I challenge you to explain [the beauty of] Euler's Equation in terms I can understand

Factual Questions
41

In case anyone else wants to understand the nature and relationship of addition, multiplication, negation, and rotation a little better, I’ll repost what I’ve written on these boards before:

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.]

42

The really beautiful thing about Euler is its application to electronics: you can take a hideously convoluted trigonometry problem, and turn it into a relatively easy algebra problem.

43

So, Indistinguishable, I like the explanation of complex numbers as two-dimensional rotations, but I’m trying and failing to extend it to three dimensions, since after all, I can make half of a 180 degree turn in much more than two different ways. At first I thought I’d end up with quaternions, but that doesn’t work, since one of the three rotations ends up leaving my stick unchanged. In fact, if I’m not mistaken, that means that what I’m dealing with isn’t even a group, since inverses aren’t unique. But obviously it has some sort of mathematical structure. Is there anything as simple as complex numbers that can apply here?

44

For quaternions you need to think of the “stick” not only having a “forward” direction but also having an “up” direction.

45

Right, but what if my stick doesn’t have an “up” direction, but is rotationally symmetric about its axis? Effectively, I’m looking at a set of equivalence classes on the quaternion group.

46

Isn’t talk of complex numbers representing rotation really an after-the-fact justification (and that’s why it falls down when a physicist asks about 3 dimensions, or 5 or 12)?
I mean, as far as I can tell, mathematicians like complex numbers mostly because you need complex numbers to solve polynomial equations, so mathematicians who play with equations tend to automatically use complex numbers rather than just reals. [And then complex math turned out to be a useful way to efficiently represent objects with a magnitude and phase, so electrical engineering grabbed it]. And it seems the mathematicians who were actually thinking about representing points and translations in 2 or more dimensions didn’t invent complex numbers; they invented vectors and associated objects.

But if someone wants to enlighten me, go ahead.

47

My recollection is that Hamilton tried and failed to come up with a parallel to complex numbers using only 3 dimensions. He had to go to four dimensions. Doing a quick web search turns up http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/QLetter/QLetter.html

48

The unfortunate thing about rotations and scalings in 3d is that they don’t always add up to combinations of rotations and scaling anymore. For example, consider the linear operator L = “Rotate 180 degrees around the Z-axis [i.e., negate the X and Y coordinates]”. If I add this to the identity, what happens? Well, it sends any point <X, Y, Z> to <-X, -Y, Z> + <X, Y, Z> = <0, 0, 2Z>. That is, L + 1 = 2 * (Project onto the Z-axis). But this is no longer just a combination of scaling and rotation; there’s that projection in there, which can’t be represented in terms of scaling and rotation.

So, what happens is, if you want to talk about the additive arithmetic of rotations in 3d, in the way we’ve been doing, then you can’t avoid talking about the arithmetic of arbitrary linear operators in 3d. Which is perfectly fine; you can talk about the ring of arbitrary linear operators on 3d, with pointwise addition and multiplication as composition. This isn’t the nicest ring from the point of view of analogy to ordinary arithmetic; left or right inverses may fail to exist or be unique, it’s not commutative, all kinds of things. It’s far from a field. But it is a ring…

You could also talk about the ring of arbitrary linear operators on 2d space in the same way, of course, and it also has the same inverse failures, non-commutativity, etc. It just happens to be the case, in 2d, that the subring generated by rotations is a field, a property badly lost in higher dimensions.

Quaternions can be thought of, among other ways, as linear transformations upon 4-dimensional space. [Technical digression, for those who might care: In fact, every ring S (extending, say, the reals) can be thought of as a subring of the linear transformations upon some (real) vector space, by Cayley’s theorem/the Yoneda lemma, interpreted appropriately. Basically, scalars are equivalent to linear operators on 1-dimensional space, and if S extends R, then any vector space over S is automatically a vector space over R, so that S is isomorphic to the ring of linear operators on 1-dimensional S-space, which faithfully embeds into the ring of linear operators on this same space considered as over the scalars R]. There is a connection to rotations of 3d space, because of the connection between SO(3) and three-dimensional real projective space, which is the rays through four-dimensional space. But so far as I can see, the natural home for understanding quaternion arithmetic is in four-dimensional space.

49

You say “justification” like it needs justification beyond being true and useful. Who cares what order history discovered ideas in? History makes a hash of everything. The topic of this thread, for example. The easiest way to take the veil of mystery off of it is to recognize that all it is talking about is rotation. Historically, that was not understood, and so it has acquired an undue mystique. But let us not be shackled by the superstitions of the ancients, eh?

Yes, it’s not immediately obvious that the free ring extension of the reals with a square root of -1 is in fact the subring of the linear transformations of two-dimensional space generated by the rotations. But it’s true. It’s even easy to see as true; not quite obvious if no one ever told you, but quite clear enough once exposed to the idea. Besides, scaling and rotation are basic concepts deserving of study regardless; they don’t need justification as related to solutions of quadratic equations, though the fact that they have that relation is certainly nice and worth observing.

