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

Factual Questions
21

We say that a quantity grows exponentially if it multiplies at a constant rate, in the sense that over any two intervals of the same length, it multiplies by the same amount. For example, a quantity which multiplies by 2[sup]t[/sup] over any t seconds is growing exponentially: over any one second interval, it doubles; over any three second interval, it octuples; over any half second interval, it multiplies by sqrt(2), etc. So this sort of getting larger and larger at faster and faster speeds is a prime example of exponential growth.

But, also, a quantity which rotates at a constant speed is also growing exponentially; in any two intervals of the same length, it rotates by the same amount. For example, a quantity which rotates by 12 degrees a second: over any one second interval, it multiplies by a 12 degree rotation; over any three second interval, it multiplies by a 36 degree rotation; over any half second interval, it multiplies by a 6 degree rotation, etc. So rotation is also a prime example of exponential growth.

Now, if a quantity is growing exponentially in this sense, then its velocity (rate of change) at any moment is proportional to its current value. In fact, this gives a precisely equivalent way of defining exponential growth: a quantity is growing exponentially just in case there is some constant ratio between its velocity and its value. We call this the “growth constant” of that quantity. For example, if I were driving west from New York in such a way as that the further I got, the faster I drove, so that at any moment my speed was always 7 times my distance from New York per hour, then the distance between me and New York would be growing exponentially, with a growth constant of 7/hour.

Given two different exponentially growing quantities X and Y, we might want to compare how fast they grow. One way to do this is to say “If the amount X multiplies by over any interval is the same as the amount Y multiplies by over an interval p times as long, then X is growing p times as fast as Y”. So, for example, if X octuples every second, while Y doubles every second, then X is growing three times as fast as Y, since the amount X multiplies by over any second is the same as the amount Y multiplies by over any three seconds. Naturally, to grow p times as fast as a quantity which multiplies by b every second is to multiply by b[sup]p[/sup] every second. This is how we’re going to define exponentiation.

Switching to our other way of looking at exponential growth, “The amount X multiplies by over any interval of time is the same as the amount Y multiplies by in an interval p times as long” is equivalent to “The growth constant of X is p times the growth constant of Y”. So, if the growth constant of X is p times the growth constant of Y, and Y multiplies by b over any second, then X multiplies by b[sup]p[/sup] over any second. Again, this can serve as our definition of exponentiation. b[sup]p[/sup] is the amount you multiply by every second, if you grow exponentially with a growth constant p times as large as that of a quantity which multiplies by b every second.

Finally, what is e? By e we mean the amount a quantity multiplies by over any second, if it grows exponentially with a growth constant of 1 per second. Which means, for any exponentially growing quantity with growth constant K/sec, it grows K times as fast as something multiplying by e every second; that is, having growth constant K/second is the same as multiplying by e[sup]K[/sup] every second.

In particular, consider a vector V rotating at a radian per second. This is exponential growth, and as argued before, the growth constant will be 90 degrees rotation/second; that is, at any moment, the velocity of the vector will be equal to its current value per second, rotated 90 degrees. Accordingly, the amount this vector multiplies by over any second will be, using our definitions, e[sup]90 degrees rotation[/sup]. Which is to say, e[sup]90 degrees rotation[/sup] = 1 radian rotation. Or, using the notation “i” for 90 degree rotation, we have that e[sup]i[/sup] = rotation by one radian, as before. This is Euler’s theorem.

Basically, the reason e is so simply related to rotation is because e is all about exponential growth, and rotation is a very simple kind of exponential growth. When we say e[sup]p[/sup], we mean the amount a quantity multiplies by over a second, if its rate of growth is always p times its current value per second. This makes sense even when p is not just a scaling by some factor, but has some rotational content as well, which is to say, it makes sense not just for positive p but for arbitrary complex p. There’s no need to bring Taylor series into it to define this, and indeed, it’s easier to understand and motivate on its own terms without them.

22

Which one? CPCC is a good school and has lots of PhD professors.

That’s not true. A full-time college teacher is widely called a professor in the United States. On top of that, I’d estimate that about half of the instructors I had at community college had PhDs, and far more than half in the maths and natural sciences. The community college I went to is part of the largest community college district in the country, and in my opinion one of the best. A great many incredible people who teach there would be offended by your assertion that they aren’t professors. I should go back and tell my old mentor who got his PhD from Stanford that he isn’t a professor. Or maybe the one who left to get his PhD from Harvard and has since returned.

