The Reality of Imaginary Numbers
A few years back I was tutoring a psych student in some pre-req math needed for a stats class. We were talking about number systems when I mentioned the imaginary numbers and she burst into laughter.
“What’s so funny?”
“Imaginary numbers!?! You’re joking right? That’s not real math.”
The problem with being completely immersed in math is you don’t realize when you sound absolutely ridiculous.
“It is!” I said, “And when you add them with real numbers they become complex numbers!”
“What? Now you’re telling me numbers can have complexes too?!”
Guess the term “complex” has a totally different meaning to a psychology student…
Point is: don’t judge a math concept by its name. What may seem silly at first, might be an idea that changed the world of mathematics.
Why Are They Called Imaginary?
A Common Problem
During the 16th and 17th centuries mathematicians were working on formulas and methods for solving tricky algebraic equations. Back then it was pretty cool to be a mathematician. They would challenge each other to math duels and win money for solving the toughest problems.
But they kept running into a dead end.
A dead end algebra students are painstakingly aware of.
If the solutions to algebraic problems are where the graph crosses the x-axis, what are the solutions to a graph like this? Do they simply not exist? But I thought quadratics should have 2 solutions…
Italian mathematician Rafael Bombelli came across this same dead end, but instead of quitting he hacked the system.
He thought, if there is no number whose square equals -1, well then let it be. Instead of trying to solve for the square root of -1, he left it as is and moved on.
Yeah, it was a bit hacky but it worked. He applied the laws of algebra to the square root of -1 and defined what we now know as complex arithmetic.
But…Bombelli’s “solution” wasn’t popular.
Up until this point math was purely tangible. There was either a practical application or the problem could be visualized with geometry or a graph.
The square root of -1 had neither. It was nonsense.
Descartes’ Dubs them Imaginary
Just like you might be feeling incredulous towards imaginary numbers, so were Bombelli’s peers.
One of those skeptical mathematicians was Rene Descartes. He coined the term imaginary in his book La Geometrie:
“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
Descartes stressed that this is an alternate system, a way of solving a “what-if” scenario. These imaginary roots, although useful, aren’t real in the sense that they are not true solutions on a graph.
They’re imagined solutions.
Gauss Clears the Chaos
Mathematicians accepted Descartes’ perspective and the term imaginary stuck. Soon mathematicians began using Bombelli’s rules and replaced the square root of -1 with i to emphasize its intangible, imaginary nature.
It took over a century and a serious hard hitter mathematician to clear up this confusion surrounding imaginary numbers.
“That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.”
— Friedrich Gauss
Gauss argued that imaginary numbers aren’t made up, in fact they make perfect sense AND they can be visualized.
Yay, what a relief!
The problem is that we’ve been looking looking for them in the wrong place. They aren’t part of the reals at all. They exist alongside, or lateral to the reals. You can think of them as another dimension, an extension, of the real number line.
From the Number Line to the Complex Plane
Extending our number system isn’t as strange an idea as it might sound.
Mathematicians had already been doing this for centuries. It began by adding zero to the natural numbers to make the whole numbers. Then we added rational numbers, the concept of negative, and even irrational numbers like π, e, and the square root of 2 to describe numbers with no pattern and no end.
All of this easily fit into our number line, but what about imaginary numbers?
Where do imaginary numbers go on the number line?
Short answer is: They don’t.
Imaginary numbers are an extension of the reals. We represent them by drawing a vertical imaginary number line through zero.
These two number lines together make the complex plane.
The Key is in Rotation
The simplest way to understand imaginary numbers is to interpret multiplication of +1, -1, and √-1 (or as Gauss says direct, inverse and lateral units) as rotation about the complex plane.
Multiplying by +1
You can think of multiplying by +1 as either a 0˚ or 360˚ rotation about the origin, since either way lands you right back where you started.
Multiplying by -1
Multiplying by -1 can be interpreted as rotating 180˚ counter clockwise around the origin. For example, if I start at 2 and multiply by -1, I end up at -2 which is 180˚ counter clockwise. And if I multiply -2 by -1, I end up back at positive 2.
Multiplying by i or √-1
Now here’s the cool part. We need a 90˚ rotation in order to interact with our new axis. That’s where imaginary numbers come into play.
Notice that if I multiply 2 by i, I get 2i which is a 90˚ ccw revolution.
If I multiply 2i by i, I get 2i² which doesn’t look like -2, but check this out → i² is actually equal to -1.
So 2i² = 2(-1) or -2, another 90˚ counter clockwise rotation.
Similarly, -2 times i equals -2i, another quarter turn.
And finally, -2i times i equals -2i² or -2(-1) which equals 2.
We could keep multiplying by i and revolving around the plane which is why imaginary numbers yield a pattern that repeats every 4 iterations.
Turns out imaginary numbers make sense after all ;)
What I love most about imaginary numbers is it took centuries for us to understand and accept this paradigm shifting concept. But because mathematicians kept working towards understanding something purely theoretical at the time, we now have the tools pivotal to modern fields like electrical engineering and quantum mechanics.
So if imaginary numbers still feel a bit strange and mysterious, know that you’re in good company because many brilliant minds struggled in comprehending this ground breaking idea too.
Next Up→ Working with Complex Numbers
In the next post, I’ll show you how you can use some basic geometry to dive deeper into the complex plane and uncover rotations that yield complex numbers 🙌🏻
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