The Wikipedia article doesn’t do a very good job of getting it across to the layman.

Things like this intrigue me, but I’m sad to say they’re lost on me as of where my understanding of mathematics stands now:

I want to get the hype. I want to understand it, but reading about it makes me feel like I’m reading about the turbo-encabulator.

Here is my background in pure math: I went through college algebra, and a class called college mathematics, which did a little bit of several different disciplines. Had a tough time with both of them because I have a bad habit of asking “why” and math professors, in my experience, aren’t very good at answering that one.

The first key idea to understand is that complex numbers are nothing more than rotations and scalings. If you know that two left turns make a U-turn, then you already understand that i^2 = -1, even if you’re not aware that these mean the same thing. See here.

Once that’s in place, it’s not so hard to understand Euler’s theorem. All it really says is that as you swing a stick in a circle, the direction in which the end of the stick is moving at any moment is 90 degrees rotated from the direction in which the stick is pointing. Is that beautiful? Well, it’s certainly true. And if you put it in certain symbols, it looks mystical and lofty and so on. But it’s a terribly simple theorem; so simple small children understand it, just without understanding that they understand it. All the stuff about it reaching down into the very depths of existence and so on is rot; Gauss had the right idea. Again, see here.

I’m being very lazy by just linking to those, but I figure, why duplicate work? I’ll be happy to answer any questions you have after reading those.

I speak as someone who’s interested in math but has no talent for it.

I think the reason that Euler’s equation is held by some to be a paragon of mathematical, if not absolute, beauty, is that, without it being designed to do so, it links huge and and apparently disparate parts of mathematics. Moreover, it does it in a simple, ergo beautiful, and at least superficially unexpected, way. Indeed, at first blush, the equation would seem to suggest a profound and deep relationship among geometry, analysis, and algebra - apparently distinct branches of math, each with its own developmental history, ontogeny, and, if you will, “purpose”.

Phrased somewhat differently, the three constants it connects (or five if you include ‘-1’ and ‘0’) were each derived, independently, in their own way, and in their “own” area, without explicit consideration of the others. Each fits and works ‘perfectly’ in those areas of mathematics to which it applies. To then find that these profoundly important mathematical quantities are dependent on one and other, and that one defines the others and vice versa, has an almost mystical, and certainly beautiful, quality.

Contributing to its beauty for some, is the realization that this unexpected and “accidental” interconnectedness may have an implication on the age-old question of whether math is “invented” or “discovered”. IMO, the Euler equation helps to “prove” that mathematics and its truths exist independently from the mind and are “out there”, waiting to be discovered. This makes the equation even more beautiful to my lay eyes.

The equation follows from an analysis of the Taylor series of sine(x), cosine(x), and e[sup]x[/sup] around the point 0. The derivative of sin(x) is cos(x), the derivative of cos(x) is -sin(x), and the derivative of e[sup]x[/sup] is e[sup]x[/sup]. Also, cos(0) = 1, sin(0) = 0, and e[sup]0[/sup] = 1.

So, the expansion of cos(x) is:
1 - x[sup]2[/sup]/2! + x[sup]4[/sup]/4! - x[sup]6[/sup]/6! + …
And the expansion of sin(x) is:
x - x[sup]3[/sup]/3! + x[sup]5[/sup]/5! - x[sup]7[/sup]/7! + …
And the expansion of e(x) is:
1 + x + x[sup]2[/sup]/2! + x[sup]3[/sup]/3! + x[sup]4[/sup]/4! + …

Now note what happens when we calculate e[sup]ix[/sup] using the Taylor expansion of e[sup]x[/sup]:
1 + ix + (ix)[sup]2[/sup]/2! + (ix)[sup]3[/sup]/3! + (ix)[sup]4[/sup]/4! + …
Now, i[sup]2[/sup] = -1, i[sup]3[/sup] = -i, and i[sup]4[/sup] = 1, so simplifying the equation gives us:
1 + ix - x[sup]2[/sup]/2! - ix[sup]3[/sup]/3! + x[sup]4[/sup]/4! + …

Note how the 1st, 3rd, and 5th term are the first 3 terms of the expansion of cos(x), and the 2nd and 4th term are i multiplied by the first 2 terms of the expansion of sin(x). By induction, we can say that the following terms will match with other terms. So effectively, e[sup]ix[/sup] = cos(x) + i sin(x).

Finally, we look at the values of cos(pi) and sin(pi). Cos(pi) = -1, and sin(pi) = 0. So, plugging in x = pi, the equation becomes:

The Wikipedia’s explanation seems very clearcut – it manages to include five very important elements of math in a simple and thus elegant way. All five variables – E, i, pi, 0 and 1 are important mathematically and finding that they fit together into one simple equation is impressive. It’s even more impressive if you consider the equation includes real, irrational, and imaginary numbers yet yields a real number result.

