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Comment by gumby | Hacker News Reader

“Imaginary” was an unfortunate choice of word. Positive integers and rationals feel “legit”* to most people, even (or especially) the non-mathematically trained.

But “imaginary” numbers are no more or less legit than, say, transcendental numbers. But in middle school you learn this confusing thing and maths starts to fall off the rails.

* Had to find a synonym for “real” here.

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"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." - Gauss

I vote for “complete” numbers in place of “complex”, and we just don’t reference “imaginary” at all — it goes from “real” to “complete” when you add the missing root of unity.

My vote as an amateur mathematician would be “rotational” numbers. Everything having to do with complex numbers secretly (or not so secretly) has to do with angles and rotations, and it would be easy to describe to a class of middle schoolers that if you want to do math to describe how things rotate, it makes sense that you need a 2D space.

Twirly numbers, to me.

This just made it click for me why imaginary numbers are used in electrical engineering

Tristan Needham’s word is “amplitwist”, meaning a scaling composed with a planar rotation.

A complex number should be thought of as the quotient of plane vectors: a quantity z = v/u which when multiplied by vector u yields v: that is zu = v.

(Important note: multiplication of a complex number by a vector is not commutative, zu ≠ uz.)

For more: http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

Well, IMO their rotational feature is kind of the accidental, happy and mysterious side-effect that starts from the "simple" computational trick of defining sqrt(-1) being i, and taking the calcuslations from there.

I would philosophically say "Completeness" is a more important property (since now negative roots have a meaning attached to them). But their rotational aspect is perhaps more mind blowing.

I disagree but I don’t have an incredibly concrete argument, only an intuition that in the case of complex numbers, our mathematical formalism for solving equations proceeded our mathematical intuition about cyclical processes.

But "completeness" already has a (different) definition, like when we say the reals are complete but the rationals are not

edit: to match with existing terminology you might call the reals the "completes" and the complex numbers the "algebraically-closeds" but... meh.

How are imaginary complete? It's not hard to make a superset of the complex numbers.

algebraically, see: fundamental theorem of algebra (proved by Gauss in his PhD thesis)

What's your point? You can define some other property which complex numbers don't satisfy.

Sure. What is my burden of proof exactly?

That's what I've finally settled on in my mind. Since we use imaginary numbers to model and measure real phenomenon, they're not imaginary in the sense that I would usually use that word. There's got to be a better name... Transcendental complement maybe.

> Positive integers and rationals feel “legit”* to most people, even (or especially) the non-mathematically trained.

Probably because they are understandable quantities in their daily lives whereas complex numbers never are. I really doubt the name has anything to do with it.

> But “imaginary” numbers are no more or less legit than, say, transcendental numbers.

I disagree? The ratio of a circumference to a diameter is clearly a positive number between 3 and 4, and it sure feels more legit than something whose square is a negative number.

Well it's not that the square is a negative number but rather that you were only taught to do arithmetic with 1x1 matrices so you don't have a nice abstraction in your head to convince yourself. It's similar to people who insist that 0 is not a natural number because they haven't studied abstract algebra and don't care about the underlying algebraic structure (nevermind that peano arithmetic starts by postulating zero and succ...)

Why would abstract algebra suggest 0 is a natural number? The natural numbers aren't so much as a group, with or without 0. Peano starts with 0 because it's easier that way, but starting with 1 is easier for other things.

when I did my math degree in college the profs and grad students called them "complex numbers." I think they know there's a problem so they want to get the students who would go on to teach used to not calling them "imaginary"

That doesn't quite get at it, though -- a complex number has an imaginary and real part. I guess we could call them "complex numbers with a zero real part," but

1) that's quite a mouthful

i) it implies that the other part is something other than real

-1) it seems weird to describe something sort of... subtractively like this

Eh, it's fair to call i a complex number. I mean C includes the imaginary line.

ai is a complex number and an imaginary, but not all complex numbers are imaginary numbers (like square/rectangle). a+bi is a complex number, but not an imaginary number. And a is a complex number and a real number. Which is the problem with trying to reduce it this way, you lose the ability to distinguish (by name) the two components of a complex number. What are the two parts of a complex number called if we drop real and imaginary?

> What are the two parts of a complex number called if we drop real and imaginary?

Real and imaginary! I'm not saying we should never use the work imaginary, I'm just saying that what makes i interesting is not that it's an imaginary number, but that it's a complex number. The imaginaries on their own aren't a number system at all, and rarely come up in isolation. Objecting in introducing sqrt(-1) as a complex number seems a little silly.

It is equally fair to call i and 1 complex numbers.

I don't think that's a very fair comparison. You wind up with 1 when trying to make every number system from the naturals up. The imaginaries basically never come up except when using complex numbers in general, and certainly don't in the high-school and undergrad classes where people first learn about them.

FTA: it’s complex as in “military-industrial complex,” speaking of combination—of real and imaginary parts—rather than complication

From first introduction in middle school math all the way through undergraduate university math to reading this tonight at age 55, no teacher or professor or book I recall has ever stated that "complex" meant "multi-part." I always thought of it as "complicated." Yes we learned "real part" and "imaginary part" but that was never connected to the name "complex." An exercise left to the reader, I suppose.

Math education has a long way to go.

Yes, as Rich Hickey has pointed out in programming talks, to “complect” originally meant to twist, braid, or weave together. It is a textile metaphor.

I didn't find it confusing until I got to branch cuts.

Try Needham's "Visual Complex Analysis" - it's so aggressively intuition-y (and visual, duh) that you'll either join the cult or run for the hills (with good reason).

If I were in charge of math, I’d call them placeholder numbers, or something like that. They’re there to get you to a solution, but the solution to a legit problem will be a legit number.

It allows you to use the same algorithm to solve polynomials, and stuff like that, but you end up with these placeholders that will vanish along the way to the final solution.

Except for quantum mechanics, where solutions to legit problems are complex, just to name one

The wave function can be viewed as another placeholder which gets you to your “legit” probability distributions over observables.

Only if you go in for a hidden-variable theory, which is absolutely cool but means you have to give up a lot of properties you'd really expect the universe to have

Yeah we should stop using the word imaginary, and just call them iNumbers (sorry Apple)

I like to call them "double-barreled numbers". I never liked the term "imaginary", because that gives mystical connotations and even hamper understanding, in my opinion.

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