The Mystery of the Complex Plane

Think about when you learned algebra for the first time. On that seemingly uneventful day of middle school, the letter “x” is carefully scrawled onto the class whiteboard, and all of a sudden the world of functions, systems of equations, and incomprehensible word problems smashes its way into your mind like a battering ram of novel information. How did you cope with it all? Well, the answer for me and many other students is that it’s useful. It can seem needlessly complicated at times, but algebra is taught with a clear goal in mind: Given some known values, find the others, an idea that is useful even in day-to-day problem-solving and has obvious scientific applications.

The same wasn’t true for my class when we learned about imaginary numbers. After being inundated with formulas and definitions I could accept that they had some neat arithmetic quirks, but they were still confined to exactly what the name implied: The imagination. We didn’t need to be able to hold i apples in our hands, but without some kind of model in our heads that gave them a habitat in the rest of the mathematical ecosystem most of us let the information wash away after that one bizarre lesson. It was only after they were reintroduced in my Precalculus class earlier this year that I was forced to confront this enigma again, but this time I wanted answers. And after spending hours on my own growing to learn and love the complex numbers system, hopefully, I can lay out the journey and the very real world where these numbers lie.

First off, the number itself. “I” is defined as the square root of -1. If that sounds ridiculous, don’t worry: You’re joined by some of the greatest mathematicians of the 17th century.

“For the rest, neither the false nor the true roots are always real, sometimes they are only imaginary, that is to say one may imagine as many as I said in each equation, but sometimes there exists no quantity corresponding to those one imagines.”

— Rene Descartes

That was Descartes, the man who developed the coordinate system and countless other mathematical concepts we use today, refusing to acknowledge the validity of this novel idea. If even revolutionary mathematicians had such an attitude toward complex numbers, why do we even bother with them?

A Nonexistent Problem

In Algebra 1 you may see these numbers come up when dealing with quadratics, the truth is that the parabola below does a far better job at disproving the existence of these numbers than it does at corroborating them:

\( y=x^2+1 \)

This parabola clearly has no roots, instead increasing towards infinity on both sides of the y-axis. And it feels completely pointless to solve for the x-intercepts and declare the result as a new number system.

The quadratic formula presents a window into the “how” of complex numbers: From here we were instructed in my class to perform the 4 basic operations, calculate complex conjugates, and learn fancy words like the “modulus” and “argument.” But it does so at the expense of the “why”: What is the point of this detour into this imaginary number system, and can we actually better understand math as a result? This failure to make math matter is what causes so many frustrated students— myself included for a long time— to write off this knowledge as a waste of time and leave their Algebra class with a sour taste in their mouths. Instead, let’s look at the real story of why imaginary numbers were invented, and how they provide a natural extension of the math we know and love today.

Complex Numbers, Real Solution

The story of imaginary numbers starts off not so different from the version told in the classroom. It’s the late 1500s, and the famous Italian mathematician Gerolamo Cardano is puzzling over a powerful formula of his own derivation, one that would spit out the roots of a cubic polynomial given some basic facts about the expression.

x_1=\sqrt[3]{\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}} - \sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}

(for a cubic of the form \( x^3+px+q \))

This formula already looks more unpleasant than its quadratic counterpart, but things got even worse when Cardano discovered that there were cubics that had very real roots that couldn’t be solved by the neat formula:

\( x^3-15x-4 \)

The equation above is of the form needed to solve with the cubic formula when \( p=5 \) and \( q=4 \), but try plugging those numbers into the expression: We end up with the following:

\( \sqrt[3]{-2+\sqrt{4-125}}+\sqrt[3]{2+\sqrt{4-125}} \)

\( =\sqrt[3]{-2+\sqrt{-119}} + \sqrt[3]{2+\sqrt{-119}} \)

We’ve hit the same dead-end as with the parabola at the beginning, but here it actually matters because it’s keeping us from very real and useful solutions, including the seemingly distant value of \( x=4 \). However, this wasn’t good enough for Cardano, and he wrote the result off as absurd and remained stumped as to a “true” solution of the cubic.

It’s here, when confronted with an actual problem instead of a nonsensical thought experiment of imaginary roots, that the story can continue. A contemporary of Cardano named Bombelli decides to look a bit further into Cardano’s formula, but instead of stopping with the inexplicable negative roots he takes a leap of faith, treating the square roots as usable in computation and simplifying the formula, and here he uncovers a wild truth: By defining these impossible square roots as Cardano described and solving the resulting equation with some fancy algebra, we can actually tease out this very solution of \( x=4 \)! It’s here that Bombelli lays out the fundamental rules of complex arithmetic, describing how to add and subtract complex expressions as a first step into understanding this new number.

This was a groundbreaking demonstration of the connection i had to the contemporary world of math, but at the time even this demonstration of the new number’s power wasn’t convincing to most mathematicians: They had the proof of concept, but they needed to understand the method.

Number Hunting

So with some affirmation of their value, let’s suspend our disbelief about i for a moment and take a second look at our paradoxical parabola. These roots aren’t visible on this 2D plane, so what if they’re not in 2 dimensions at all? How can we describe and calculate roots that intersect in another dimension? We need a language that allows us to relate the various axes of our coordinate system, and to create that we need some more info about this “imaginary” number. What are some of its properties, and do they have any geometric significance?

