"Imaginary" numbers were badly named

For years the word "imaginary" wired my brain towards something else. I heard it and quietly filed those numbers under fake: a trick mathematicians use, not something real.

How I used to think

My picture of numbers was simple, and honestly it worked. The number 2 was a 2-meter rod floating in space. The square of 2 was me taking more 2-meter rods and standing them up along the edge of the first one, one by one, until I could feel the area: a 4-square-meter square. Every positive number had a shape I could hold in my head.

2 is a rod. 2² is rods stood up along it, one by one, until the square appears.

Then I tried the same thing with −2, and the picture just broke. A rod can be long or short. How is a rod negative? If I put −2 in space, what am I even looking at?

A rod can be short or long. What does a negative rod look like?

I pushed the question further. Imagine space with no stars at all, complete darkness, and only this one rod glowing. There's no up, no east, nothing else there. How would you even tell which direction the rod is facing? And if you can't tell, what could the minus sign possibly mean?

One glowing rod, nothing else. Which way is it facing?

Imaginary numbers confused me even more, because I took the name literally. Real rods were the ones I could bring into space and build a wooden square from. So I pictured 2i as a rod that exists only in my mind. Multiply two of those imagined rods together, and do I get a real square floating in front of me? I was looking for the moment something crosses over from imagination into the real world.

My old question: multiply two rods from my head, and does a real square appear?

It took me a while to see that the whole question was wrong. Multiplication never makes matter, for any kind of number. And imaginary numbers were never about imagination. They're about direction.

The answer: the same rod, pointing different ways. The − and the i say which way.

What I figured out

The problem was never the numbers. It was how I pictured them, and the name didn't help: "imaginary" started as an insult in the 1600s, and it stuck. Change one habit in how you picture numbers, and they become as ordinary as real numbers.

The habit is this: stop picturing a number as an amount. Picture it as an arrow, with a length and a direction.

Then multiplying stops being "more stuff" and becomes something you do to an arrow: stretch it, squish it, or turn it. Direction and action are the same idea seen from two sides. The arrow for a number is just where its action takes a 1-meter rod: ×2 stretches it into an arrow of length 2, and ×(−1) turns it around into an arrow pointing the other way.

Start with stretching, since that part is familiar.

Numbers are things you do

Take a 2-meter rod. Now:

×3 stretches it to 6 meters.
×½ squishes it to 1 meter.
×1 does nothing.
×0 crushes it to a point.

So far, every number is a stretch or a squish. Multiplying two numbers is just doing one action after the other: stretch by 2, then by 3, and you've stretched by 6.

What does −2 even mean?

So, back to the rod that broke my picture. A rod has length but no direction, and negatives are about direction. Lay the rod on the ground from a starting mark and say "east is forward." Then +2 points east and −2 points west. Same rod, same length, facing the other way.

Now the dark space with one glowing rod. Is it +2 or −2? You can't tell. My instinct was right. There's no east, no forward, nothing to compare against. The sign doesn't belong to the rod.

My first thought was: but I can tell! If the rod lines up with my nose, it's vertical. If it lines up with my eyes, it's horizontal. But that only works because I put myself in the picture. My face became the reference. Take me out, and the rod is back to pointing nowhere. Turn me upside down, and my "up" becomes "down" while the rod never moved.

It doesn't have to be me, though. Anything works as a reference, even a second rod with nobody around. Add one and something real appears: the two either point the same way or opposite ways. Call the first one positive (a coin toss), and the second is forced to be + or −. A minus sign means "opposite to whatever I picked as the reference."

So as an action, ×(−1) means turn around. Not stretch, not squish: rotate 180°.

And the rule everyone memorises in school, negative times negative is positive, stops being a rule:

×(−1) once: you're facing backwards.
×(−1) twice: you turned around twice, so you're facing forwards again.

That's why (−2)² = 4. Squaring a number can't depend on which way you happened to call forward, because forward was never real in the first place.

The question that creates i

Now ask a simple question about actions, with no square roots in sight:

Is there an action that, done twice, turns you around?

Stretching twice won't do it: you're still facing forward, just bigger. Turning around twice won't do it: you're back where you started.

But a quarter turn will. Turn 90°, then another 90°, and you're facing backwards. That's the whole thing. Call that quarter turn i:

i × i = −1 means "a quarter turn, twice, is a half turn."

That's all "i is the square root of −1" ever meant. Written as √−1 it sounds like nonsense, because you're asking for an amount that multiplies by itself to give a negative. Asked as an action, it has an obvious answer.

The catch is that a quarter turn needs somewhere to turn to. Stuck on a single line, there's only forwards and backwards, and no 90° in between. The moment you allow a sideways direction, i exists. Gauss, one of the greatest mathematicians ever, said exactly this: they should have been called lateral numbers. Sideways numbers. Not imaginary ones.

Stretch and turn, together

Once you have sideways, a number like 3 + 4i is just a point you can walk to: 3 forward, 4 sideways. As an action, multiplying by it stretches things by 5 and turns them a little (about 53°).

Every "complex number" is a stretch and a turn bundled together, and multiplying two of them just does one action after the other. Stretches multiply, turns add up. That's why electrical engineers, signal processing and quantum physics all use them: anything that rotates or oscillates is easier to describe as turning.

Up to 3D: quaternions

Back in my dark space, one more thing bugged me. The glowing rod doesn't only have to lie across my view or stand up in it. It could point straight at me, and then all I'd see is a dot of light. That's a third direction, toward me and away from me. Complex numbers only cover a flat sheet: forward and sideways. Is there a number for that direction too?

There is, and someone spent years looking for it. William Rowan Hamilton tried to stretch the complex-number trick into three dimensions and kept failing. In 1843, walking past Broom Bridge in Dublin, he finally saw the answer and scratched it into the stone: you need three different quarter turns, i, j and k, one for each way you can turn in 3D, with

i² = j² = k² = ijk = −1

These are quaternions. My next question was: fine, but does anyone actually use this? Yes, and they're probably running in your pocket right now. When your phone works out which way it's tilted, when a game camera glides around a character, when a drone or a satellite keeps itself pointing the right way, there's often a quaternion doing the turning.

They won over the obvious approach (three separate angles for tilt, turn and roll) because of a problem called gimbal lock. At certain positions two of those angles line up and you lose a direction of control. Quaternions don't have that weak spot.

If you want to see one in real code, open three.js, the main 3D library for the web, at src/math/Quaternion.js and search for these functions:

setFromAxisAngle: builds a quaternion from "turn this many degrees around this axis." Notice it computes halfAngle = angle / 2. The quaternion is applied twice, once on each side of the arrow it's turning, so each one only needs to do half the turn. Same idea as the quarter turn done twice.
multiplyQuaternions: Hamilton's rules written out as four lines of arithmetic.
slerp: smoothly blends one orientation into another.

To build the intuition before reading code, play with eater.net/quaternions, interactive explainers by 3Blue1Brown and Ben Eater.

The takeaway

Amounts answer "how much?" Actions answer "how much, and which way?"

A positive number stretches or squishes.
−1 turns you around.
i turns you a quarter of the way.

Nothing about that is imaginary. It just needed a better name.

References

Euler's formula with introductory group theory by 3Blue1Brown. The best visual version of this post's idea: numbers as actions that stretch, squish and rotate.
Visualizing quaternions, interactive explainers by 3Blue1Brown and Ben Eater.
three.js Quaternion.js, quaternions in real code.