Not picking on you, just picking on the ubiquity of this Taylor series demonstration:

I hate, loathe, detest the Taylor series demonstration of Euler’s theorem. This, more than anything else, has I think caused people to view the result as a marvelous, mystical coincidence, rather than the trivial, straightforward geometric fact it is. It completes and tragically destroys the intuitive understanding of what is going on here.

It is as bad as proving 2/(1 - x[sup]2[/sup]) = 1/(1 + x) + 1/(1 - x) by the fact that the nth term of the Taylor series on the left is 2x[sup]n[/sup] if n is even and 0 if n is odd, while the nth term of the Taylor series of 1/(1 + x) is (-1)[sup]n[/sup]x[sup]n[/sup] and the nth term of the Taylor series of 1/(1 - x) is x[sup]n[/sup], thus adding up appropriately. Certainly, one could give such a demonstration, but to establish such a simple identity in this way is bizarrely, unnecessarily roundabout. To carry out the demonstration in this way is to lose sight of where the Taylor series came from, the simple defining properties of (in this case) division. These properties can be used to establish the result much more directly (in the ordinary algebraic way); sure, the Taylor series happen to therefore follow the identity as well, but to translate the argument into that context and out of the context in which it naturally lives is pointlessly obfuscatory.

Similarly, you yourself gave an account of where the Taylor series come from; e[sup]x[/sup] is its own derivative, and thus e[sup]ix[/sup] has derivative i times itself. And, at the same time, cos(x) + i * sin(x) has derivative i times itself. Accordingly, as they satisfy the same first-order differential equation with the same starting value [1 on input 0], they must be equal. This already is a much cleaner argument than needless invoking of the Taylor series.

But even that is further than ideal from making clearest the underlying intuition. For where did the derivatives of cosine and sine come from in the first place? From simple properties of rotation, which cosine and sine simply serve as components of. But for Euler’s theorem, this separation into separate components is also a needless distraction. Let r be rotation by one radian (so that r[sup]2[/sup] is rotation by two radians, r[sup]1/2[/sup] is rotation by half a radian, etc.). Now, we ask ourselves, what is ln(r)? That is (drawing upon the defining property of the natural logarithm, the thing that makes it “natural”), what is the constant ratio between the rate of growth of the exponential function r[sup]t[/sup] and the value of r[sup]t[/sup] itself? In other words, what is the derivative of r[sup]t[/sup] when t = 0? In other words, as a vector of unit length rotates at a rate of one radian per unit of time, how is its velocity related to its value at any moment?

Well, two things: because we’re in radians, the speed at which this vector is moving is equal to the speed at which its angle increases, which is 1. And because it’s rotating around a circle, where the tangents are perpendicular to the radii, the direction in which this vector is moving is, at any moment, 90 degrees rotated from its current value. Thus, combining the magnitude and direction information about the velocity, we have that ln(r) = scale by a factor of 1 and rotate by 90 degrees = just rotate 90 degrees. But, of course, we have a standard name for 90 degree rotation: i. So, ln(r) = i. In other words, r = e[sup]i[/sup]. And thus, r[sup]theta[/sup] (rotation by theta radians) = (e[sup]i[/sup])^theta = e[sup]i * theta[/sup].

In particular, the defining property of 2π (aka, τ) is that it is the number of radians in a full revolution. Thus, as rotation by π radians is half a revolution, i.e., negation, i.e., -1, we have that -1 = e[sup]iπ[/sup], if you care about that sort of thing. Perhaps more importantly, we have, for a full revolution R, that R = e[sup]i2π[/sup]. All these are just rescaled ways of saying what was in the above paragraph, that r = e[sup]i[/sup].

And, yes, if you like, you can break rotation into its parallel and perpendicular components, aka, cosine and sine. The defining properties of cosine and sine are that r[sup]theta[/sup] = cos(theta) + sin(theta) * i, and thus, by the above, the right hand side here is another expression for e[sup]i * theta[/sup]. But unless you have some particular need to break things apart into these separate components, it is often better to simply speak in terms of rotation directly.

Now, once one fully understands Euler’s theorem, one recognizes that, in the end, all it is really saying is that the direction in which a rotating stick’s end is moving is always 90 degrees rotated from the direction in which the stick is pointing, a trivial geometric fact. Hardly anything to find deeply, double rainbow-ly amazing anymore. But understanding things is better than finding them amazing.

