Imaginary numbers

1  Introduction

We are familiar with finding the square root of a positive number i.e. Imaginary_files\Imaginary_MathML_0.jpgwhere Imaginary_files\Imaginary_MathML_1.jpg is greater than or equal to zero (usually abbreviated by Imaginary_files\Imaginary_MathML_2.jpg). There are two solutions to every problem of this type i.e. Imaginary_files\Imaginary_MathML_3.jpg. However, when Imaginary_files\Imaginary_MathML_4.jpg is negative (written Imaginary_files\Imaginary_MathML_5.jpg) 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 Imaginary_files\Imaginary_MathML_6.jpg, the symbol ‘Imaginary_files\Imaginary_MathML_7.jpg’ is used to represent Imaginary_files\Imaginary_MathML_8.jpg. (In engineering, an alternative, ‘Imaginary_files\Imaginary_MathML_9.jpg’ is used to avoid confusion with the symbol representing current.) The symbol Imaginary_files\Imaginary_MathML_10.jpg (or Imaginary_files\Imaginary_MathML_11.jpg) 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 Imaginary_files\Imaginary_MathML_12.jpg by a real number Imaginary_files\Imaginary_MathML_13.jpg i.e. Imaginary_files\Imaginary_MathML_14.jpg is written as either Imaginary_files\Imaginary_MathML_15.jpg or Imaginary_files\Imaginary_MathML_16.jpg. Any real multiple of Imaginary_files\Imaginary_MathML_17.jpg is called an imaginary number.

1.1  Examples

1. Multiply Imaginary_files\Imaginary_MathML_18.jpg by 3.


Multiplication by 3 is equivalent to adding Imaginary_files\Imaginary_MathML_19.jpg to itself twice i.e. Imaginary_files\Imaginary_MathML_20.jpg. This is written as either Imaginary_files\Imaginary_MathML_21.jpg or Imaginary_files\Imaginary_MathML_22.jpg.

2. Divide Imaginary_files\Imaginary_MathML_23.jpg by 6.


Since Imaginary_files\Imaginary_MathML_24.jpg is Imaginary_files\Imaginary_MathML_25.jpg, this can be re-written as Imaginary_files\Imaginary_MathML_26.jpg.

2  Powers of an imaginary number

This is performed in the same way as with real numbers, i.e.Imaginary_files\Imaginary_MathML_27.jpg .

2.1  Example

Evaluate Imaginary_files\Imaginary_MathML_28.jpg



3  Square roots of a negative number

In the same way that Imaginary_files\Imaginary_MathML_30.jpghas two solutions i.e. Imaginary_files\Imaginary_MathML_31.jpg, so Imaginary_files\Imaginary_MathML_32.jpg also has two solutions i.e. Imaginary_files\Imaginary_MathML_33.jpg. This can be shown by multiplying Imaginary_files\Imaginary_MathML_34.jpg by itself:


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

3.1  Example

Find the square roots of Imaginary_files\Imaginary_MathML_36.jpg.


Imaginary_files\Imaginary_MathML_37.jpg Imaginary_files\Imaginary_MathML_38.jpg.