# Complex Numbers/Introduction

Notice: Incomplete

## Prerequisites

All four operations (+, -, *, /) for the real numbers

The purpose of parentheses in math

Box method for multiplication and/or FOIL

## Meaning of Complex Numbers

### What does i mean?

i is a 'fake', or imaginary number such that i*i=-1. We need to make up a number because 1*1=1 and -1*-1=1.

### Complex numbers

Complex numbers are numbers that 'look like' all of the following:

-1-2i

-3+10i

3-2i

4+3i

## Math of Complex Numbers

### Take away

To 'take away' complex numbers, you 'take away' similar things, so (1+2i)-(3+4i)=(1-3)+(2-4)i=-2-2i

### Times

To 'multiply' complex numbers, you 'multiply' similar things, so (1+2i)(3+4i)=(1)(3)+(1)(4i)+(2i)(3)+(2i)(4i)

#### With Box Method

Using the example of (1+2i)*(3+4i), we get:

 1 2i 3 1*3=3 2i*3=6i 4i 1*4i=4i 2i*4i=8i*i=-8

Going across both diagonals, we get:

(3+-8)+(6+4)i=-5+10i

#### With FOIL

To multiply complex numbers, you multiply using FOIL, like this: (1+2i)*(3+4i)=(1*3)+(1*4i)+(2i*3)+(2i*4i)=3+4i+6i-8=-5+10i

### Divide

To divide complex numbers, you turn it into a multiplication problem, like this: ${\displaystyle {\frac {(1+2i)}{(3+4i)}}=(1+2i)*({\frac {3}{(3*3)+(4*4)}}-{\frac {4}{(3*3)+(4*4)}}i)}$ , which can be solved normally. This works because ${\displaystyle (3+4i)({\frac {3}{(3*3)+(4*4)}}-{\frac {4}{(3*3)+(4*4)}}i)=}$

${\displaystyle ({\frac {3*3}{(3*3)+(4*4)}}+{\frac {4*4}{(3*3)+(4*4)}})+({\frac {4*3}{(3*3)+(4+4)}}-{\frac {3*4}{(3*3)+(4*4)}})i={\frac {3*3}{(3*3)+(4*4)}}+{\frac {4*4}{(3*3)+(4*4)}}={\frac {(3*3)+(4*4)}{(3*3)+(4*4)}}=1}$  and dividing is just multiplication by the reciprocal.

 3 4i ${\displaystyle {\frac {3}{(3*3)+(4*4)}}}$ ${\displaystyle {\frac {3*3}{(3*3)+(4*4)}}}$ ${\displaystyle {\frac {4*3}{(3*3)+(4*4)}}i}$ ${\displaystyle -{\frac {4}{(3*3)+{4*4}}}i}$ ${\displaystyle -{\frac {3*4}{(3*3)+(4*4)}}i}$ ${\displaystyle {\frac {4*4}{(3*3)+(4*4)}}}$

### Conjugate

The conjugate of a complex number is the number you get if you replace the + linking the real and imaginary parts with - or vice versa. Some examples include:

Complex number Conjugate
1+2i 1-2i
1-2i 1+2i
-1+2i -1-2i
-1-2i -1+2i