# What are imaginary numbers?

Oct 3, 2015

The Real numbers can be represented as a line.

The Complex numbers can be represented by a plane whose $x$ axis is the Real numbers.

Imaginary numbers are the numbers on the $y$ axis of that plane.

#### Explanation:

Suppose we start with Whole numbers $0 , 1 , 2 , 3 , \ldots$

We can add them them quite happily and we always get another Whole number.

We can solve problems like $x + 2 = 5$, but when we try to solve problems like $x + 5 = 2$ we find our Whole numbers are insufficient.

So we can introduce the idea of a negative number and expand our idea of what a number is to include all of the Integers:

$\mathbb{Z} = \left\{0 , 1 , - 1 , 2 , - 2 , 3 , - 3 , \ldots\right\}$

We can add and multiply any two Integers and we always get an Integer.

We can solve problems like $2 x + 6 = 0$, but when we try to solve problems like $6 x + 2 = 0$ we find our Integers are insufficient.

So we can introduce the idea of a Rational number and expand our idea of what a number is to include all numbers of the form $\frac{m}{n}$ where $m , n \in \mathbb{Z}$ and $n \ne 0$.

...

To cut a long story short, in order to be able to solve problems like ${x}^{2} + 1 = 0$ we introduce the imaginary unit $i$, with the property ${i}^{2} = - 1$.

A square root of a negative Real number is a pure imaginary number.

In fact, we define the principal square root of a negative Real number as:

$\sqrt{x} = i \sqrt{- x}$

Any Complex number $z \in \mathbb{C}$ can be represented as $z = a + i b$ where $a$ and $b$ are Real numbers. $a$ is called the Real part of $z$ and $i b$ the Imaginary part. This can be pictured as the point $\left(a , b\right)$ on a plane.

Please note that Imaginary numbers are no more imaginary than Real numbers. Sir Isaac Newton seemed to dislike working with negative numbers, which he called "imaginary" numbers.