E^(pi i) = -1?

Factual Questions
61

This is circular, though. Most people encounter imaginary numbers in an algebraic context because that’s how math curriculum presents them. There’s no reason that modern math instruction should mirror historical progress.

If, instead, the standard progression of math instruction followed an explanation more like @Chronos 's where each new set of numbers was added/discovered to close various operations, then we get to talk about how square roots aren’t closed with single-dimensional numbers.

Yeah, I also first encountered i as the solution to sqrt(-1), but for years it was just a weird arbitrary thing that didn’t obviously map to anything that made sense, and years later the explanation of i as rotational made a lot of sense. But we don’t have to teach it that way.

62

That’s because they are complex integers not just the imaginary ones. Like the Gaussian prime 1 + i.

From one point every number except the positive whole numbers arose from our imaginations.

63

Even those. We generalized from the universe the concept of “one thing”, “two things”, “three thing”, etc.

64

“God made the natural numbers; all else is the work of man”.- Leopold Kronecker

65

This.

Agree that @Chronos’ post was a model of clarity and how this stuff ought to be introduced / motivated.

66

My first encounter with imaginary and complex numbers was with the Mandelbrot Set. I read a book by Arthur C. Clarke that talked about it.

I became a bit obsessed with this infinite fractal thing, and taught myself about complex numbers in order to understand it better.

Several years later, when I encountered them in actual school, the way that we used them was very boring, and most of my classmates didn’t understand them.

I do wonder if teaching about fractals and such as a way of at least giving a use for complex numbers would be of value.

67

I would say rather that God made the real numbers, and that it is the counting numbers which are the human invention. Sure, I might think that I have 17 sheep in my flock, and my neighbor has 23 in his flock. But maybe my sheep are larger than his, and so I actually have more pounds of mutton. Maybe one of my sheep lost a leg in a wolf attack-- Is that a fractional sheep? What if I have conjoined twins among my flock? Humans want things to be discrete individuals, but that’s not the way the world actually works.

68

Hmmm. You don’t hold with what this new guy Einstein says about photons, do you?

69

Sometimes some things act like discrete particles, but they’re not.

70

We treat things in the real world as continuous and infinitely divisible, but when you get down to the molecular level, or the atomic level, or the quantum level, that breaks down. According to the latest understandings of modern physics, is anything in the real world a real continuum?

71

I won’t deny that the pedagogy could be improved, but it feels like you’re just reversing the problem. Ok, so you start with the focus on rotation. Now you get to algebra and the quadratic formula. Great! Except that all that focus on rotations is lost; you’re really just looking for a mechanical solution to x^2+1=0. You leave the students confused–why all that emphasis on rotation when it doesn’t appear in some major applications?

Also, while some of the simpler aspects are explainable (as you say, two 90 degree turns is a 180 turn, just like turning twice to get -1), actually justifying the argument in all its detail needs somewhat higher level math. The usual justification of Euler’s formula needs series expansions and thus calculus (I don’t see how that would ever come before algebra). It’s possible to do a more hand-waving justification with just trig, but that’s still at a fairly high level. The pieces interconnect in a way that doesn’t leave much freedom to reorder things.

In my education, even when the rotation-like aspects we emphasized (phasors, complex impedance, etc. in engineering classes), complex numbers were still treated somewhat mechanically, as in that you go through the steps and get an answer. That could have been improved, certainly.

So I dunno. I think the complex numbers deserve a better treatment, but I don’t see any obvious way to do better than the current approach, which is to introduce new features on demand. You need algebra to do calculus, but you need calculus to understand series expansions or even the simpler “e^ix is just moving at right-angles to your current position” argument. Further depth like their appearance in Fourier transforms requires even more math.

Personally, I only really grokked them when I did my software-defined radio work. To the point now where I no longer imagine waveforms existing just in a 2D plane; they are always paths through 3D space, and usually a helix. It was pretty mind-blowing when I finally got it, and previous education wasn’t a great deal of help.

72

At the lowest level, nothing is continuous and nothing is discrete. Instead it’s a complicated mess. Sometimes it’s conceptually easier to treat things as continuous. Sometimes it’s easier to treat things as discrete. But in reality, it’s neither.

73

Wibbly wobbley timey wimey… stuff.

74

Who says you need to introduce Euler’s identity as soon as you introduce imaginary numbers? Of course you’re not going to be saying that e^(ix) = cos(x) + i*sin(x) until you have at least trigonometry and pre-calculus, and probably calculus.

You might as well rail against the teaching of natural numbers using counts of apples and sheep and such, because all of that focus on apples is lost once you start doing algebra. So what?

