1 - Introduction to Complex Numbers - The Complex Plane and The Imaginary Unit i

The imaginary unit, known as the imaginary number i, is defined as the square root of -1.

Definition of a complex number

Any number that can be written in the form is a complex number, where a and b are real numbers.

The Complex Plane

The complex plane is an ordinary plane, but the x axis represents the line of real numbers, and the y-axis represents the imaginary number line, so, multiples of i.

In the Politecnico di Torino course of Mathematical Analysis 1, The Complex Plane can sometimes be named "The Gaussian Plane". They're the same thing.

Real and Imaginary Parts

Consider the complex number

This complex number has a real number a, and a multiple b of the imaginary number i.
In essence, this real number a represents the real part of this complex number z, and it is denoted by:
This other number b, as you probably guessed, represents the imaginary part of this complex number z, and it is denoted by:

Do not confuse the imaginary part for the same notation that denotes the image of a function!

Plotting complex numbers in the complex plane

You would do this just like you would any point (x,y), but, for the x coordinates you will use the real part.
For the y coordinate you will use the imaginary part b.