Imaginary Numbers Are Real

Math as a field has existed since the beginning of history. From the moment humanity started building civilization, numbers existed, but not all the numbers we know today existed. We started with the integers: $1, 2, 3, 4, 5, ...$

These numbers were enough until we started running into problems calculating ratios. Around that time, fractional numbers, Fractions, came into existence to solve some of the problems we were facing.

Later we discovered that not every number can be written as a fraction, like $\sqrt{2}$ for example. This period was a bit rough, because some cultures considered whole numbers sacred, believing that everything in the universe could be written as whole numbers and fractions.

The Evolution of Numbers

There’s an old story that tells of the Greeks, where a group followed a philosophical and religious school called the “Pythagoreans”. They discovered something that shocked them: the length of the diagonal of any square with a side length of 1 equals $\sqrt{2}$. It can’t be written as a fraction. An irrational number. The person who discovered this was named “Hippasus”, and the story goes that the Pythagoreans killed him for saying so (most likely by drowning him).

And this is the pattern we’ll keep seeing throughout this story: every time a new kind of number was discovered, people resisted it at first.

After that, our next new invention of numbers was negative numbers and zero. These numbers also faced resistance, because if you really think about it: what does it even mean to have -5 of something?

People tried as hard as they could to avoid writing anything as negative. So if you had, say, $a + b - c = 0$, you’d write it as $a + b = c$ instead.

I have 5 apples. There’s nothing in the universe that’s negative something of something.

Anyway, that’s the new system of numbers we have. Positive numbers, negative numbers, irrational numbers, and zero.

The Equation

We stayed like this for a while. And mathematicians, as usual, kept solving more and more complex equations. Until one of them arrived at an equation that no one could solve with any existing numbers, and was left standing in front of it not knowing what to do. That equation was:

Now we have a problem. Roots are the values that, when you plug them into the equation, make the equation equal zero. Someone once proved this:

Polynomials of degree $n$ have exactly $n$ roots.

This proof is so important that it was named the Fundamental Theorem of Algebra. They called it that because it’s one of the most important facts that all of algebra is built on.

This proof says that this equation must have two numbers that make it equal zero. No argument.

But if you look at the graph, you won’t find any point that equals zero. That means the Roots exist, just not among the numbers we know. That’s exactly the problem.

If we look at the function, we won’t find any point where it equals zero.

See how this function never touches the X axis at all:

If we try to solve:

We get some very strange numbers.

Imaginary Numbers

The root of negative one? Even worse, negative the root of negative one? Well, there are two solutions to the equation, but unfortunately neither one makes sense. These numbers kept showing up in mathematical solutions all the time. Some people got used to them, and others assumed that whenever these numbers showed up, it meant there was no solution to the equation they were solving.

We called these numbers imaginary numbers. One of the worst names on the planet, because if you tell someone a number is “imaginary,” they’ll assume it “doesn’t exist” or that it’s some kind of hack you use just to solve your equations. Some people have suggested we call them “lateral numbers” or “elevated numbers” instead, for reasons that will become clear soon.

Since these numbers keep showing up everywhere, what if we added them to our number system? It wouldn’t be the first time we’ve had to adjust our system. But we can’t just add anything willy-nilly. If these numbers are actually going to be useful, they need to obey the existing laws of algebra.

It looks like our strange new numbers work perfectly fine with every mathematical operation we’ve tried. But how do we write them?

People agreed on a way to write imaginary numbers. Every imaginary number can be written as a real number multiplied by $\sqrt{-1}$. This number is so special that we gave it its own symbol: $i$¹.

The Complex Plane

Now, we want a way to add our numbers to the system so that we can write functions and graph them. We’ll sacrifice our Y axis and put our imaginary numbers on it like this:

This is our new coordinate system, with real numbers on the horizontal axis and imaginary numbers on the vertical one:

Notice something about this system.

Take $1$ as an example. What happens if you multiply this number by $i$, or by $\sqrt{-1}$?

You’d get $1 \times \sqrt{-1}$, which is $\sqrt{-1}$. Now what if you multiply $\sqrt{-1}$ by $\sqrt{-1}$? You’d get $\sqrt{-1}^2$, which is -1. What if you do it again? You’d get -$\sqrt{-1}$. Multiply once more and it goes back to 1.

And this pattern repeats forever.

See how multiplying by $i$ rotates the numbers in a full circle:

Complex Numbers

Now we have two kinds of numbers: real numbers, whose set is denoted $\Reals$, and imaginary numbers, whose set is denoted $\Im$. The set of imaginary numbers looks a lot cooler, I know. And when we combine them together, we get a third kind called complex numbers, $\mathbb{C}$.

We found a way to combine these two sets by placing real numbers on the X axis and imaginary numbers on the Y axis. But how exactly do we place a point on this graph of ours?

The notation everyone uses is to add the two numbers together. This makes the algebra for these numbers much easier.

For example, if we want the point at $(2,3)$, we’d write it as $2 + 3i$. Combining a real number with an imaginary number like this gives you a complex number.

For example, the point $2 + 3i$ is placed on the graph like this:

Operations on Complex Numbers

Now that we know what complex numbers are, let’s look at how we work with them. We won’t be talking about real numbers alone, or imaginary numbers alone. We’ll talk about the complex whole, both parts together.

The basic rule is simple: real with real, and imaginary with imaginary.

Addition

Exactly like adding any two things of the same kind. Real numbers together, and imaginary numbers together.

Subtraction

The exact same idea.

Multiplication

Multiplication is a bit more interesting. We multiply just like we would any two binomials, but we need to remember that $i^2 = -1$.

