Introduction to Complex Numbers

In mathematics, real numbers represent quantities that can be placed on a number line—numbers like 3, $\sqrt{2}$ , and $\log 23$ . These numbers can be compared, ordered, and used to measure or count physical quantities.

However, certain operations in mathematics yield results that do not exist on the real number line. One such operation is the square root of a negative number.

What's the Square Root of a Negative Number?

Consider how useful square roots are: $\sqrt{25} = 5$ , $\sqrt{2} \approx 1.41$ , and so on. But what about $\sqrt{-64}$ ? Is it 8? Is it more than 8? Less than 8?

We learned about the Trichotomy Property of real numbers. 8 is a real number. If $\sqrt{-64}$ is a real number, than we must be able to say one of these things. Either they are equal, either 8 is greater, or is it lesser, but only if both are real numbers. So which is it?

The answer is: $\sqrt{-64}$ is not a real number. It does not exist on the number line. No real number multiplied by itself equals negative 8 or any negative number for that matter.

There is no $a \in \mathbb{R}$ such that $a^2 = -64$

Asking if $\sqrt{-64}$ is greater or less than 9 is like asking if a dice is greater than the chemical formula of sugar—it simply doesn't make sense in that context.

The Birth of Complex Numbers

In higher mathematics, it became useful to represent these "impossible" numbers in a structured way. This led to the creation of complex numbers.

We know that we can express the square root of a number as the product of the square roots of its factors.

$\sqrt{6} = \sqrt{3 \cdot 2} = \sqrt{3} \cdot \sqrt{2}$

Similarly, by that logic we can say

$\sqrt{-64} = \sqrt{64 \cdot -1} = \sqrt{64} \cdot \sqrt{-1}$

This approach lets us reuse the square root of the negative part— $\sqrt{-1}$ —as a common unit in all such cases.

Imaginary Numbers

The square root of -1 is defined as a new kind of number called an imaginary number, denoted as $i$ and read as iota.

$i = \sqrt{-1}$

Any multiple of $i$ is also imaginary: $2i$ , $\dfrac{3}{2}i$ , $\sqrt{7}i$ .

Complex Numbers

A complex number is a number that has a real part and an imaginary part. It's written in the form:

$z = x + iy$

Here, $x$ and $y$ are real numbers, and $i$ is the imaginary unit.

Examples of Complex Numbers

$1 + i$
$\dfrac{3}{2} + 7i$
$\sqrt{3} + \sqrt{2}i$

Classifying Complex Numbers

Based on the values of $x$ and $y$ , complex numbers can be classified as:

Real: $z = x + 0i$ (imaginary part is zero)
Purely Imaginary: $z = 0 + iy$ (real part is zero)
Complex: $z = x + iy$ with both $x \neq 0$ and $y \neq 0$

For $z=x+iy$

$\begin{aligned} \text{If } y = 0, &\Rightarrow z \text{ is real} \\\\ \text{If } x = 0 \text{ and } y \neq 0, &\Rightarrow z \text{ is purely imaginary} \\\\ \text{If } x \neq 0 \text{ and } y \neq 0, &\Rightarrow z \text{ is complex} \end{aligned}$

Some clarification is needed here. Generally, all real numbers are a subset of complex numbers. In a way, 2 is a complex number whose imaginary part is 0 like $2+0i$ . Similarly $2i$ is also a complex number with its real part 0 such as $0+2i$ .

However, in general, specificity takes precedence over semantics. Keep in the back of your mind that all real numbers and imaginary numbers are also complex numbers, but it is better to specifically call 2 a real number and $2i$ an imaginary number.

Activity: Match The Columns!

Reorder the Type of Number to match them with the correct Condition.

Condition

Type of Number

$z = x + iy \text{ where } y = 0$

$z = x + iy \text{ where } x = 0 \text{ and } y \neq 0$

$z = x + iy \text{ where } x \neq 0 \text{ and } y \neq 0$

True or False Activity!!!

Read each statement carefully and decide if it is true or false.

The number $\sqrt{-9}$ is a real number because squaring $-3$ gives $-9$ .

Every real number is also a complex number.

The sum of two imaginary numbers is always a real number.

The product of a nonzero real number and a nonzero imaginary number is always imaginary.

The square root of $-1$ is less than the square root of $1$ .

Activity: Sort Into Categories!

Drag and sort the given items into Real, Imaginary, and Complex. Think carefully — everything has its place!

Unsorted

Real

Imaginary

Complex

Score: 0 / 15 (0%)

Why Use Complex Numbers?

With real numbers, we can only talk about one aspect of something, one dimension. But higher mathematics is greatly helped by matrices, vectors, tensors, cartesian planes, and other tools including complex numbers to talk about more than one thing, more than one dimension. A complex number has two pieces of information, an x and a y value and thus two dimensions.

Mathematicians play around with these abstract concepts of matrices, and complex numbers to find useful applications such as the four below.

1. Electrical Engineering

The general forumula for Impedance (impedance is a property of current like resistance and capacitance) in an AC Circuit, is

$\ Z = R + jX$

Where $j = i$ . Electrical engineering prefers $j$ for the same imaginary unit since $i$ can be misinterpreted as current. In physics you will learn that we write Impedance in a complex form because it is useful for analyzing some AC circuits.

2. Signal Engineering

We have the Fourier Transform

$F(\omega) = \displaystyle \int_{-\infty}^{\infty} f(t) \, e^{-i\omega t} \, dt$

Note the random imaginary unit iota in the exponent. Fourier transform is the foundation of modern signal processing and everything to do with waves. You will definitely work on this in any engineering field where any type of wave has to be studied and analyzed.

3. Theoretical Physics

We have the Schrodinger equation.

$i\hbar \dfrac{\partial \psi(x,t)}{\partial t} = \left( -\dfrac{\hbar^2}{2m} \dfrac{\partial^2}{\partial x^2} + V(x,t) \right) \psi(x,t)$

These equations are not to scare or and there is no expectation for you to know of them. But, the Schroginder equation is a powerhouse in Physics. Note the $i$ at the very start.

4. Trigonometry

A simpler example might be Euler's formula.

$e^{i\theta} = \cos{\theta} + i \sin{\theta}$

This formula uses complex numbers to somehow turn trignometric functions that go up and down and oscillate in a single exponential equation with the transcendental number e.

Summary

I cannot tell you why complex numbers are useful without showing you how these individual examples work because of their exact nature. A true understanding would only happen if you try to do signal processing with and without complex numbers and see the comparison. And explaining that is beyond the scope of this course and too time consuming. I hope just showing that you will use complex numbers in your engineering and sciences futures is enough at this stage.

End of Lesson