23

(In fact, it makes sense even much more generally than the complex numbers; we can define e[sup]p[/sup] in the same way for any linear transformation p from some vector space to itself, as the amount a vector multiplies by over a second if it grows exponentially in such a way as that its velocity is always p times its current value per second; in particular, when one chooses to represent such linear transformations in coordinates as square matrices, then this gives what’s known as the matrix exponential. Complex numbers are just the particular special case where one is concerned with scalings and rotations of two-dimensional space.)

24

I just checked out the CPCC math page. Zero people have the job title professor. (http://appserver.cpcc.edu/edirectory/departments.asp?dept=404&subdept=121#listing)

You are using the term professor incorrectly. It’s a job title and not a way to refer to any PhD who teaches a class. Even at universities every PhD is not a professor. Some carry the title lecturer.

25

“F#ckin’ magnets, how do they work?”

This is all Euler’s ever really meant to me: http://i54.tinypic.com/2zp09l0.png

26

Take this to another thread. I just looked at the directory of the school I went to, and you’re wrong, but I don’t want to talk about it here. I’m still trying (and failing) to really grasp Euler’s Equation.

27

Well, I guess I’ll take your word for it. Consider my ignorance fought.

28

Try this explantion from the BetterExplained blog.

29

Very lucid, Indistinguishable, thanks. But I find the last bit that I’ve quoted hard to follow (particularly as it relates to my own question about exponentiation) – could you rephrase without using SI units :wink: this is math we are talking about after all.

The reason I referred to defining exponentiation without series expansion is because of my own understanding of the definition of exponentiation:
x[sup]1[/sup] := x
x[sup]2[/sup] := xx
x[sup]3[/sup] := xx*x
and so on.
Similarly to how the gamma function is a generalization of the factorial, exponentiation is defined via x[sup]n-1[/sup] = x[sup]n[/sup] / x, and so it can be extended to include x[sup]0[/sup] = 1, negative and non-integer exponents. With a series expansion it is easy to unambiguously extend exponentiation into the complex plane, but otherwise we must supply a definition of x[sup]i[/sup] (again, similarly to the gamma function). And what definition should be supplied? Therefore it is still ultimately not clear to me how Euler’s equation is trivial, at least to a layman who is trying to invoke the understanding of exponentiation he was taught in school.

30

There is a big difference between understanding Euler’s Equation and appreciating the beauty of it. I was a long time math teacher of gifted students. I think the beauty bit goes right back to the way you were taught math at school. If it was all learning methods and getting the right answer, then that is what I call training, not teaching.

To love math, and appreciate the beauty, you need to feel math, play with it, explore and let the numbers talk to you. You need to discover primes and patterns and infinity, for their beauty, not for any purpose. You need to get that from someone who is passionate about math for its own sake, not for ‘why’. That may sound nuts, but it is the reason that I can get high on math when many other subjects leave me cold. The most important word in teaching kids math is ‘play’. The loss of pure math, and the introduction of making every topic for a practical purpose, destroyed the beauty of math for those capable of feeling it.

If you want to follow up on this, may I recommend the classic, A Mathematician’s Apology by G. H. Hardy, the edition with the introduction by C. P. Snow. It’s an easy, short read, not much math, but a lot of feeling.

The other book which I think gets it across really well is the novel, The Curious Incident of the Dog in the Night-Time by Mark Haddon. A strange suggestion, unless you have read it - then my logic is clear.

So I can’t accept your challenge to explain the beauty of Euler’s equation because the answer isn’t anything to do with ‘why’. There’s no why to music. I am sure you already understand the actual math. I admire you searching out why people find this equation so beautiful. I do hope you actually discover that one day.

Once you have the feeling for math and have experienced the joy of playing with it, I can assure you that the answer to ‘why’ is there as well.

31

I also wanted to note that using a Taylor expansion (or using a DE to derive Euler’s equation) gets into circularity issues – the very foundations of the complex analysis that is required assumes Euler’s equation in re-writing x + iy. In either case it is ultimately necessary to supply a definition of x[sup]i[/sup] (right?)

