Complex Numbers and i

Motivation: x²+1 = 0, how can we express the roots?

We must use the imaginary number i which is:

i²=-1 and i is the imaginary unit

i³=-i

i⁴=1

Imaginary doesn’t directly imply non-existent as some may believe

Imaginary numbers are used in combination with real numbers to form something called a complex number z where z=a+bi

a and b are the real and imaginary part of z where a = Re(z) and b = Im(z)

z is real if b=0

z is purely imaginary when a=0

Two complex numbers can be equal only if their real and imaginary parts are equal

The complex number is useful for describing two dimensional variables

The real part of the complex number quantifies one dimension, and the imaginary part quantifies another

Example: Complex numbers are useful for representing two dimensional variables where both dimmensions are physically significant.
Think of it as the difference between a variable for the length of a stick (one dimension only), and a variable for the size
of a photograph (2 dimensions, one for length, one for width). For the photograph, we could use a complex number to
describe it where the real part would quantify one dimension, and the imaginary part would quantify the other.
The key point to remember is that imaginary numbers are often used to represent a second physical dimension. (picomonster)
The key point to remember is that imaginary numbers are often used to represent a second physical dimension.

Photo Example