75

You don’t, but then the notion of imaginary numbers as turning-numbers seems to be even more disconnected. I’m all in favor of having more fun in math, and the introduction of a special rotating number could fit into this category, but eventually you want it to connect to the rest of math. Algebra comes before trig because you need to be comfortable with symbolic manipulation. But then you’re faced with the first “real world” application of complex numbers having no seeming connection to the turning-numbers.

Do you have a specific proposal in mind with how the classes should be ordered? Not trying to be defensive here; I just don’t understand how to introduce imaginary numbers as turning-numbers without it seeming out of the blue. Apples and sheep are used because they’re concrete examples of a general concept, and students generally like to see something concrete. But imaginary numbers as turning-numbers is a conceptual thing–totally different in my mind. Introducing a concept only to ignore it seems strange (except as fun).

76

Sorry to post this so late in the game (this thread), but I just could not remember where I had seen it first. It turns out that it was on Wikipedia, discussing division by zero and instances where it is defined. It seems to me, non-mathematician that I am (but still very interested in the field), the thing that is undefined is just moved to another issue in those instances. I don’t know if that makes sense to you mathematicians.

77

Pretty much. As noted earlier, you can add a kind of 1/0=∞ number to your system. But then you lose other elements of your arithmetic.

In particular, it is always the case with the real numbers that if you start with any real numbers a and b, then a-b is a real number. But that’s not true with ∞ - ∞, which must be undefined. So you’ve managed to give 1/0 a defined result, at the expense of other operations becoming undefined.

That doesn’t mean it’s not useful in some cases, but it does mean you pay a cost.

The same is not true of adding the imaginary number i to your system. That is a universal win. You can do more things with the math and you lose nothing.

78

Thanks. That’s actually pretty cool, IMHO. I don’t get the angst some show about calling some numbers imaginary. It’s a good name, just like flavor for quarks. As long as the thing names is useful, all’s well.

79

That’s a good comparison. Color, too. Maybe some people get confused initially, but not after spending 10 minutes on the subject.

80

Another bit of expansion on a topic that was brought up earlier: groups.

At a very high level, a group is a set of actions you can perform. Actions on what? That doesn’t actually matter that much! In fact, we’ll just ignore that for the moment. There are a few rules: we can combine actions together to form a new action, which is also in the set of actions. For every action, there is an inverse action. If you combine an action with its inverse, you get a null action, called the identity. And if you combine an action with the identity, you get the same action back. There are some minor extra rules, but that’s basically it.

The relevance here is that there is an additive group of the (integers, rationals, reals, complex numbers). The action is to add a number (Add to what? Doesn’t matter. You can start with 0 if you want, but you don’t have to). So, “add 3” is an element of the group. So is “add 4”, and the combination “add 7” also is. Every action has an inverse, so “add 3” has an “add -3”, which if you combine gives an “add 0”, which is the identity element. And “add 3” combined with “add 0” is “add 3”, just as we wanted.

It’s easy to verify that the rules all work for the common number systems. Well, except for one: the whole numbers. The whole numbers don’t include the negative numbers, so there’s no inverse to “add 3”. But everything with a negative works.

If there’s a group for addition, then how about multiplication? Well, almost. We can only have a multiplication group if we specifically exclude 0, because 0 has no inverse. But all the other numbers are fine, because if you multiply together any two non-zero numbers, the result is also non-zero. There’s no chance of zero somehow “infecting” the group by arising from the result of some operation. Oh, and you can’t do it with the integers, because 3 has no inverse (it would have to be 1/3, which isn’t an integer). Rationals/reals/complex are fine.

Now, we can add 0 anyway, we just don’t have a multiplicative group afterward. Manually adding an inverse of 0 doesn’t help, because there’s no way to do it without ruining the other rules (including your additive group). That’s not the end of the world, but groups have some nice properties and we’d like to keep them if possible.

There are lots of other groups. The set of moves on a Rubik’s cube is one. So is the set of times on a clock. The Rubik’s cube is a finite group, since there are only so many moves you can make. The times on a clock are finite if you limit yourself to hours, but could be infinite if you allow any time division.

The addition/multiplication groups are actually more powerful than just a plain group: they’re abelian groups. That means that the order of operations doesn’t matter: +3, +4 is the same as +4, +3. Same for x3, x4 and x4, x3. However, this is not true of the Rubik’s cube! Turning the left face, then the top face is not the same as the top face, then the left face. So the moves on a Rubik’s cube still form a group, but not an abelian group.