**Imaginary numbers**

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1 Introduction

We are familiar with finding the square root of a positive number i.e. where is greater than or equal to zero (usually abbreviated by ). There are two solutions to every problem of this type i.e. . However, when is negative (written ) there is no real number which, when multiplied by itself, gives an answer which is a negative number. In order to overcome this gap in the number system and enable a solution to be found to the equation , the symbol ‘’ is used to represent . (In engineering, an alternative, ‘’ is used to avoid confusion with the symbol representing current.) The symbol (or ) is treated in exactly the same way as any other number. All of the normal arithmetical operations can be carried out using it. Multiplication of by a real number i.e. is written as either or . Any real multiple of is called an **imaginary** number.

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1.1 Examples

1. Multiply by 3.

Solution:

Multiplication by 3 is equivalent to adding to itself twice i.e. . This is written as either or .

2. Divide by 6.

Solution:

Since is , this can be re-written as .

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2 Powers of an imaginary number

This is performed in the same way as with real numbers, i.e. .

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2.1 Example

Evaluate

Solution:

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3 Square roots of a negative number

In the same way that has two solutions i.e. , so also has two solutions i.e. . This can be shown by multiplying by itself:

.

This result can be extended and generalised to give the square roots of any negative number.

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3.1 Example

Find the square roots of .

Solution:

.