I suspect I have probably failed to actually meet the challenge I set myself in this thread (to make clear to the OP and everyone else that the equation “(e[sup]i[/sup])[sup]π[/sup] = -1” actually straight-up just plain means “If you were to keep staring at someone and walking to your left, you will trace out a circle clockwise around them, so that in the time it takes for you to walk the length of half that circle, you’ll end up on the other side of them, just as far as you started”, only in symbols that you may not already recognize how to interpret to mean this). But I would like to hear from the OP or anyone else who was confused by my explanations (including the linked posts) any comments about particularly confusing parts or unaddressed issues, suggestions for improvement, etc.
I would really like to help bring the world to a point where mathematicians stop presenting the ideas around Euler’s theorem in such extraordinarily silly, obfuscating ways as they almost always do, but to succeed in this plan, it would be helpful to have feedback on what is and is not working as I try to explain the arithmetic of rotation.
Well, slightly more slavishly following the symbols, the equation “(e^i)^π = -1” means “If you were to keep staring at your house and walking to your left, with the speed at which you walk at any moment being equal to your current distance from home per unit of time, and to do this for a number of units of time equal to the ratio between half a circle’s circumference and its radius, then you will trace out at constant speed half a circle clockwise around your house, ending up on the other side of it, just as far as you started.” As the color-coding indicates, the “i” is interpreted as the 90 degree clockwise rotation between the direction from your house to you and the velocity at which you walk, the “e^x” is interpreted as “If you were to keep staring at your house and walking [with the velocity determined by x as applied to the displacement from your house to you]”, the “^π” says to do this long enough to cover the distance of half a circle, and the final “-1” is the result that you will indeed end up halfway around that circle. The meaning of the colored equation is precisely that of the colored sentence; if you understand why the sentence is true, then you understand why the equation is true, since they’re the same thing. The only minor thing beyond that that there is to grasp is why we should use the particular symbols in that equation to describe that sentence.
(I hope this gloss and key is helpful, but I fear it isn’t.)
Speaking for myself I have to say that I think you are doing a really fantastic job of explaining and demystifying the equation. Thanks.
For me, the only thing that continues to bother me is the “how do you raise e to the sqrt(-1) power – what does than mean?” I have to keep telling myself that it’s just a definition. It makes more sense for me to think of exp(ix) as cos(x) + i*sin(x) – that way the rotational aspect is obvious and manifest – the only problem is that that is Euler’s formula. I understand the derivation using a DE, but as I’ve said earlier I don’t buy it when applied in the complex plane, again because of the same circularity issues. So ultimately I’m unsatisfied so far with your explanations. I already knew what Euler’s formula was saying in terms of rotations, but for me the problem is the step of interpreting exponentiation of complex numbers as rotations. Just defining it that way is fine and dandy, but using Euler’s formula in order to construct the definition is tautological.
Suppose you have some process where your rate of change is proportional to your value at any moment; in other words, there is some constant such that your current rate of change is always equal to that constant times your current value. We call this exponential growth, and that constant is called the “force of interest” (or “growth constant” or “logarithmic return” or other such names).
Exponential growth has the (in fact, equivalent) property that over any unit of time, you multiply by the same amount. So suppose you have some quantity which is growing exponentially, and over any 1 unit of time it quintuples. Then over 2 units of time, it quintples twice; it multiplies by five and then multiplies by five again, with the total effect of multiplying by 25. And over 3 units of time, it quintuples three times; it multiplies by 5[sup]3[/sup]. In general, over t units of time, it multiplies by 5[sup]t[/sup].
Well, nothing special about 5 there. Whenever you have some exponentially growing quantity, the amount it multiplies by over t units of time is the t-th power of the amount it multiplies by over 1 unit of time.
Ok, so what? This still doesn’t help us with imaginary exponents. Well, let’s re-express this a bit:
Suppose you have some exponentially growing quantity, and you want it to grow faster. What do you do? You increase its force of interest (which determines its rate of change). If you double its force of interest, it does everything twice as fast: it’ll get done in 1 unit of time what it used to get done in 2 units of time. If you multiply its force of interest by 8, it’ll get done in 1 unit of time what it used to get done in 8 units of time. In general, if you multiply its force of interest by p, it’ll get done in 1 unit of time what it used to get done in p units of time.
So what? So if it used to multiply by b over 1 unit of time, and you multiply its force of interest by p, then it’ll now multiply by b[sup]p[/sup] over 1 unit of time.
