Complex Numbers

Introduction

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Complex numbers are numbers made up in two parts. These two parts are made from two different types of number, the first of which is called a Real Number, and the second part is called an Imaginary Number.

Complex numbers can be commonly seen from quadratic equations, phasor maths in electronics or to more complex vector maths used in simulations.

The diagram below shows a basic example of a complex number.

What are Real Numbers?

Real Numbers are numbers that 'exist'. These can include whole numbers as well as decimals.

Real numbers also include both irrational numbers and constants such as pi or e.

In essence, a real number is essentially any number than can fit on a number line.

Examples of Real Numbers

There are countless ways in where you would find Real Numbers, but a common situation you would see them labelled as Real Numbers is when finding the roots to an equation.

For example by calculating the discriminant (denoted as 'D'), such as found in the quadratic formula, you can find how many possible values of 'X', or Real Roots, there are. For values where the determinant is negative, it is said for there to be no Real Solution as negative numbers don't have square roots.

For example:

  • If D > 0 : There are two Real Roots

  • If D = 0 : One Real Roots

  • If D < 0 : No Real Solution

What are Imaginary Numbers?

Using the quadratic formula, we know how to find the Real Roots of an equation, but what happens when there are no real roots to an equation?

Well, it is often said that when the determinant is negative, there would be two Imaginary roots.

To be able to calculate the roots of an equation thought to have previously been impossible, famous mathematicians such as Euler created the constant 'i', the imaginary unit, to represent the square root of -1.

This means that substituting the root of a negative number, for a positive number multiplied by 'i', we can have a solution of x in these situations.

By collecting terms of 'i', we can also simplify an answer and even sometimes get a result a Reel Number, for example by multiplying terms of 'i' together. Below is an example of multiplying the square roots of two negative numbers, using terms of 'i' to get a real number result.

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