how a calculator handles these in this next activity.

## Activity 8 Square roots of negative numbers

Before you find out the answer that your calculator gives to the square root of −16, think about what the answer could be.

### Comment

Remember that a positive number multiplied by a negative number (or vice versa) gives a negative result.

### Answer

What answer you get from your calculator will depend upon the calculator you are using.

Most will simply give you an error message!

This is because to obtain −16 you have to multiply 4 and −4, so the root of −16 is both of these different numbers at the same time. Hence, most calculators will show an error message.

More advanced graphics calculators will give the answer for the square root of −16 as 4*i*.

In this context, the *i* is shorthand for and is known as an ‘imaginary number’. This is the concept that mathematicians have used to get around the quandary of having two answers simultaneously to the same question. This is not the same as the issue of a square root of a positive number having two possible answers. That is either a positive or a negative number, not both a positive and a negative number.

You may well be thinking that this just more maths for maths sake, but many fields within technology and science require the use of imaginary numbers to provide solutions to real problems.

For now though, it is enough to know that there is no ‘real’ answer to the square root of a negative number. This is another area that you will learn more of if you continue studying mathematics.

This section also wraps up the extended study of exponents, or powers. In everyday life there is not much need for the particular skills and ideas that you have been studying here, but they do form part of the basic concepts that are used in more complex maths and therefore also other subject areas that use maths. This makes scientific notation, roots and how to perform calculations efficiently with powers of the same base number some of the fundamental areas to study if you want to continue with any maths related area in the future.