We came at this conclusion by thinking about traditional exponents, for which we can express exponentiation as iterated multiplication. But now we’ll take it as a definition to help us interpret more general exponents. To calculate b[sup]p[/sup], first take exponential growth which multiplies by b over 1 unit of time; this has some force of interest (which we call ln(b)/unit of time). Multiply that force of interest by p (so that we’re now talking about exponential growth with a force of interest of p * ln(b)/unit of time), and then the resulting exponential growth is the kind that multiplies by b[sup]p[/sup] each unit of time, thus defining b[sup]p[/sup].
So to calculate e[sup]p[/sup], we need to think about the kind of exponential growth which multiplies by e over every unit of time, figure out its force of interest, multiply that force of interest by p, and then look at how much that new kind of exponential growth multiplies by over every unit of time.
Well, e is actually, by definition, the amount you multiply by over a unit of time with a force of interest of 1/unit of time. So e[sup]p[/sup] is the amount you multiply by over a unit of time with a force of interest of p/unit of time; that is, e[sup]p[/sup] is how much you multiply by over a unit of time if your rate of change is always p * your current value/unit of time.
So to calculate (in the context of 2d vectors) the value of e[sup]90 degree rotation[/sup], we just have to figure out how much you multiply by over a unit of time if your rate of change is always 90 degree rotation * your current value/unit of time. That is, if the direction in which a vector is changing is always 90 degrees rotated from its current direction, and the speed at which that vector is changing is always equal to the size of that vector/unit of time, then what happens? Well, what happens is that the vector rotates at a radian per unit of time. Thus, e[sup]90 degree rotation[/sup] = rotation by one radian.
Does the above account of exponentiation help clarify anything? Note that I didn’t invoke Euler’s formula at any point to define exponentiation; I defined exponentiation first, by reasoning based on exponential growth and iterated multiplication (the traditional concept of exponentiation which was being generalized), and then used this definition of exponentiation to derive Euler’s formula.
I do have a follow-up question, but first: is an expression s[sup]v[/sup] (where s is a scalar and v a vector) a well-defined operation? If so, perhaps you could just point me to an article.
No, but s[sup]L[/sup], where s is a positive scalar and L is a linear operator on vectors, is well-defined, by the very definition, I gave in that post. (See below)
You presumably ask about a vector as the exponent because you are thinking about complex numbers as vectors. Don’t think about complex numbers as vectors; think about complex numbers as certain operations which transform 2d vectors into other 2d vectors. (Just as, even for more familiar arithmetic, you shouldn’t think of, say, 5 as a length; there’s no particular stick in the world with length 5. 5 is an operation which transforms lengths into other lengths (or vectors into other vectors, or other such things, depending on the context); 5 yards is the length you get by applying 5 to the length of a yardstick, 5 feet is the length you get by applying 5 to the length of a ruler, etc.)
s[sup]L[/sup] means “Consider exponential growth which multiplies by s over every unit of time; this has some force of interest, which we call ln(s)/unit of time. If we multiply that force of interest by L, we describe a new kind of exponential growth; exponential growth with force of interest ln(s) * L/unit of time. This new exponential growth describes multiplying by s[sup]L[/sup]/unit of time. In other words, s[sup]L[/sup] is the amount a quantity multiplies by over 1 unit of time if that quantity’s velocity is always ln(s) * L * its current value/unit of time”. (There’s not actually any need here for the base to be a scalar; we can replace exponential scaling with some other kind of exponential growth, and this definition still works perfectly fine.)
So, for example, don’t think of i as the vector “Straight-up with length 1”. What direction counts as up? What length counts as 1?
No, think of i instead as the operation “Rotate my input vector by 90 degrees”.
If I stand somewhere, and I ask you “Excuse me, can you point me in the direction of the vector 3 + 4i?”, no one can help you. It depends on your coordinate system. But if I stand somewhere, and I ask you “Excuse me, can you tell me what 3 + 4i would do to a vector, such as the displacement from me to that tree over there?”, then people can help you [well, once you fix a direction of rotation]. Then people can tell you, “Oh, sure. Let’s see, rotating clockwise parallel to the ground, that’d change the displacement from you to that tree into the displacement from you to that barn over there”, or whatever.
