---
published: true
title: "2 - Complex Numbers"
category: further-maths
tags: FP1
layout: post
---

# Imaginary numbers

**Imaginary numbers** are numbers defined in terms of their relation to \`\sqrt{-1}\`, which is known as \`i\`. Some equations which seemed impossible before are now completely doable. For example:

\`x^2 = -4\`

To solve this, you just work out the square root of 16 (4) and make it a coefficient of i:

\`x = sqrt{4}i = 2i\`

# Complex Numbers

**Complex numbers** are numbers with both a real and imaginary part. They can be squared and manipulated just like any other binomial - just keep in mind that while you are expanding these. They are expressed as:

\`x + yi\`

In this equation:

* The **real part** is the part without an \`i\` coefficient.
* The **imaginary part** is the part with an \`i\` coefficient.

## Handy Expressions

### Addition

Addition is the same as ever, the real part is the addition of the real parts while the imaginary part is the addition of the imaginary parts.

\`(a + bi) + (c + di) = (a + c) + (b + d)i\`

### Subtraction

Subtraction, similarly, has the real part as the subtraction of the real parts and the imaginary part is the subtraction of the imaginary parts.

\`(a + bi) - (c + di) = (a - c) + (b - d)i\`

### Multiplication

\`(a + bi)(c + di) = (ac - bd) + (bc + ad)i\`

### Division

\`\frac{a + bi}{c + di} = (\frac{ac + bd}{c^2 + d^2}) + (\frac{bc - ad}{c^2 + d^2})i\`

## Complex Conjugates

A **complex conjugate** to a **complex number** has the same real part but the opposite imaginary part to that number, as in, if:

\`z = x + yi\`

Then its complex conjugate

\`\overline{z} = x - yi\`

The important equations formed from these properties are that:

\`z + \overline{z} = 2x\`

\`z \times \overline{z} = x^2 + y^2\`

## Equating real and imaginary parts

If...

\`a + bi = c + di\`

.. then due to the nature of complex numbers it is definite that:

\`a = c\`

\`b = d\`
Add FP 2 / Complex Numbers
Add FP 2 / Complex Numbers