32

Sure. I used the word “second” in that explanation, but what I really meant was “unit of input”, for whatever notion of unit of input you want. Anyway, though, let me rephrase what I said to make this clearer:

Suppose you have some function f from, let’s say, the semipositive reals [0, infinity) to the complex numbers [or, really, any “smooth” ring] with the property that f turns addition into multiplication; that is, f(x + y + …) = f(x) * f(y) * … That is, f is continuous, f(0) = 1, and f(x + y) = f(x) * f(y). Basically, f is meant to represent some kind of exponential growth; f(t) is the amount a quantity grows by after t many units of time. For example, f(t) may equal 2[sup]t[/sup].

In fact, any function with these properties can be thought of as exponentiation with some base. Thus, we’ll call such functions “exponential”. Note, though, that f(1) doesn’t uniquely determine the entirety of the function; just its value at integer arguments. For example, just from f(1) = 7, we can’t tell if f(0.5) is the positive or the negative square root of 7. In fact, there will be many different exponential functions whose value at 1 is 7. That’s alright. We’ll still think of these all as exponentiation with some base, but the base carries slightly more information than just f(1). [See this discussion].

Now, as I said before, exponential growth can alternatively, equivalently, be characterized in terms of growth constants; that is, instead of saying that f turns addition into multiplication, we can just say that the logarithm of f turns addition into addition, which is to say that the logarithm of f is linear, which is to say, ln(f(t)) = kt for some constant k, which is to say, f(0) = 1 and df(t)/dt = k * f(t) for some constant k. We’ll call this k the “growth constant” of f.

If f(t) = b[sup]t[/sup], and g(t) = (b[sup]p[/sup])[sup]t[/sup], then g basically represents exponential growth at a rate p times as fast as f. This will also correspond to the growth constant of g being p times the growth constant of f. Thus, we can define exponentiation this way: if f is an exponential function with growth constant k, then the exponential function corresponding to the p-th power of the base of f is the one with growth constant p * k.

In other words, if we know how to calculate b[sup]t[/sup] for semipositive t, then we know the exponential function with base b. It will have some growth constant, the derivative of b[sup]t[/sup] with respect to t at t = 0, which we call ln(b). Then to determine b[sup]p[/sup], we want the base of the exponential function with growth constant p * ln(b). Thus, b[sup]p[/sup] = g(1), where g(0) = 1 and the derivative of g is p * ln(b) times itself.

In particular, e is meant to be the base of the exponential function with growth constant 1. That’s its raison d’etre in the world; it’s the amount a function multiplies by over an interval of length 1, if that function is its own derivative. So e[sup]p[/sup] means g(1), where g is the function satisfying the property that g(0) = 1 and the derivative of g is p * g. That is, e[sup]p[/sup] is the value at 1 of the exponential function with growth constant p (i.e., whose derivative is p times itself).

In particular, e[sup]i[/sup] is g(1), where g is the function satisfying the property that g(0) = 1 and the derivative of g is i * g. In other words, it’s the amount a quantity multiplies by if the derivative of that quantity is always i * that quantity. I.e., remembering that i means 90 degree rotation, it’s the cumulative effect of one unit of evolution under the rule “Keep your derivative equal to your value rotated 90 degrees”.

Which, as we saw before, describes rotation by one radian, and thus e[sup]i[/sup] = r.

33

My last two posts give a general definition of b[sup]p[/sup] for any p, which includes i as a special case. Reiterating, b[sup]p[/sup] = g(1), where g is the function satisfying g(0) = 1 and such that the derivative of g is p * ln(b) * g.

(And remember, ln(b) is in turn defined as the ratio between the derivative of b[sup]t[/sup] and b[sup]t[/sup] itself. This may seem circular, but it’s not: in order to determine ln(b), we only need to know how to raise b to real powers. Then, we can use this to determine how to raise b to arbitrary powers, as above.)

[And finally, e is defined by ln(e) = 1. Saying e[sup]x[/sup] = y is just another way of saying that ln(y) = x.]

34

Using the word “itself” there was poorly thought out… That last bit should read “and the derivative of g is p * ln(b) * g”.

35

That article is very good. I mean, I have my minor nits with some of the presentational choices, but overall, I am happy (and surprised) to see somebody else making many of the same points I’ve been trying to make.

36

I’ve been subscribed to that blog for a couple of years. He doesn’t post as frequently as I’d like but hey, that’s what RSS is for. When my niece was learning the Pythagorean theorem, I used this article to help her really “grok” the concept or at least appreciate some of its applications rather than learning by rote.