Now, it’s true, if you have a vector which you are considering as pointing to the right with length 1, and you take rotations to be counterclockwise, then applying i to that vector will end up with a vector pointing straight-up with length 1. Sure. And it’s also true that in a sense, complex numbers form an abstract vector space, and on the most natural coordinate system we could assign to that abstract vector space, i would have the coordinates <0, 1>. Yes. But that’s not helpful to understanding multiplication and exponentiation of complex numbers. The most helpful perspective to understanding multiplication and exponentiation of complex numbers is to think about complex numbers as linear operators on vectors, rather than merely as vectors themselves.
Yeah, I think so too, and I think I often underestimate the difficulty of this kind of thinking for those who aren’t yet accustomed to it. By which I mean, not the difficulty of learning to call operations by the word “numbers” (it’s just a label), but the difficulty of learning to appreciate that one can perform calculations on operations, manipulating and combining them in various ways just as if they were entities in their own right (because, of course, they are!).
For example, in certain contexts, I like to also explain vectors as translation operations on points, with addition of vectors as composition of translations, so that complex numbers become operations on operations on points, with multiplication as composition of operations on operations on points and addition as pointwise composition at the underlying level of mere operations on points. For me personally, I find this helps makes many things much clearer, but I suspect that if I were to carry on in this way for long among non-mathematicians, I would be quickly lynched.
I’m still here, and still struggling, but not ready to blame it on anyone else but myself now. I’m waiting to get a chance to sit down with a pencil and paper and re-read the thread. Probably Monday or Wednesday night.
I don’t mean to be a brat – I’m sincere in saying you are really doing a great service here. But when you say “think of i as an operation on a vector”, I think "in the expression e[sup]i[/sup] what is the vector that i is operating on? e? e is a scalar. Or are you interpreting the scalar e as e multiplied by the x-hat coordinate in the complex plane and so treating it as a vector, even though you just told us not to think of “i” as a vector but an operator? Do you see how this is confusing and contradictory? I suggest really stepping back and trying to approach this again, but without the attitude that it is all so obvious. So you may have to slow down a notch; I have a solid background in math of this level, but find some of what you are saying sloppy, and not as obvious as you imply by your not connecting all of the dots.
BTW, my follow up question/suggestion was going to be, in order to make this all a lot clearer, for you to give an example like the above, but instead of a 90 degree rotation, use a 45 degree rotation, and explain all of the steps exactly as before. In other words, explaining esup/sqrt(2)[/sup], in detail using your above logic.
That’s perfectly fine. As I said, it is sometimes difficult for me to appreciate what difficulties others will have in understanding some of the things I say.
You can speak about operations on vectors abstractly, without having to say which vector they operate on. I can speak about “90 degree rotation”, without having to mention a particular vector to rotate 90 degrees, just as I can speak about the function f(x) = x[sup]2[/sup] without having to specify a value for x.
In the expression “e[sup]i[/sup]”, both “i”, and “e[sup]i[/sup]” denote operations on vectors. “e” is also an operation on vectors. There aren’t actually any vectors around in that expression, just operations on vectors.
“e” is the operation that takes a vector and subjects it to exponential growth for 1 unit of time at a force of interest of 1/unit of time. So e is an operation which turns vectors into vectors, with the familiar property that e(v) = 2.71828… * v. [When I write e(v), I mean “e applied to the vector v”, as in application of a function to an argument; e is a particular kind of function from vectors to vectors]
“i” is the operation that takes a vector and rotates it 90 degrees. So i is also an operation which turns vectors into vectors, defined by i(v) = the result of rotating v by 90 degrees.
“e[sup]i[/sup]” is the operation that takes a vector and subjects it to exponential growth for 1 unit of time at a force of interest of i * 1/unit of time, which means, at any moment, the rate of change of the vector is i applied to the vector, per unit of time. And this has the effect that e[sup]i/sup = the result of rotating v by 1 radian.
e, i, and e[sup]i[/sup] are all functions from vectors to vectors. If you like, we can rewrite our identity as (e[sup]i[/sup])(v) = the result of rotating the vector v by radian, as above, writing out explicitly an input vector v for these functions.
I think this is an excellent idea. (I’ll make this very long and hopefully detailed, but I hope this will not give the misleading impression that this is a very tricky problem to tackle; once you are familiar with this kind of thinking, one would not give so long a description as I am about to.)
What is e[sup]45 degree rotation[/sup]? Well:
e is the operation on vectors which subjects them to exponential growth at a force of interest of 1/unit of time for 1 unit of time. (Reminder: this means that at any moment, the rate of change of the vector is 1 * the current value of the vector, per unit of time).