That is one application of complex numbers, sure. It is not the only one, or even necessarily the most important one.

Yes, as noted in this post, electrical engineers can benefit from the use of complex numbers… as can anyone who wants to talk about scaling and rotation (in a particular direction). Why are complex numbers useful for talking about magnitude and phase? Because magnitude and phase are just other words for scaling and rotation.

Believe it or not, historically, vectors were invented by studying quaternions, which were discovered in an attempt to generalize complex numbers. That having been said, this is another example of history making a hash of everything; vectors are such a simple and fundamental notion, they should have been discovered (and should certainly now be taught) first, with complex numbers best understood by reference to them, as (say it with me) scaling and rotation (in a particular direction).

50

I agree … (sorta).

For the purposes of the specific equation in this specific thread (Euler equation) the explanation of complex numbers as “rotations” on an Argand Diagram is useful.

However, I don’t think it’s a very helpful explanation for understanding complex numbers at a deeper level. Yes, “rotations” helps one master using complex numbers as a tool. It also helps one to be comfortable with seeing i as a clever bookkeeping device for magnitude and angles.

Imaginary numbers were first used as “possible” solutions to cubic equations. It was 200 years later that the insight to map a geometric interpretation emerged. Well, we had thousands of brilliant mathematicians within that 200-year period that didn’t initially see imaginary numbers as “valid” – and – for the ones that did – they themselves didn’t see a geometric equivalence. It seems that mathematical intuition failed them and duplication of that hazy thinking are still present today’s students.

I think mathematicians often don’t see the cognitive hurdles that average people have so explanations that seem very obvious will still be unsatisfactory. I don’t think we should be surprised by this. Imagine if we were to go back in time (1637) and try to convince mathematician Rene Descarte the validity of sqrt(-1). We could attempt to explain the “rotation” on a X-Y plane. I’m not so sure he would be convinced (and keep in mind that Mr Descarte isn’t exactly the dumbest guy on the planet). Maybe if we spent several months with him showing the future scenarios of wave analysis, signal analysis, Fourier transforms, etc would he finally accept the usefulness of i. The problem is that Mr. Descarte was from an era where math was tied back to empirical sensory things (what some folks call “reality”). A few hundred years later we tried to recast mathematics from pure logic: mathematics is a purely a mental game of consistent rules which sometimes has applications in reality.

When some mathematicians say negative numbers and complex numbers are as “reality” as anything else, it seems like there’s a semantics gap. They are using the word “reality” as a synonym for “useful proven bookkeeping device” whereas most non-mathematicians are thinking of “reality” as countable objects and basic Euclidean shapes such as circles and squares.

There’s quite a step change from thinking of math as symbols substituting for reality vs a pure mental game of abstractions and logic. This is an interesting difference in thought process because clearly, there were some ancient mathematicians that understood i without the Argand Diagram. If there was no Complex Plane diagram, what exactly was it that they “understood”? Well, they understood that sqrt(-1) could be included as part of a consistent set of rules without introducing any contradictions. No Argand Diagram was needed.

51

Why should you have to resort to wave analysis, signal analysis, Fourier transforms, etc.?

Why not just say to old Descartes, “Let’s study the arithmetic of rotation, because rotations are useful things to study. Who-ho-ho… look at this! A half turn rotation, along a given direction in a two-dimensional space, is as good as negation. Which means a quarter turn rotation, when carried out twice, becomes negation. Isn’t that an interesting thing? Looks like, in the arithmetic of rotation, a quarter turn is the square root of -1! Fancy that.”

Yes, you can understand the complex numbers as the free ring extension of the reals with a square root of -1, some formal construction of abstract algebra. And this is a fine thing to think about… if such things are what you’re interested in.

But rotations are also a fine thing to think about. And one many people are interested in. And, more to the point, one vast numbers of people already have great intuitions about. So why not teach students about the ring of scalings and rotations before teaching them about the quotient of the polynomial ring R by the ideal generated by x[sup]2[/sup] + 1? Why not present material in the terms by which it is most easily understood? Particularly if it helps obviate some of the ridiculously persistent air of mystery surrounding such material.

All of this is to the side of questions about “reality”. I’m not saying the understanding of complex numbers as scalings and rotations of 2d space is necessary to give them “reality”. I completely agree that mathematics studies whatever abstract games people want to study. I’m just saying that the arithmetic of scalings and rotations of 2d space is very natural an object of study and pedagogically far preferable as a presentation of ideas such as complex numbers, Euler’s theorem, etc., than our current method of teaching these, which apparently continues for many students to leave them shrouded in a ridiculously persistent but wholly unnecessary air of mysteriousness.

