Maths

A Beginners Guide to Complex Numbers

Unveiling the Fascinating World of Complex Numbers and Discover How The Concepts Work

Introduction

Complex numbers open a window to another realm of mathematics, with applications in various fields of study. Despite its name, complex numbers are actually quite simple when we are able to grasp their concept (although the same could be said about any topic). In this article, I aim to help you reach that understanding.

Imaginary numbers

Imaginary numbers are a number system that has emerged over the centuries in conjunction with contributions from many mathematicians. The concept of imaginary numbers was first introduced to provide a solution to many unsolvable equations.

The simplest example of this is the equation $x^2+1=0$. At first glance, it might seem like an easy solve. We just have to do some rearranging to find $x$.

But where do we go from here? When a positive number is squared, the result is positive. When a negative number is squared, the result is still positive. Unfortunately, to solve this equation we need to find a number that gives a negative number when squared. That's where imaginary numbers come in.

The system of imaginary numbers is based on the imaginary unit, $i$. By definition, $i^2=-1$. With this, we can solve for the square root of any negative number.

What is a Complex Number? - Basics

To put it simply, a complex number is a number that has elements of both real and imaginary numbers. Complex numbers can be written in the form $a+bi$, where $a$ and $b$ are constants. Because of this, both real and imaginary numbers fall under the category of complex numbers, since real numbers can be written in the same form when $b=0$, and the same goes for imaginary numbers when $a=0$.

Other than a few changes, the way we deal with real numbers can mostly be applied to complex numbers. For example,

Addition and subtraction between complex numbers (this one is obvious)

Multiplication between complex numbers is done in the same way we expand brackets with two or more terms (note that $i^2=-1$ and hence $bdi^2=-bd$)

Division by complex numbers is not as intuitive, but we can use complex conjugates to remove the imaginary element from the denominator (this is similar to rationalizing denominators such as $1+\sqrt{2}$)

Square Rooting Imaginary Numbers?

Now that we have solved the mystery of $\sqrt{-1}$, I have another question for you. Try solving for $\sqrt{i}$. Seems like another unsolvable problem, but don't worry, we don't have to create another number system to solve this one. Complex numbers will do the trick.

The first step to solving $\sqrt{i}$ is to assume that the solution is a complex number - that is to say, it can be written in the form $a+bi$.

Thus, we can form this equation. If we can solve for $a$ and $b$, this proves that our initial assumption is correct

Step 1: Square both sides and expand the right side

Notice how we can write each side of the equation as a complex number, as shown below.

Step 2: Rearrange into $a+bi$

Step 3: Both the real and imaginary elements of the equation must be equal

Solve for $a$ and $b$

Hence, we have solved that $\sqrt{i}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i, or -\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i$

Another method that can be used to solve many complex problems, such as $x^3=1$, is the use of the quadratic formula. We all know the problem of $x^3=1$ gives us $x=1$, but there are also complex solutions.

Since we know $x=1$ is a solution, we can factor out $x-1$ from $x^3-1$

From the remaining factor, we can solve for the 2 complex solutions

Notice that using the other method of the substitution $x=a+bi$, we will get the same result.

Although both these methods give the same solutions, the method of factorising and using the quadratic formula is quite a bit simpler. On the other hand, for the problem of $\sqrt{i}$, the quadratic formula clearly cannot be applied. It is very important to know which method is best for certain complex problems, whether it is substitution of $x=a+bi$ or the quadratic formula, or even other methods that will be covered later.

Argand Diagrams

An Argand diagram is a way to express complex numbers on a Cartesian plane, with the real element on the $x$-axis, and the imaginary element on the $y$-axis. For example, $3+5i$ would be represented by the point $(3,5)$.

Argand diagrams also unlock the modulus-argument form, where a complex number can be written in the form $r(cos θ + i sin θ)$, where $r$ is the absolute value of the complex number (which is equivalent to the distance between it and the origin), and $θ$ is the counterclockwise angle of the point from the positive $x$-axis. Below is a simple diagram that explains this notation.

The diagram shows a complex number $A = a + bi$ on an Argand diagram. Note that $a=r cos θ$ and $b=r sin θ$

Euler's Formula

Euler's formula is one of the most well-known rules when it comes to complex equations. This formula provides us with another way to represent complex numbers. To derive this, we have to use the Taylor expansion for the following functions.

First, substitute $x=iθ$

We can simplify this by using $i^2=-1$ and $i^4=1$, then grouping the real and imaginary parts separately.

If you take a look at the Taylor expansions above, you might notice something.

Euler's formula

With Euler's formula, we can also easily show Euler's identity, which you might have seen before.

Summary

To recapitulate, complex numbers are a fundamental extension of the real number system that incorporates imaginary numbers. By understanding the representation of complex numbers in different forms, the arithmetic operations on them, and the geometric interpretation of complex numbers on the complex plane, you can develop a keen intuition about this important mathematical concept.

While the rules and formulae may seem unfamiliar at first, with practice complex numbers can become a powerful way to solve many problems in engineering, physics, and mathematics that can't be easily solved with real numbers alone. By mastering the basics covered here, you will have a solid foundation in complex numbers and be able to apply them confidently in various contexts. Overall, complex numbers will enrich your mathematical language and allow for a deeper understanding of many real-world systems.

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