Notice that $6i^2$ turned into $-6$, meaning an ordinary real number. This isn’t a mistake, this is exactly what’s supposed to happen. The imaginary and real numbers mix together during multiplication in a perfectly consistent way.

The Elephant in the Room

We now understand the basic operations on complex numbers. So here’s a question: where are the roots we were talking about here? If our equation is supposed to equal zero at two points, where are those two points?

Look at our equation:

How many variables are in this equation of ours? We have one, $x$, which is our input, and we have $f(x)$, which is our output. If $x$ equals 2, for example, then $f(x)$ equals $2^2 + 1$, which is $5$.

If we bring in our imaginary numbers, then we now have an extra variable. So the number of variables we have becomes 3. Two of them are input, and only one is output.

Our function is now called a complex function, because it takes a complex number rather than a real one.

To understand the idea, let’s take a simpler function as an example:

$$f(x + iy) = x + 5$$

Notice that our imaginary part is now $y$, not $x$.

So how would we graph this function? First, let’s plug in a few numbers.

$$\begin{aligned} f(\textcolor{#4488ff}{0} + \textcolor{#ff4444}{0i}) &= \textcolor{#4488ff}{0} + 5 = \textcolor{#44cc88}{5} \\ f(\textcolor{#4488ff}{1} + \textcolor{#ff4444}{1i}) &= \textcolor{#4488ff}{1} + 5 = \textcolor{#44cc88}{6} \\ f(\textcolor{#4488ff}{2} + \textcolor{#ff4444}{2i}) &= \textcolor{#4488ff}{2} + 5 = \textcolor{#44cc88}{7} \\ f(\textcolor{#4488ff}{3} + \textcolor{#ff4444}{3i}) &= \textcolor{#4488ff}{3} + 5 = \textcolor{#44cc88}{8} \end{aligned}$$

As you can see, we have 3 numbers changing. 3 Variables. If we want to graph this function of ours, we’ll need to draw it as a 3D graph, and it’ll look like this:

Notice that the output of this function depends only on the real part, which is $x$. No matter how much you change $y$, the output won’t change, because the nature of this function’s output is that it depends only on the real part, even though it takes the imaginary part as input as well.

Let’s go back to our original function, which we agreed would be $f(x + iy) = (x + iy)^2 + 1$.

Let’s plug in numbers the same way we did with the previous function, but this time with some random numbers.

$$\begin{alignedat}{2} f(\textcolor{#4488ff}{2} + \textcolor{#ff4444}{1i}) &= (\textcolor{#4488ff}{2} + \textcolor{#ff4444}{1i})^2 + 1 &&= (\textcolor{#44cc88}{4} + \textcolor{#ffcc00}{4i}) \\ f(\textcolor{#4488ff}{1} + \textcolor{#ff4444}{3i}) &= (\textcolor{#4488ff}{1} + \textcolor{#ff4444}{3i})^2 + 1 &&= (\textcolor{#44cc88}{-7} + \textcolor{#ffcc00}{6i}) \\ f(\textcolor{#4488ff}{3} + \textcolor{#ff4444}{2i}) &= (\textcolor{#4488ff}{3} + \textcolor{#ff4444}{2i})^2 + 1 &&= (\textcolor{#44cc88}{6} + \textcolor{#ffcc00}{12i}) \\ f(\textcolor{#4488ff}{5} + \textcolor{#ff4444}{4i}) &= (\textcolor{#4488ff}{5} + \textcolor{#ff4444}{4i})^2 + 1 &&= (\textcolor{#44cc88}{10} + \textcolor{#ffcc00}{40i}) \end{alignedat}$$

Now we have a problem. Our output is made up of two numbers, and our Input is made up of two numbers as well. That means these count as 4 Variables. In other words, we’d need 4 dimensions to graph this equation. Since we live in a three-dimensional world, we can’t graph this output of ours in a single drawing.

Our solution is to “model” this output of ours into a single number that represents the distance from the origin.

So instead of the equation with its output looking like this:

$$\begin{alignedat}{2} f(\textcolor{#4488ff}{1} + \textcolor{#ff4444}{3i}) &= (\textcolor{#4488ff}{1} + \textcolor{#ff4444}{3i})^2 + 1 &&= (\textcolor{#44cc88}{-7} + \textcolor{#ffcc00}{6i}) \\ \end{alignedat}$$

It’ll look like this:

$$\begin{aligned} f(\textcolor{#4488ff}{1} + \textcolor{#ff4444}{3i}) &= (\textcolor{#4488ff}{1} + \textcolor{#ff4444}{3i})^2 + 1 \\ &= \sqrt{\textcolor{#44cc88}{-7}^2 + \textcolor{#ffcc00}{6}^2} \\ &= \textcolor{#44cc88}{\sqrt{85}} \end{aligned}$$

And just like that, our output is now a single number, and we have 3 variables we can graph.

Just one small adjustment: instead of writing the function like this:

There’s a standard where we define $z = x + iy$ and write it like this instead:

And since the output of the function is a length, we’ll add the magnitude symbol:

$$|f(z)| = |z^2 + 1|$$

That’s the complete transformation of our function.

Notice now that our Imaginary axis is $y$, not $x$ as it was in the original equation.

Also notice that our whole equation equals zero when our real part, $x$, equals zero, and our imaginary part, $y$, equals $i$ or $-i$, in other words, at the point $0 + i$ and $0 -i$.

Footnotes

1. You’ll sometimes see it written as $j$, particularly in engineering fields, since $i$ is reserved for current. ↩