Cyclic Powers

We know that \( i=\sqrt{-1} \), and consequentially \( i^2=-1 \). What happens when we raise it to higher powers? Carrying out the math starting from \( i^0=1 \) we see the following:

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

\( i^2 =\sqrt{-1}*\sqrt{-1}=-1 \)

\( i^3=-i \)

\( i^4=1 \)

Notice the cycle: we end up back at 1 after just 4 exponentiations! Every 4th power brings us back to 1, and this allows us to compute powers of \( i \) regardless of their size. So knowing this, what would \( i^12 \) be? What about \( i^363 \)?

The first one is straightforward: 12 is divisible by 3, so that means the cycle repeats itself all the way to 12 and lands on 1:

\( i^{12}=i^{12/4}=i^3=-i \)

What about 363? We know it isn’t divisible by 3, but if we look around it we can see that 360 is. So \( i^{363}=i^{360}*i^{3} \), and that means that the leftover \( i^3 \) is all that we need to worry about, making the final answer -i. Using this logic we can come up with the general rule for the powers of I:

For \( i^x \), the remainder of \( x/4 \) is the power \( i \) is raised to, either 0, 1, 2, or 3.

Next, we know adding 2 real numbers is possible, and looking at our definition of \( i=\sqrt{-1}, \) we know we can add together imaginary numbers just like we add radicals normally:

\( i+i=2\sqrt{-1}. \)

Continuing to leverage our radical properties, we know we can’t combine a square root and a real number without simplifying the radical first, and since \( i=\sqrt{-1} \) is as simple as we can go, we can write the combination of a real number “a” and the imaginary number “bi” as \( a+bi \), the standard notation for complex numbers.

So… Where Are the Roots?

This is a striking algebraic feature. Symmetries of numbers like this usually have strong geometric parallels, so let’s go on the hunt for some, examining the cycles of \( i \) and deducing where the imaginary numbers might lie with respect to the real ones. For starters, we know that even powers map to real numbers, so we can place the first 2 in the cycle on the number line:

\( i^0=1 \)

\( i^2=-1. \)

With subsequent even powers going to the same locations due to the cyclic nature of imaginary powers. Graphing the two on the number line below, we notice that the two are on opposite ends of the number line. Extrapolating based on this logic, where does \( i^1, \) go?

Looking at our number line, it may seem reasonable that our lone imaginary number should be somewhere between the movement from 1 to -1, and we can make the conjecture that, since the movement for each power should be equal, \( i^1, \) should be directly in the middle for symmetry. How can we verify this?

Complex Numbers as Rotation

Let’s write our statement more formally: if we have a point \( (a,b), \) we know that to rotate it 180 degrees we have to make both coordinates negative, in other words \( (-a,-b), \). This is the 180-degree rotation we see above, and to get to the middle we want a 90-degree rotation, which corresponds to \( (-b, a). \) The image below demonstrates this, rotating the blue arrow 90 degrees from the original red vector and displaying the new coordinates.

Let’s see if this logic applies to the powers of i. With our complex number in formal notation: \( a+bi \) if we multiply our expression by \( i^2 \) we get the following:

\( i^2(a+bi)=ai^2+bi^3=-a-bi. \)

This is exactly what we predicted for a 180-degree rotation of our endpoint \( (a,b)! \)

Trying this for \( i(a+bi)=ai+bi^2=-b+ai, \) we see it has the same correspondence to a 90-degree rotation: Multiplying a complex number by I rotates it by 90 degrees!

The Domain of Complex Numbers

Remember, all real numbers are complex numbers, just with a complex component of 0. (for example, 1 is just the complex number \( 1+0i \)), so this means that \( i^1 \) is simply 1 rotated by 90 degrees, \( i^2 \) by 180 degrees, \( i^3 \) by 270 degrees, and \( i^4 \) bringing it back around to 1 as it circumnavigates this expanded plane. This has effectively given us 2 degrees of freedom for what direction an arbitrary number grows: Horizontally as its real component scales and vertically as the imaginary component scales. This means that our number line truly is insufficient to encompass this expanded domain of numbers. Adding an imaginary axis is the only logical step to solve this problem, and so we arrive at the image below.

Using the notation we established earlier to package the real and imaginary coefficients a and b into a point, each distinct complex number can be written as a point in this new plane. We call this the complex plane, and it is the (mostly) complete number system that we have today.

After taking a deep dive into the geometry of these numbers, hopefully, they’re starting to feel a lot less imaginary. What started as a cheap trick is now a full-fledged and incredibly powerful coordinate system. The imaginary numbers’ unique algebraic properties allow us to build a language of rotations and reflections with i as the lever, rotating the number plane by 90 degrees and placing the imaginary number line perpendicular to their real counterparts.

What’s Next?

One pressing question still lingers: What happens to the roots of the parabola? To answer that will require us to plunge far deeper into the rabbit hole of geometry, complex functions, and higher dimensions, but thanks to the complex plane now we know where to start looking. In the next few posts, we’ll be waist-deep in this new plane, learning about the syntax and functions of the number system so we know what we’re working with. Looking forward to seeing you in the weeds!