37

Thanks for your patient explanations Indistinguishable. I will try to synthesize what you’ve said and answer the OP’s question (for myself at least) in my own language:

Exponentiation’s defining characteristic is this property:
x[sup]n[/sup]*x[sup]m[/sup] = x[sup]n+m[/sup]
In abstract terms: f(a)*f(b) = f(a+b)

Imaginary numbers act multiplicatively as operators that rotate by an angle in the complex plane (for simplicity I only consider imaginary numbers of unit length). Successive rotations by angle1 and by angle2 are equivalent to one rotation by (angle1+angle2). Therefore the multiplication of complex numbers satisfies the defining characteristic of exponentiation (above), and we therefore expect to be able to re-write them somehow as exponents of the angles they rotate by.

Now, all this is only true for imaginary numbers of unit length, so really we are talking about numbers of the form:
cos(angle) + i*sin(angle)
(writing it like this is just a way of conveniently being sure the length of the vector in the complex plane is equal to 1)

Using calculus, it is easy to show that:
cos(angle) + isin(angle) = e[sup]iangle[/sup]
Indistinguishable also has some really compelling arguments in his posts for why this makes sense intuitively. In any case it works out such that ‘e’ is the special number such that e[sup]iangle[/sup] rotates a vector exactly by ‘angle’ and without stretching or squeezing it – representing rotation around a perfect circle. It should therefore be no surprise that pi (which is related to circles) and e are related in a simple equation like e[sup]ipi[/sup] = -1 (which is really just describing a rotation by 180 degrees).

I am only skeptical about the formal validity of the step showing that:
cos(angle) + isin(angle) = e[sup]iangle[/sup]

Ignoring the fact that the proof of equality is only valid up to a constant (and therefore from it one can only derive that e[sup]i*pi[/sup] = -1 + C), it seems clearly tautological to me, since, at least in the complex analysis classes I vaguely remember from college, Euler’s formula is assumed in deriving the validity of the rules for differentiating on complex functions. And as I mentioned earlier, there is also the problem of getting one’s mind around what x[sup]i[/sup] means in terms of “how do you multiply x by itself an imaginary number of times”? The same problem crops up for exponents of negative numbers and non-integers. It all ultimately comes down to definitions: 1) you must define exponentiation to mean something non-intuitive if the symbol is to be applied to complex numbers, and 2) you must define that C=0 above (or equivalent, depending on your derivation)

38

I’m still struggling with this earlier explanation from Indistinguishable. Why is rotation considered analogous to multiplication, rather than addition? It seems to me that, in that example, you’re merely adding 12 degrees per second, which is not exponential growth.
I guess I don’t understand what “it multiplies by a 12 degree rotation” means.

39

Actually, I guess that BetterExplained link does, well, explain it…
This is all very cool. We covered complex numbers at school for some time, but I don’t remember the *point *of them ever being explained like this.

40

I said before that any function that turns addition into multiplication can be thought of as exponentiation with some base. Well, similarly, any function that turns addition into addition (i.e., distributes over addition; i.e., is “linear”) can be thought of as multiplication by some constant. That is, if f is continuous, f(0) = 0, and f(x + y) = f(x) + f(y), then we can think of f as multiplication by some constant. In particular, if f is the operation on vectors which rotates them by some fixed angle theta, then f will satisfy these properties. [Thinking of vector addition as given by triangles, the observation that rotation distributes over addition is the observation that when you rotate a triangle, it’s still a triangle]. Thus, rotation by the angle theta is multiplication by some constant.

(Note for the detail-oriented: the above discussion spoke of multiplying vectors by rotations. This is just like how we can multiply vectors by scalars to resize them. But also, just as we can multiply scalars by scalars as well, to compose resizings into their cumulative effect, we can just as well multiply rotations by rotations, to compose them into their cumulative effect. Indeed, for any linear operations between vector spaces, we can speak of multiplying them to compose them into their cumulative effect (since composition also satisfies the above property, of distributing over additions on either side). That is, if f takes inputs of type A and produces outputs of type B, and g takes inputs of type B and produces outputs of type C, then we can think of g * f as the composite function which takes inputs of type A and produces outputs of type C. [When presented in terms of co-ordinates, this is called matrix multiplication]. Note that, in the general case, the order of composition matters; however, if one limits themselves to talking about scalings and rotations of vectors in 2d space [i.e., the complex numbers], then everything commutes.)