52

One other note: Argand diagrams are all well and good. But if all they’re used for is to say “Complex numbers are vectors in a plane, and here are some magic rules for how to multiply such vectors: multiply their magnitudes (as measured in some reference units) and add their angles (as measured from some reference direction)”, then this is still just so much unexplained voodoo. “Why on earth would I want to study such a crazy operation?”. That it does anything nice at all seems like, well, cleverness. But “clever” can be the opposite of “intuitive”.

The proposal is not to think of complex numbers as vectors but rather as operations on vectors. Then multiplication is just composition of operations, performing one after another. And this is perfectly in keeping with how multiplication works in previously familiar cases (positive or semipositive numbers, which are just scalings, with multiplication being performing one scaling after another; signed numbers, which add the possibility of flipping/half-turns, which are also multiplied by sequential execution). Now it’s not some crazy operation somebody randomly chose to look at and which surprisingly turned out to be useful; it’s the most natural thing in the world to study, the idea of doing one thing after another.

Of course, operations on vectors can themselves be treated as the inhabitants of some abstract vector space, since they can be scaled and added pointwise. Complex numbers do indeed comprise a 2-dimensional vector space. But this element of their nature sheds little light on understanding their multiplicative properties. If all the Argand diagram is used to show is the magic vector multiplication rule above (which is how it was always presented to me as a young lad), there is a seriously negligent failure to provide readily available insight.

53

Equivalence classes on the quaternion group? I think what you’re saying here is “Equivalence classes of rotations, where two rotations are equivalent if they send this particular stick here to the same thing.” But those equivalence classes might as well just be represented by the vector they send your preferred stick to. Which leaves you with the vectors of 3d space, not the rotations of 3d space.

I think perhaps I failed to apprehend some confusion in your previous post. The rotations of three-dimensional space do indeed form a group under composition; yes, some of them will leave this particular stick unturned, and some will happen to coincide on that particular stick, but taking rotations as equal just in case they are the same in general, across all sticks, we do indeed have a group here. The point is not to try to think of rotations as represented by particular vectors and then to multiply vectors; to point is just to multiply rotations themselves. [Again, the 2d case is nicer; in two dimensions, non-zero vectors form a torsor over the group of non-zero products of scaling and rotation (that is, you can divide vectors by vectors to get the unique complex number sending the one to the other), a property which is lost in higher dimensions, but which allows one to identify the linear operations of interest with particular vectors they output, once one has selected a reference input vector.]

Anyway, analyzing the rotations of 3d space as a multiplicative group under composition works gangbusters*. The problems arise only when one wants to add rotations (in the pointwise sense), and discovers that the results are no longer rotations and no longer have all the nice properties of rotations, as mentioned in the previous post.

*: Well, I think you’re probably already quite familiar with this, and thus would worry that my explaining it to you is somehow patronizing or at least unhelpful. Except things you’ve said in your last two posts seem to indicate some miscommunication of what I’m saying about viewing complex arithmetic in terms of rotations, so I thought I’d better clarify.

54

I can’t believe no one else has posted this, which pretty much sums up my thoughts on the subject.

For most people (excludes mathematicians, physicists, and electrical engineers), the constants of pi, e, and i are, at best, barely related. To find out at the end of the movie that they somehow all tie together in such a simple formula is kind of an evil plot twist.

55

I certainly failed to apprehend some of the confusion, myself. When I say that I was trying and failing to generalize complex numbers, all of that trying and failing was done inside my skull. If I had tried it on a whiteboard, I probably would have been better able to express my thoughts, including the confusion in them. In particular, I probably would not have made that “equivalence classes on quaternions” statement.

56

Yes, but that’s because most people have been taught about those last two constants very poorly. It’s not so much an evil plot twist as having missed vital opening scenes because of a scratch on the DVD.

When “i” is recognized as just a fancy name for “quarter of a full turn”, the relationship between i and π has zero mystery; who would be surprised at a relationship between a quarter turn and π? They’re both basic aspects of geometry, and indeed, even more specifically, the geometry of rotation. [The observation that this i happens to square to -1 is, in fact, not even all that important to its appearance in Euler’s theorem; the theorem itself is really just directly about 90 degree rotation. If it were commonly worded as “e[sup]90 degree rotation * π[/sup] = -1”, there would perhaps be less confusion as to its nature]

As for e, I doubt 1 in 100 laypeople were ever given or retained a decent description of what e is about, but its entire purpose in life is to describe the result of continuously repeating some action (i.e., “exponential growth”). More specifically, e’s purpose is to describe the relationship between various basic ratios in exponential growth. And as π is a basic ratio in the context of the continuously repetitive action of rotation, it shouldn’t be too hard to imagine that there would be a natural relationship between e and π as well.

