A REALLY Brief Intro to Complex Numbers

August 18, 2013 MathyNick Blog No comments

Venn Diagram of Number Sets from Keith’s Think Zone

Imaginary numbers and complex numbers interest me a lot. Aside from the properties of infinity ( $\infty$ ), complex numbers $\mathbb{C}$ are, well… complex! They’re very strange and yet they open a whole new world of numbers beyond the Real numbers $\mathbb{R}$ .

Let’s find out what real numbers and complex numbers are.

The Real numbers $\mathbb{R}$ are the set made up of all the Rational numbers $\mathbb{Q}$ and Irrational numbers together.
To make our transition into the complex numbers, we need to define what $\sqrt{-1}$ is since it is not a real number. That is to say, there is no real number $i$ so that when we square it, it is equal to $-1$ , ( $i^2=-1$ ). So we make up a new number to make it work. We define the Imaginary unit, $i$ to equal $\sqrt{-1}$ , ( $i= \sqrt{-1}$ .
An Imaginary number is a specific kind of complex number written in the form $bi$ where $b$ is a real number and $i$ is the imaginary unit.
The Complex numbers $\mathbb{C}$ are the set that involves a combination of real and imaginary numbers. They are written in the form $a+bi$ , where $a$ and $b$ are real numbers.