45 degree rotation is the operation on vectors which, well, rotates them 45 degrees.
e[sup]45 degree rotation[/sup] is the operation on vectors which subjects them to exponential growth for 1 unit of time at a force of interest which is 45 degree rotation * the force of interest of e. That is, e[sup]45 degree rotation[/sup] is the operation on vectors which subjects them to exponential growth for 1 unit of time with a force of interest of 45 degree rotation/unit of time. That is, e[sup]45 degree rotation[/sup] is the operation on vectors given by “Start with whatever input vector you like. At any given moment, the velocity at which it changes will be its current value, rotated 45 degrees, per unit of time. Evolve for one unit of time under this process of change; what you end up with is the output vector”.
(In the rest of this post, I’ll stop writing the "per unit of time"s and so forth where they strictly belong, since I fear they probably prove more distracting than helpful to most readers)
So what happens if you have a stick with one fixed end, and the stick can grow and rotate, and specifically at any moment the velocity of the moving end of the stick is equal in size to the current stick length, but points in the direction 45 degrees rotated from the direction the stick is pointing (the direction from the fixed end to the moving end)? Well, first, using the direction information, we see that the endpoint of the stick traces out a spiral, such that the tangent to the spiral at any point is 45 degrees rotated from the direction the stick was pointing in when it hit that part of the spiral. And we can also calculate the speed at which the stick traces out this spiral, and then conclude that what happens in one unit of time is the stick scales up in length by some amount, and rotates by some amount, and this combination of scaling and rotation is what will be e[sup]45 degree rotation[/sup].
As it happens, the easiest way to calculate the speed at which this stick is moving along the spiral is to separate the two things which happen to it (it grows and it rotates), and calculate the speed at which each of those two things happens. It grows because its endpoint is always moving a bit outwards; that its endpoint is also always moving a bit to the side is irrelevant to its growth. It rotates because its endpoint is always moving a bit to the side; that its endpoint is also always moving a bit outwards is irrelevant to its rotation.
As you noted, we can specifically break down 45 degree rotation into (1 + i)/sqrt(2). That is, to rotate a vector by 45 degrees, we can make two copies of it: one exactly the same (and thus parallel to our original vector), and one rotated by 90 degrees (and thus perpendicular to our original vector). If we then add these two together, and scale the result down by a factor of sqrt(2), we end up with the result of rotating our original vector by 45 degrees. That is, (rotate v by 45 degrees) = (the result of keeping v the same + the result of rotating v by 90 degrees) multiplied in size by 1/sqrt(2), which we can express by saying 45 degree rotation = (1 + i)/sqrt(2). The parallel (i.e., real) component of 45 degree rotation is 1/sqrt(2), and the perpendicular (i.e., imaginary) component of 45 degree rotation is i/sqrt(2).
So the endpoint of our stick is always moving outwards at a speed of 1/sqrt(2) times the current sticklength (because of the parallel component of 45 degree rotation), and always moving sideways at a speed of 1/sqrt(2) times the current sticklength (because of the perpendicular component of 45 degree rotation). The first of these tells us how fast the stick grows; the second tells us how fast the stick rotates.
Because the endpoint of the stick is always moving sideways at a speed of 1/sqrt(2) times the current sticklength, the stick is essentially rotating at a speed of 1/sqrt(2) radians [per unit of time].
Because the endpoint of the stick is always moving outwards at a speed of 1/sqrt(2) times the current sticklength, the stick’s length is exponentially increasing with a force of interest of 1/sqrt(2); that is, the stick’s length multiplies by e[sup]1/sqrt(2)[/sup] [over every unit of time].
Thus, in conclusion: e[sup]45 degree rotation[/sup] is the operation on vectors which causes them to simultaneously grow and rotate (tracing out a spiral along the way), where the vector multiplies in size by e[sup]1/sqrt(2)[/sup] and rotates by 1/sqrt(2) radians.
That is, e[sup]45 degree rotation[/sup] = e[sup]1/sqrt(2)[/sup] * (rotation by 1/sqrt(2) radians). This is algebraically reflected as e[sup]45 degree rotation[/sup] = e[sup](1 + i)/sqrt(2)[/sup] = e[sup]1/sqrt(2)[/sup] r[sup]1/sqrt(2)[/sup], where r = e[sup]i[/sup] = rotation by one radian.
I’ve written up a bullet point version of the argument here, in case anyone is having trouble seeing the big picture, although I suspect more people are having trouble understanding the details instead.