Of course, most people have not been taught these concepts in ways making these definitional connections clear. But they should be! The idea that these are disparate concepts far from having any obvious connections to each other is hogwash; its perpetuation is willful ignorance. (Alas, mathematicians are the ones most complicit in playing this sort of thing up, it apparently being more exciting to fantasize about having stolen brief glimpses of a forbidden realm of transcendent Platonic beauties than to actually understand things.)

57

No, those seem to be set on “stun.”

58

First off, I need to correct a word I wrote earlier because it changes the meaning significantly and by correcting it, it continues my line of thinking from that post to this response here.

I mistakenly wrote: “I think mathematicians often don’t see the cognitive hurdles that average people have so explanations that seem very obvious will still be unsatisfactory.”

I meant to write “unsatisfying” instead of “unsatisfactory” which results in “explanations that seem very obvious will still be unsatisfying.”

Since the time-travel scenario I brought up is a thought experiment, this is something we can’t definitively resolve but nevertheless, it’s fun to continue trying to get inside the skull of Descarte. In my opinion, it has relevance to the difficulty today’s students have with abstract concepts of math (such as sqrt -1).

I believe you and I are looking at this issue differently. You see it as a “math” problem and if you present some kind of coherent picture “rotations equal logical chain of operations, etc” then that alone should convince Descarte. I don’t see it as a math problem – it’s really more of a psychology problem. I could see Descarte understanding and following each component of your logic – but accepting it as some kind of satisfying “truth” is a different thing altogether. It was Descarte himself who coined the term “imaginary” as a derogatory label. Can we get inside his head and figure out why he would call it that? I think we can safely assume that he understood very well (at least the mechanics of it) how other mathematicians were using sqrt(-1) – I don’t think that was his stumbling block at all. Therefore, presenting “90 degree rotations” would leave him saying, “ok, I see the 90 degree argument, and at the surface level, I even agree with it, but so what?”

Because if you pile on more and more practical scenarios, you may be able to overcome the psychological hump that rejects the idea. Eventually one of those useful scenarios could finally trigger the emotional acceptance of the idea.

Don’t many of the major math ideas get accepted this way? The ideas of:

irrational numbers
0 (zero)
negative numbers
imaginary numbers
non-Euclidian geometry

Each of those ideas could have been presented with coherent logic at the time of their introduction but even a consistent argument is not enough. When that happened, the ancients would criticize the results as not “legitimate” or “imaginary.” These derogatory labels are not attacking the mechanical logic, it’s dismissing something else entirely… maybe the “truth” or “reality” of the manipulations … it’s something emotional or psychological. When I look at history, it seems to take dozens of practical examples possibly over hundreds of years (and maybe also for the old mathematicians to die off) for the idea to really gain acceptance.

Maybe an analog situation would be Albert Einstein’s discomfort with quantum mechanics. It seems he understood the cold logic behind the ideas of other physicists such as Bohr and he even conceded that they were getting predictive answers from their calculations. However, something about it still unsettled him.

The change in attitudes resulting from the “math as pure logic” works of Russell, Whitehead and the success of quantum mechanics has given us a new framework to accept new math ideas. Even though Godel shut some of the doors regarding math as “pure abstraction”, it still allows modern mathematicians not to have the same high emotional barrier to new ideas that Descarte had. Today, we are now able to say “mathematics is whatever abstract games with rules and patterns that people want to study.” It’s a bold statement that the ancient Greeks wouldn’t have said.

All this comes back to my experience with today’s math students (especially the ones that will not go on to become mathematicians.) The vast majority of people still (at least subconsciously) want to cling to some kind of physical tangible reality of the math concepts they are trying to learn. Many math professors on the other hand have already accepted math as abstract logic at face value but mistakenly assume that the students they teach have made a similar transition.

59

One other comment…

My comments on the matter deal more with the teacher’s side and how they perceive (and get frustrated by) students not “getting it.” My example of Descarte’s rejection of i is an attempt to get mathematicians to be sympathetic to non-mathematician’s brains not being sufficiently primed to absorb increasingly abstract math ideas.

I agree that we shouldn’t be shackled by history. Just because ancient folks had difficulties with irrational and complex numbers, it doesn’t mean we have to suffer the same way. If there are new or better pedagogical tools to teach ideas such as complex numbers, we should certainly use them. My notes take into account that today’s state of math education is what it is (I was being descriptive) and several cognitive difficulties arise from that. You make a compelling case for changing the curriculum (you’re being prescriptive) and my comments wouldn’t apply to that alternative scenario.

60

I think the biggest problem Descartes would have is not in accepting i as an operation, but in accepting operations as numbers. I know that was a bit of a stumbling block for me, too, until I realized that all numbers are operations to begin with: I can’t have five, but I can have five apples, or five minutes, or five pens, or whatever. Five is an operation which can act on (among other things) apples, minutes, or pens.