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

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

Add
To add complex numbers, you add similar things, so (1+2i)+(3+4i)=(1+3)+(2+4)i=4+6i

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: 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: $$\frac{(1+2i)}{(3+4i)}=(1+2i)*(\frac{3}{(3*3)+(4*4)}-\frac{4}{(3*3)+(4*4)}i)$$, which can be solved normally. This works because $$(3+4i)(\frac{3}{(3*3)+(4*4)}-\frac{4}{(3*3)+(4*4)}i)=$$

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

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: