Linear algebra: Complex numbers

Linear algebra: Complex numbers

1. Imaginary numbers

There are numbers namely -1, 0, 2, 1000 called real number.  There are imaginary numbers 
which are thought to be impossible. Consider imaginary unit i (Greek character)

Example:

if x2=1,  then x=i or -i

There are some properties of imaginary numbers:

i x i=-1 or i2=-1, -i x i=1

-1 x i=-i or i x (-1)=i

i3=-I, i4=(i2)2=(-1)2=1

2. complex numbers

A complex number is denoted as

z=a+bi

a and b are real numbers, and i is imaginary number. a is real part of z denoted as Re(z)=a. 
b scaling i is imaginary part of z denoted as Im(z)=b.

The conjugate complex number z is the mirror image of z reflected.

Proof 1: a complex number plus its conjugate number is a real number.

Proof 2: the product of a complex number and its conjugate number is a real number.

Example1: