Er, complex numbers come from x^2+1, not x^2-1.

Sigh, yes. Assume I mistyped.

A true statement, but what’s the relevance here?

You specifically asked why I was focusing on that.

Except it isn’t. You are simply wrong about that being “negative numbers talked about in a roundabout way”. Firstly, j^2=1 does not uniquely specify j=-1; as I’ve been saying, j could also be 1. Is 1+2j equal to -1, or is it 3? This isn’t a way of talking about negative numbers at all!

i^2 = -1 doesn’t uniquely specify the imaginary unit either. Obviously there are two solutions to any square root.

There appears to be a fundamental miscommunication between us as to what I’m trying to express, and I don’t think it’s worth the effort to correct it. As I said, I think this would be an interesting experiment to see if it could get kids past the initial “this is stupid and complicated” barrier, by bridging with their existing knowledge of negative numbers. Maybe it wouldn’t work, of course, but who knows?

Let’s imagine a (heavily compressed) version of how complex numbers could be introduced…

This is why I’m pretty sure we’re fundamentally miscommunicating – the example that you provide after this quoted section is precisely the thing that I think causes most kids to tune out, because there’s doesn’t seem to be a point to introducing new mathematical baggage.

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The focus on “x^2 – 1″ is because that’s the definition of imaginary numbers.

Er, complex numbers come from x^2+1, not x^2-1. My point is, just changing the sign from +1 to -1 does not necessarily make a good analogue. Using x^2-2 would be better, if you pretend you don’t know about irrational numbers.

I’ve always been under the impression that “i^2 = -1″ is more canonical/correct to say than “i = sqrt(-1)” (though both are fine to say casually).

A true statement, but what’s the relevance here?

You’re still slightly missing my point – it’s meant to be just a rhetorical trick to help bridging. You pretend you’re teaching the complex numbers, then at the end reveal “Hey, that complicated stuff I just taught you? It’s not complicated at all – it’s just negative numbers, but talked about in a somewhat roundabout way. Now, if we just add a single negative sign, then what you’ve just learned also applies to the complex numbers.”

Except it isn’t. You are simply wrong about that being “negative numbers talked about in a roundabout way”. Firstly, j^2=1 does not uniquely specify j=-1; as I’ve been saying, j could also be 1. Is 1+2j equal to -1, or is it 3? This isn’t a way of talking about negative numbers at all!

Secondly, you can’t introduce negative numbers as a more complicated thing if you already have negative numbers. If you want to build up negative numbers from something simpler, great! If you want to build up negative numbers from a system of numbers that already includes them, you are doing something that makes no sense. If you don’t start with “Let’s pretend we don’t know about negative numbers”, then anything you do regarding a “roundabout way of representing negative numbers” is simply pointless.

(And, once again, the key feature of negative numbers is that they provide additive inverses to the positive numbers, not what they square to. And, as I mentioned above, it doesn’t uniquely specify them.)

The aim, hopefully, is to break down the “why are we learning this useless complicated crap?” barrier that most people throw up when they first learn about complex numbers, by showing that they aren’t complicated by the analogy with negative numbers.

You are proposing breaking down the barrier of “Why are we learning things that are complicated and unnecessary?” by introducing something that is more complicated and more unnecessary. Well, OK — it isn’t more complicated (since it’s not like you’re actually talking about split-complex numbers). It’s just more unnecessarily complicated.

What you are suggesting fails at the introduction. The point of complex numbers is that they do things real numbers cannot. Let’s imagine a (heavily compressed) version of how complex numbers could be introduced…

“So, complex numbers! In the real numbers, -1 doesn’t have a square root. But suppose it did have a square root! Let’s call it i; it’s some new type of number, not a real number. A complex number will be a number of the form x+iy, where x and y are real numbers.” (Again, obviously this is heavily compressed.)

Now, let’s analogize that:
“So, complex numbers! In the real numbers…” — what? What problem are you solving? You are replacing something that may seem unnecessary by something that is very obviously unnecessary.
“But suppose we did have a number that squared to 1!” — Huh? We already have such numbers. We call them 1 and -1.
“Let’s call it j. It isn’t some new type of number, it’s a real number.” — Then why are we talking about it? You started off saying you were going to introduce a new type of number. Now you’re not. Which is it? Until you get to this new type of number, you’re wasting my time.
“A complex number will be a number of the form x+jy, where x and y are real numbers.” — There’s already a name for numbers of that form. They’re called “real numbers”. Any real number can be written in that form. You are not telling me anything substantial and new here, you are just presenting overcomplicated ways to talk about things I already know.”

This is, start to finish, just a poorly thought-out and ill-informed idea. I say “poorly thought out” because you don’t seem to have considered such basic issues as “Why would students care about this more than they would complex numbers?” (If they have any sense, they’ll care about it less.) Or “How do I get them to go along with the idea that this is a new type of number when it’s plainly not?” I say “ill-informed”, because if you knew more mathematics, you would not only have recognized the problems here, but also thought of a closer analogue; at the least, you’d know how to actually adjoin -1 to a system that doesn’t have it.

If you want an analogue of C that a.) acts similarly in terms of the algebra they have to do and will help them get used to that, b.) actually builds a new number system on top of an old one, doing something the old one can’t, c.) gets them used to the idea of introducing some new thing and treating it abstractly, while d.) also allowing them to think of that new thing concretely because it’s something they already know exists in the real numbers, thus sparing them for now worries like “What the hell is an imaginary number, really?”, and e.) doesn’t run into problems of “you have to make sure to distinguish the number from the representation” as well as f.) doesn’t run into any craziness like polynomials having more roots than their degree or being unable to divide (which I realize are not actually part of your proposal but are within spitting distance of it), I strongly recommend my example of Q(√2) (or any other real quadratic field, if you prefer). (At least, if they’re familiar with the notion of “rational number” and that √2 is irrational. Otherwise… maybe go with adjoining -1 as I described it? :-/ I like that one a lot less.)

Simply put, your proposal sacrifices a lot of essential points just to achieve a.) and d.) above, while not even doing all the things you claim it does (e.g. it will seem more unnecessary, not less). I really don’t think you’ve thought this through and I really don’t think you know what you’re doing here.

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The focus on “x^2 – 1″ is because that’s the definition of imaginary numbers.

Er, complex numbers come from x^2+1, not x^2-1. My point is, just changing the sign from +1 to -1 does not necessarily make a good analogue. Using x^2-2 would be better, if you pretend you don’t know about irrational numbers.

I’ve always been under the impression that “i^2 = -1″ is more canonical/correct to say than “i = sqrt(-1)” (though both are fine to say casually).

A true statement, but what’s the relevance here?

You’re still slightly missing my point – it’s meant to be just a rhetorical trick to help bridging. You pretend you’re teaching the complex numbers, then at the end reveal “Hey, that complicated stuff I just taught you? It’s not complicated at all – it’s just negative numbers, but talked about in a somewhat roundabout way. Now, if we just add a single negative sign, then what you’ve just learned also applies to the complex numbers.”

Except it isn’t. You are simply wrong about that being “negative numbers talked about in a roundabout way”. Firstly, j^2=1 does not uniquely specify j=-1; as I’ve been saying, j could also be 1. Is 1+2j equal to -1, or is it 3? This isn’t a way of talking about negative numbers at all!

Secondly, you can’t introduce negative numbers as a more complicated thing if you already have negative numbers. If you want to build up negative numbers from something simpler, great! If you want to build up negative numbers from a system of numbers that already includes them, you are doing something that makes no sense. If you don’t start with “Let’s pretend we don’t know about negative numbers”, then anything you do regarding a “roundabout way of representing negative numbers” is simply pointless.

(And, once again, the key feature of negative numbers is that they provide additive inverses to the positive numbers, not what they square to. And, as I mentioned above, it doesn’t uniquely specify them.)

The aim, hopefully, is to break down the “why are we learning this useless complicated crap?” barrier that most people throw up when they first learn about complex numbers, by showing that they aren’t complicated by the analogy with negative numbers.

You are proposing breaking down the barrier of “Why are we learning things that are complicated and unnecessary?” by introducing something that is more complicated and more unnecessary. Well, OK — it isn’t more complicated (since it’s not like you’re actually talking about split-complex numbers). It’s just more unnecessarily complicated.

What you are suggesting fails at the introduction. The point of complex numbers is that they do things real numbers cannot. Let’s imagine a (heavily compressed) version of how complex numbers could be introduced…

“So, complex numbers! In the real numbers, -1 doesn’t have a square root. But suppose it did have a square root! Let’s call it i; it’s some new type of number, not a real number. A complex number will be a number of the form x+iy, where x and y are real numbers.” (Again, obviously this is heavily compressed.)

Now, let’s analogize that:
“So, complex numbers! In the real numbers…” — what? What problem are you solving? You are replacing something that may seem unnecessary by something that is very obviously unnecessary.
“But suppose we did have a number that squared to 1!” — Huh? We already have such numbers. We call them 1 and -1.
“Let’s call it j. It isn’t some new type of number, it’s a real number.” — Then why are we talking about it? You started off saying you were going to introduce a new type of number. Now you’re not. Which is it? Until you get to this new type of number, you’re wasting my time.
“A complex number will be a number of the form x+jy, where x and y are real numbers.” — There’s already a name for numbers of that form. They’re called “real numbers”. Any real number can be written in that form. You are not telling me anything substantial and new here, you are just presenting overcomplicated ways to talk about things I already know.”

This is, start to finish, just a poorly thought-out and ill-informed idea. I say “poorly thought out” because you don’t seem to have considered such basic issues as “Why would students care about this more than they would complex numbers?” (If they have any sense, they’ll care about it less.) Or “How do I get them to go along with the idea that this is a new type of number when it’s plainly not?” I say “ill-informed”, because if you knew more mathematics, you would not only have recognized the problems here, but also thought of a closer analogue; at the least, you’d know how to actually adjoin -1 to a system that doesn’t have it.

If you want an analogue of C that a.) acts similarly in terms of the algebra they have to do and will help them get used to that, b.) actually builds a new number system on top of an old one, doing something the old one can’t, c.) gets them used to the idea of introducing some new thing and treating it abstractly, while d.) also allowing them to think of that new thing concretely because it’s something they already know exists in the real numbers, thus sparing them for now worries like “What the hell is an imaginary number, really?”, and e.) doesn’t run into problems of “you have to make sure to distinguish the number from the representation” as well as f.) doesn’t run into any craziness like polynomials having more roots than their degree or being unable to divide (which I realize are not actually part of your proposal but are within spitting distance of it), I strongly recommend my example of Q(√2). (At least, if they’re familiar with the notion of “rational number” and that √2 is irrational. Otherwise… maybe go with adjoining -1 as I described it? :-/ I like that one a lot less.)

Simply put, your proposal sacrifices a lot of essential points just to achieve a.) and d.) above, while not even doing all the things you claim it does (e.g. it will seem more unnecessary, not less). I really don’t think you’ve thought this through and I really don’t think you know what you’re doing here.

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@Sniffnoy: The focus on “x^2 – 1″ is because that’s the definition of imaginary numbers. I’ve always been under the impression that “i^2 = -1″ is more canonical/correct to say than “i = sqrt(-1)” (though both are fine to say casually).

You’re still slightly missing my point – it’s meant to be just a rhetorical trick to help bridging. You pretend you’re teaching the complex numbers, then at the end reveal “Hey, that complicated stuff I just taught you? It’s not complicated at all – it’s just negative numbers, but talked about in a somewhat roundabout way. Now, if we just add a single negative sign, then what you’ve just learned also applies to the complex numbers.”

The aim, hopefully, is to break down the “why are we learning this useless complicated crap?” barrier that most people throw up when they first learn about complex numbers, by showing that they aren’t complicated by the analogy with negative numbers.

I’m not a math teacher, though I’ve done small amounts of teaching otherwise. This might not actually work. But if I were put in front of a math class, I’d give it a try.

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Problem #1:
“OK, let’s say we have this number j, which squares to 1–”
“So, it’s 1 or it’s -1.”
“Well, OK, but let’s say we don’t know that…”

Since you want to eventually say j=-1, you can’t say, “Well, suppose that somehow it wasn’t”. The whole time now, they’re going to be thinking of j not as an abstraction but as an unknown — it’s 1 or it’s -1 but you don’t know which. All this talking about j is going to seem pointless; why can’t we just say which it is already? It’s not like you’re introducing any new numbers here!

Problem #2:
Let’s suppose you don’t commit so heavily to the j=-1 idea, and allow it to develop more like split-complex numbers. Now you’re in a crazy world where a degree n polynomial can have more than n roots, and where there are things other than zero you can’t divide by. I mean, it’s not that crazy to a mathematician, but to new students? Hoo boy. You may have trouble getting them to accept after this that in the settings they’re familiar with, yes, a degree n polynomial really does have at most n roots.

Simply put, split-complex numbers are going to mesh even less well with their intuitions for “numbers” than complex numbers will, and may mislead them later. You want them to think of complex numbers as just another sort of number, but you also want them to think of split-complex numbers as just another sort of number, and split-complex numbers I think are just too different from what they’re used to thinking of as “numbers”.

(And if you go all the way with split-complex numbers — though I realize that’s not your actual suggestion — there’s also the problem that students at this point almost certainly don’t yet know that there’s no real unified notion of “number”, and that there are different systems of numbers, that can be considered on their own, as opposed to there being one big final thing of “all numbers”. So they’re going to be confused about how complex and split-complex numbers fit together. Explaining that they don’t is going to be difficult.)

Your idea is half one thing, half another thing, and I don’t think it forms anything sensible. And the focus on the polynomial x^2-1 is… why? What is the relevance of it? If you want to introduce j=-1, do it as I suggested — suppose that you only know about nonnegative reals, and want to add this new number j, which has j+1=0, and now you have negative numbers as well. Doing so a.) captures what’s actually important about -1 and negative numbers (that it’s the additive inverse of 1, and that they give you additive inverse more generally; not that it squares to 1, 1 already has that property), b.) doesn’t lead them into settings that will just muddy their intuition, c.) is a situation that is actually analogous in that you are adjoining some new (abstract) thing that has some new property (“squares to 1″ is not new if you already have 1 and -1) to solve an actual problem…

That said, I’m thinking now that my earlier suggestion wasn’t really the best. I think now there is a better solution which matches much better with what you want.

Namely, instead of saying “pretend we don’t know about negative numbers”, say “pretend we don’t know about irrational numbers”. Then you introduce a j with j^2=2. That’s going to be much closer analogue to the introduction of the complex numbers — you get a field; you’re introducing something new; etc. And unlike my “pretend you don’t know about negative numbers” suggestion above, this one doesn’t have uniqueness problems. And it also has the property that — like what you wanted, with the split-complex-numbers-or-sort-of, the algebraic manipulations involved are very close to what you’d do with complex numbers, it’s just j^2=2 instead of i^2=-1. And then at the end you can say, well, we know that in fact there is such a j, it’s the square root of 2! And now we can do similar formal manipulations with complex numbers, except we have -1 instead of 2 and real numbers instead of rational numbers.

I think that’s distinctly an improvement over both what you suggested and what I earlier suggested.

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The point is to hopefully establish some bridging between “normal” numbers and the complex numbers, since this is usually the first time people have ever encountered number-like things that aren’t the traditional numbers they were taught by parents or elementary school.

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In addition, I have to say that Tab’s suggestion for an introduction for complex numbers seems ill-advised, and I seriously doubt that it would cause more enlightenment than confusion. (Though it could possibly be rescued; see the bottom.)

The complex numbers are obtained from the real numbers and formally adjoining a square root of -1 (a root of x^2+1), called i. This yields another field, the complex numbers. Every complex number can be uniquely expressed as x+iy, where x and y are real.

Trying to introduce complex numbers with an analogue where j^2=1 (I’ll say j rather than i, as I want i to designate the imaginary unit) has several problems. First off — is j a number already in our system (I’m assuming we’re starting from the real numbers here), or not? If so, j could be 1 or -1; either way, it’s already a real number, so why did we introduce it? And we no longer have any sort of uniqueness — any real number can be represented as x+jy in many different ways. (And, of course it can be represented as x; you don’t even need the j.) So, sure, this may help students get used to working with expressions of the form x+jy, with x+iy being analogous, but it seems a pointless step. If they can do algebra, they can work with expressions of the form x+iy as well; the hard part is getting used to the abstract nature of it — having around a mystery symbol, for which the relevant question is not “what is it?” but rather “how does it act?”. And if the former has an answer already in the system (“it’s -1″), then this doesn’t really help with that.

The other alternative of course is that j is not already in the system — we’re formally adjoining it. Then you’d be introducing students to the split-complex numbers. Suffice it to say that I’m pretty sure that this would be considerably more confusing than the complex numbers, especially if it’s used as a lead-in to the complex numbers. I’m not going to go into detail why here unless people really want; I think it should be pretty clear.

Now, that said, the proposal’s not totally unrescuable. You could, as an analogy, discuss actually adjoining -1 to the nonnegative real numbers to get the real numbers — i.e. straight up doing negative numbers this way, starting from the nonnegative reals, not introducing things you already have or going to split-complex numbers. But note though that the key property of -1 is not that it satisfies x^2-1=0, it’s that it satisfies x+1=0; that’s the polynomial you’d be formally adjoining a root of. So yes, you absolutely could introduce real numbers as numbers of the form x+jy, where x and y are nonnegative reals, and j+1=0.

Note, though, that this has the slight complication that such representations are not unique. This might not be too big of a pitfall, but you’re going to have to watch out for it, and stress that the representation is not the number. (I mean, unless you want to talk about equivalence classes, which seems not the best idea.) And then when you get to complex numbers, if they actually understood that point, they may well ask why it is that in the complex numbers we can identify the representation with the number! I mean, this is a good question, but it makes your job a little harder if you have to have an answer ready.

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