Complex Numbers, Defined, with examples and practice problems

Overview: This article covers the definition of complex numbers of the form $$ a+ bi $$ and how to graph complex numbers.

What are complex numbers?

A complex number can be written in the form a + bi where a and b are real numbers (including 0) and i is an imaginary number.

Therefore a complex number contains two 'parts':

one that is real
and another part that is imaginary

note: Even though complex have an imaginary part, there are actually many real life applications of these "imaginary" numbers including oscillating springs and electronics.

Examples of a complex number

$$ \begin{array}{c|c} \blue 3 + \red 5 i & \\\hline \blue{12} + \red{\sqrt{-3}} & \red{\sqrt{-3}} \text{ is the } \blue{imaginary} \text{ part} \\\hline \blue 9 - \red i & \\\hline \blue{12} - \red{\sqrt{-25}} & \red{\sqrt{-25}} \text{ is the } \blue{imaginary} \text{ part} \\\hline \end{array} $$

How do you graph complex numbers?

Complex numbers are often represented on a complex number plane (which looks very similar to a Cartesian plane).

On this plane, the imaginary part of the complex number is measured on the 'y-axis', the vertical axis;
the real part of the complex number goes on the 'x-axis', the horizontal axis;