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Complex Numbers

Why do we need Complex Numbers?

Consider the equation \(x^2=4\), it solutions are \(x=2\) or \(x=-2\). However, what if it you were asked to solve a similar equation, \(x^2=-4\)? At first, you might decide to square root both sides of the equation to get:

\(\sqrt{x^2}=\sqrt{-4}\)

\(x=\sqrt{-4}\)

But what's our next step? How can we calculate or evaluate what \(\sqrt{-4}\) equals? 

When you first started learning math, you might have learned that you can't take the square root of a negative number. Throughout history, many mathematicians had the same problem with negative square roots that hindered them from making more mathematical discoveries. So, several of them came up with the concept of imaginary and complex numbers to overcome this obstacle, which we will continue talking about below.

Imaginary Numbers

Before we talk about complex numbers, we first have to define what the number \(i\) is.

In the context of complex numbers:

\(i=\sqrt{-1}\)

This also means that \(i^2=-1\) and that \(i\) is the solution to the quadratic equation \(x^2+1=0\). Sometimes \(i\) is called the imaginary unit when we are discussing complex numbers.

*Note: Sometimes you may see the imaginary unit represented as \(j\) instead of \(i\). We will be using \(i\) in this guide.

So for our previous example \(x^2+4=0\), we would have the solution \(x=\sqrt{-4}=\sqrt{4}\cdot\sqrt{-1}=2\cdot i=2i\)

Using this new knowledge, we can define a (pure) imaginary number as a product of a real number and the imaginary unit \(i\).

Examples: \(3i\), \(\frac{5}{6}i\), \(4.2i\), \(-3\frac{1}{2}i\), \(\sqrt{7}i\)

Complex Number Terminology

We call a number a complex number if it is in the form, called its rectangular (coordinate) form:

\(z=a+bi\)

where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit we discussed in the previous section. In Rectangular and Polar Form, you will learn about another way to represent complex numbers, polar form, and how to convert between rectangular and polar form.

Every complex number, \(z=a+bi\), is made up of a real part, \(Re(z)=a\), and imaginary part, \(Im(z)=b\). For example, \(5-7i\)  has the real part \(Re(5-7i)=5\) and imaginary part \(Im(5-7i)=7\), while \(2i\) has no real part (i.e. \(Re(2i)=0)\) but has imaginary part \(Im(2i)=2\).

The conjugate of a complex number, \(z=a+bi\), is:

\(\bar{z}=a-bi\)

For example, the conjugate of \(5+7i\) is \(5-7i\) and the conjugate of \(2i\) is \(-2i\). 

Tip: The conjugate of a complex number with no imaginary part (i.e. just a real number) is the same number! For example, conjugate of \(z=5\) is \(\bar{z}=5\).

The magnitude (or absolute value) of a complex number \(z=a+bi\), is written as \(|z|\) and defined as:

\(|z|=\sqrt{a^2+b^2}\)

*Note: This formula may seem familiar to you since it is just the distance formula \(d=\sqrt{(x_2^2-x_1^2)+(y_2^2-y_1^2)}\)  applied to the distance between the origin and a complex number on the complex plane (more about the complex plane here).

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