# Complex Numbers

 From: Jon Horbury Date: 19 April 1999 Subject: Complex Numbers I am doing an Electrical Engineering course and would like to find out how to add, subtract, multiply and divide complex numbers.

### Maths Help suggests:

Complex numbers have two parts:
• A real part, and
• an imaginary part.

The imaginary part is a multiple of the number j:

#### Examples

The numbers   4 + j3,   -1 + j5,   -j3   are all examples of complex numbers.

Note that mathematicians sometimes use the symbol i for the square root of -1.
Engineers use i for other quantities (e.g. electric current) and j for complex numbers.

Separately add the real parts and the imaginary parts.

#### Example

(4 - j3) + (-1 + j5) = 3 + j2.     Because (real parts) 4 + (-1) = 3   and   (imaginary parts) -j3 + j5 = j2.

## Subtracting Complex numbers

Separately subtract the real parts and the imaginary parts.

#### Example

(2 + j7) - (-1 + j3) = 3 + j4.

## Multiplying Complex numbers

Multiply out the brackets term by term. Replace j2 with -1.

#### Example

 (2 + j) × (-4 + j3) = 2 × -4 + 2 × j3 + j × -4 + j × j3 = -8 + j6 + -j4 + j23 But j2 = -1 = -8 + j6 - j4 + -3 = -11 + j2

## Dividing Complex numbers

Multiply top and bottom of the division by the Conjugate of the bottom.
(The Conjugate of a complex number is what you get when you change the sign of the imaginary part.)

#### Example

The conjugate of   1 + j2   is   1 - j2.

The bottom part always simplifies to a real number (in this case 5). So:

You can check that (4 - 3j)(1 + 2j) = 10 + 5j