It’s not uncommon to run across thoughts being voiced along the lines “Imaginary numbers (the square roots of negative numbers) don’t actually exist; at least, not in the same way that real numbers, integers, and so on exist in the world around us. After all, it’s not like you could run sqrt(-5) miles. Imaginary numbers are just a crazy mathematical fantasy which happen, somehow, to give the right results for certain calculations, even though they’re entirely physically meaningless.” That sort of thing, though the exact details vary from case to case. Occasionally, some more sympathetic soul will rejoinder with a reference to the ubiquity of complex numbers in (usually) electrical engineering (of all things!), ostensibly defending imaginary numbers’ legitimacy but, unfortunately, doing so in an unhelpful manner which continues to portray them as weird, arcane concepts of little everyday application.

No doubt, a lot of this stems from the conventional terminology, “Real” and “Imaginary” numbers. Clearly suggestive of the one being genuine in existence while the other must lie somewhere between eccentric fantasy and dangerous delusion… But the names are actually just fossilized ignorance, and, indeed, such thoughts as I opened with above are deeply misguided.

Today, even if you don’t already, you’re going to see why.

**Prerequisites for understanding this post:** Just about none. In fact, in a way, the less, the better.

Let’s think about sticks. Sticks have lengths. And we can scale these lengths; we can make a stick twice as big, or three times as big, or half as big, or 5.8 times as big, and so on. And it’s in terms of these length ratios that we actually give our measurements; we say “John is 5.8 feet tall” to mean “John is 5.8 times as big as a ruler; i.e., if you scaled a ruler by a factor of 5.8, it’d be as large as John”. And this shows us how to interpret certain numbers as actually about real-world quantities, and life is good. We might even say this shows that 5.8 “exists”, if you want to talk that way, but I’d really rather you didn’t.

And we can interpret addition and multiplication within this framework as well: multiplication means “chain the scalings one after another”: 7 * 5 = 35 because making something 7 times as large, and then making the result 5 times as large has the net effect of making what you started with 35 times as large. Addition means “carry out both scalings, then place the one stick after the other and see where you end up”: 7 + 5 = 12 because something 7 times as large as a ruler laid end to end with something 5 times as large as a ruler ends up at the same place as something 12 times as large as a ruler. So life is really good. We know perfectly well what arithmetic means now.

But wait… we’re missing something. We haven’t accounted for negative numbers. It wouldn’t seem like it means something to scale by a negative factor, so how can we make sense of them? Well, as you are probably familiar, there is a natural convention to adopt. Instead of focusing solely on lengths, we’ll now look at what direction our sticks are pointing in as well; in addition to scaling sticks up or down in size, we’ll also talk about flipping them 180 degrees around to point the other way. So, for example, -1 will mean “Turn your stick 180 degrees”, and -5 will mean “Make your stick 5 times as big and turn it 180 degrees”. But we’ll interpret addition and multiplication exactly the same way as before: -7 * 5 = -35 because “Make it 7 times as large and turn it 180 degrees” followed by “Make it 5 times as large” has the same net effect as “Make it 35 times as large and turn it 180 degrees”. And -7 + 5 = -2 because if I make two copies of my ruler, one 7 times as large but turned around, and the other 5 times as large and unturned, and place the one after the other, the ending point’s location is the same as if I’d just made a copy of my ruler which was twice as large and turned around. So life is super. Looks like negative numbers “exist” as well (but, please, try not to talk that way).

But, hell, once we’ve started talking about turning sticks, why limit ourselves to full half-circle turns? Why not look at quarter-turns, eighth-turns, 23.4 degree turns, and so on?

Why not indeed. Once we toss these in, we get… the complex numbers. All that mysterious i means is “Make a 90 degree turn”. We still interpret addition and multiplication exactly the same way as before; multiplication is still “Do these in sequence” and addition is still “Do these in parallel, lay the results one after another, and see where you end up.” In particular, as far as multiplication goes, since “Turn your stick 90 degrees. Now turn it 90 degrees again.” has the same net effect as “Turn your stick a full 180 degrees”, we see that i * i = -1. That’s it; it’s extraordinarily simple. Life is fantastic. Complex numbers “exist” every bit as much as real numbers; it’s just that the complex numbers express scaling with arbitrary rotation, while real numbers are limited to scaling with half-turn-increment rotation. [And non-negative real numbers express scaling with no rotation at all.]

At the end you conclude that “Complex numbers “exist” every bit as much as real numbers…”

I think you should say, based on what you have in the post so far, that complex numbers ‘exist’ as much as numbers in . To get the real numbers, like say , I’d want to say something like “take a length 1 ruler (i.e. don’t stretch it), then find the

exactlength of rope that you need such that a circle can be formed with radius equal to the length of the ruler. If you could somehow make a flawless circle, having measured outjustthe right length of rope, that length of that rope laid straight would be “. A similar story involving triangles and is there too, for example. Maybe you have a more general example which would capture both the real algebraic and the transcendental numbers (i.e. any irrational number), but maybe you’ll just appeal to people’s intuitions about space or something.But a great post notwithstanding my little quibble. I just happened to be talking to someone (a non-mathematician) yesterday who didn’t think numbers like existed in the same way that, say, the integers or rationals did. I tried in a number of different ways to explain how they are very natural etc., and we shouldn’t be talking about existence in that way anyway, etc. but I don’t think it worked. I didn’t even get to ‘imaginary numbers’…

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Alex Renneton June 13, 2009at 6:32 am

[…] all it’s saying, but it is. Let’s break it down, using the relevant definitions (if you haven’t yet done so, you should first review how negative and complex numbers is a way …). First, let’s recall our definitions of and : I’ll use to mean one full revolution […]

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Euler’s Identity, While True, Is Overrated, And Is Also Understood By Small Children « The Pleasant Feeling of Paradoxon October 14, 2010at 10:30 am

What is an intuitive way of explaining imaginary numbers?…Complex numbers are no mere ad hoc mathematical construct; they are actually quite natural and mundane entities that every small child already manipulates every day, whether they realize it or not. Specifically, complex numbers are certain operations: …

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Quoraon October 19, 2010at 11:30 pm

[…] is super. Looks like negative numbers “exist” as well (but, please, try not to talk that way). (source) Excuse my logical leap, here, but I find it more appropriate to conceive of the imagination as […]

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The Gordian Knot and the Imaginary Number « isomorphison April 18, 2012at 7:03 pm