# Complex Numbers

A Complex Number

A Complex Number is a combination of a **Real Number** and an **Imaginary Number**

Real Numbers are numbers like:

1 | 12.38 | −0.8625 | 3/4 | √2 | 1998 |

Nearly any number you can think of is a Real Number!

Imaginary Numbers when **squared** give a **negative** result.

Normally this doesn't happen, because:

- when we square a positive number we get a positive result, and
- when we square a negative number we also get a positive result (because a negative times a negative gives a positive), for example
**−2 × −2 = +4**

But just imagine such numbers exist, because we want them.

Lets talk a little more about imaginary numbers ...

The "unit" imaginary number (like 1 for Real Numbers) is i, which is the square root of −1

Because when we square i we get −1

i^{2} = −1

Examples of Imaginary Numbers:

3i | 1.04i | −2.8i | 3i/4 | (√2)i | 1998i |

And we keep that little "i" there to remind us we need to multiply by √−1

## Complex Numbers

When we combine a Real Number and an Imaginary Number we get a **Complex Number**:

### Examples:

1 + i | 39 + 3i | 0.8 − 2.2i | −2 + πi | √2 + i/2 |

### Can a Number be a Combination of Two Numbers?

Can we make up a number from two other numbers? Sure we can!

We do it with fractions all the time. The fraction ^{3}/_{8} is a number made up of a 3 and an 8. We know it means "3 of 8 equal parts".

Well, a Complex Number is just **two numbers added together** (a Real and an Imaginary Number).

## Either Part Can Be Zero

So, a Complex Number has a real part and an imaginary part.

But either part can be **0**, so all Real Numbers and Imaginary Numbers are also Complex Numbers.

Complex Number | Real Part | Imaginary Part | |
---|---|---|---|

3 + 2i | 3 | 2 | |

5 | 5 | 0 | Purely Real |

−6i | 0 | −6 | Purely Imaginary |

## Complicated?

Complex does **not** mean complicated.

It means the two types of numbers, real and imaginary, together form a **complex**, just like a building complex (buildings joined together).

## A Visual Explanation

You know how the number line goes **left-right**?

Well let's have the imaginary numbers go **up-down**:

And we get the Complex Plane

A complex number can now be shown as a point:

The complex number 3 + 4**i**

## Adding

To add two complex numbers we add each part separately:

(a+b**i**) + (c+d**i**) = (a+c) + (b+d)**i**

### Example: add the complex numbers **3 + 2***i* and **1 + 7***i*

*i*

*i*

- add the real numbers, and
- add the imaginary numbers:

(3 + 2i) + (1 + 7i)

= 3 + 1 + (2 + 7)**i**

= 4 + 9i

Let's try another:

### Example: add the complex numbers **3 + 5***i* and **4 − 3***i*

*i*

*i*

(3 + 5* i*) + (4 − 3

*)*

**i**= 3 + 4 + (5 − 3)

**i**= 7 + 2

**i**On the complex plane it is:

## Multiplying

To multiply complex numbers:

**Each part of the first complex number ** gets multiplied by **each part of the second complex number**

Just use "FOIL", which stands for "**F**irsts, **O**uters, **I**nners, **L**asts" (see Binomial Multiplication for more details):

- Firsts:
**a × c** - Outers:
**a × d***i* - Inners:
**b***i*× c - Lasts:
**b***i*× d*i*
| |

(a+b ) = ac + adi + bci + bdii^{2} |

Like this:

### Example: (3 + 2i)(1 + 7i)

^{2}

^{2}= −1)

And this:

### Example: (1 + i)^{2}

^{2}

^{2}= −1)

### But There is a Quicker Way!

Use this rule:

(a+b**i**)(c+d**i**) = (ac−bd) + (ad+bc)**i**

Example: (3 + 2i)(1 + 7i) = (3×1 − 2×7) + (3×7 + 2×1)i = −11 + 23i

### Why Does That Rule Work?

It is just the "FOIL" method after a little work:

**i**)(c+d

**i**) =ac + ad

**i**+ bc

**i**+ bd

**i**

^{2}FOIL method

**i**+ bc

**i**− bd (because

**i**

^{2 }= −1)

**i**(gathering like terms)

And there we have the (ac − bd) + (ad + bc)**i** pattern.

This rule is certainly faster, but if you forget it, just remember the FOIL method.

### Let us try i^{2}

Just for fun, let's use the method to calculate i^{2}

### Example: i^{2}

We can write i with a real and imaginary part as 0 + i

^{2}= (0 + i)

^{2}= (0 + i)(0 + i)

**i**

**i**

**−1**

And that agrees nicely with the definition that i^{2 }= −1

So it all works wonderfully!

Learn more at Complex Number Multiplication.

## Conjugates

We will need to know about conjugates in a minute!

A conjugate is where we **change the sign in the middle** like this:

A conjugate is often written with a bar over it:

### Example:

5 − 3**i** = 5 + 3**i**

## Dividing

The conjugate is used to help complex division.

The trick is to **multiply both top and bottom **by the** conjugate of the bottom**.

### Example: Do this Division:

\frac{2 + 3i}{4 − 5i}

Multiply top and bottom by the conjugate of 4 − 5**i** :

\frac{2 + 3i}{4 − 5i}×\frac{4 + 5i}{4 + 5i} = \frac{8 + 10i + 12i + 15i^{2}}{16 + 20i − 20i − 25i^{2}}

Now remember that i^{2} = −1, so:

= \frac{8 + 10i + 12i − 15}{16 + 20i − 20i + 25}

Add Like Terms (and notice how on the bottom 20**i** − 20**i** cancels out!):

= \frac{−7 + 22i}{41}

Lastly we should put the answer back into a + b**i** form:

= \frac{−7 }{41} + \frac{22}{41}**i**

DONE!

Yes, there is a bit of calculation to do. But it **can** be done.

## Multiplying By the Conjugate

There is a faster way though.

In the previous example, what happened on the bottom was interesting:

(4 − 5**i**)(4 + 5**i**) = 16 + 20**i** − 20**i** − 25**i**^{2}

The middle terms (20**i** − 20**i**) cancel out! Also **i**^{2} = −1 so we end up with this:

(4 − 5**i**)(4 + 5**i**) = 4^{2} + 5^{2}

Which is really quite a simple result. The general rule is:

(a + b**i**)(a − b**i**) = a^{2} + b^{2}

We can use that to save us time when do division, like this:

### Example: Let's try this again

\frac{2 + 3i}{4 − 5i}

Multiply top and bottom by the conjugate of 4 − 5**i** :

\frac{2 + 3i}{4 − 5i}×\frac{4 + 5i}{4 + 5i} = \frac{8 + 10i + 12i + 15i^{2}}{16 + 25}

= \frac{−7 + 22i}{41}

And then back into a + b**i** form:

= \frac{−7 }{41} + \frac{22}{41}**i**

DONE!

## Notation

We often use **z** for a complex number. And **Re()** for the real part and **Im()** for the imaginary part, like this:

Which looks like this on the complex plane:

## The Mandelbrot Set

The beautiful Mandelbrot Set (pictured here) is based on Complex Numbers. It is a plot of what happens when we take the simple equation The color shows how fast | |

Here is an image made by zooming into the Mandelbrot set | |

And here is the center of the previous one zoomed in even further: |