Often, a rational result, i.e. |e[sup]x·i·π[/sup]|=1, and if x is an integer, it gives a real, rational result. You also don’t mention that both e and π are transcendental numbers, that is, they can’t be found as the root of any combination of algebraic operations, and can only be determined to a an arbitrary precision by calculating successive terms of a sequence in a series, or exactly by the use of a limit of sequences, i.e. the precursor to differential calculus.

This is interesting and perhaps a bit mysterious, until you understand that e is a unique number such that a first order function (i.e. e[sup]x[/sup]) such that the derivative of the function is itself (i.e. d/dx e[sup]x[/sup] = e[sup]x[/sup]) and that the derivative of e[sup]x[/sup] evaluated at x->0 is 1, with an x-intercept at -1. Thus, if you transform the function through a rotation, it will plot out a unit circle in the complex plane. This is the entire basis of trigonometric functions, which provide the most frequently used continuous periodic functions, i.e. those that describe cyclic or wave phenomena, whose derivatives and integrals are also periodic or have periodic components. These are used every day in nearly every branch of engineering, economics, and natural science, so it isn’t just that Euler’s formula is beautiful in some mathematically abstract sense, but it is actually highly practical in terms of its application.

I have to agree with Indistinguishable, though; explaining it via the Taylor series expansion is awkward and non-intuitive. The best way to explain is to draw the unit circle and show what functions of e[sup]i[/sup] do on it.

I wanted to address this specifically, because it gets at the heart of what algebra is and why it is (usually) taught very badly, i.e. as a repetitive series of steps to go through to get a result without a comprehensive approach to provide an intuitive basis, which is unfortunate because the entire purpose of algebra is to provide a system by which to formalize abstractions like “functions” and “sets”. Algebra is very simply a grammar for describing mathematical operations. In terms of elementary algebra, it has the equivalent of a subject (independent variable), predicate (function) and adjectives (operators), adverbs (coefficients), and prepositions (constants). (A similar metaphor can be used for other systems of algebra.) Once you learn how to “speak” the grammar of algebra (and compared to English, it is quite simple, with only a handful of operations), it makes the business of handling complex equations or systems of equations much easier, in the same way that learning how to read allows you to take complex ideas that are too difficult to absorb as a gestalt verbally and break them down into a flowing, unified theme. The answer to a question like, “Why?” may be “Because this is the rule for doing this operation,” but more likely the answer can be found by talking just a bit about conic functions, or how individual terms fall out because they’re just combinations of independent functions, just like 40 can be divided into 5 and 8.

If we taught reading the same way we teach basic mathematics, i.e. purely by a mechanical rote without context or application, we’d have a nation of illiterates. That isn’t to say you can learn to actually apply mathematics like algebra without having to crunch through problem sets to ingrain the operations into your brain, but you need to be taught what you’re doing in a way that makes sense in a constructive, progressive fashion. That’s what the “New Math” was supposed to accomplish, only it was implemented in an arse-backward fashion largely by people who didn’t understand the concepts using tools and texts that were poorly edited and vetted.

I’m curious where you went to school that these courses were taught by professors. That would be unusual at many institutions.

I’m in my last year of my math PhD and am currently teaching calculus. The main focus of every lecture is answering “why?” Furthermore, I’m currently writing my dissertation which is page after page of answering “why?” In my experience mathematicians are not only good at answering “why” it is pretty much our entire job.

I took those particular courses at community college. I took stats, but never any “pure math” at university.

Someone at my level needs it explained in English, though. I’m guessing your paper looks more like Punoqllads’ post, which I cannot make any sense out of.

I agree with everything you’ve said, but I don’t think you’ve quite done justice to the OP’s question. Sure, Euler’s theorem is a tautologically a trivial geometric fact – if you accept as trivial the definitions of rotation and their relationship to e (and accept as trivial that e and pi are themselves uninteresting and non-mysterious or beautiful numbers). Speaking for myself, I can’t completely get my mind around why this transcendental number that relates the rate of change of exponentiation to its value, is so simply related to rotation in the complex plane. Part of the problem is in trying to understand exp(i*angle) in the context of how exponentiation is defined: how do you multiply a number with itself an imaginary number of times, etc (especially without bringing series expansion into the picture)…

In that case, I’m willing to bet that you’ve never had the opportunity to ask a math professor “why?” It’s unlikely that any of those courses were taught by a mathematics professor.

I apologize for my tone and I’m not trying to pick a fight here. I’m American and have taken math classes at community college in North Carolina. These classes were generally taught by instructors with at most a master’s degree in math. I’ve only ever had one instructor at CC with a PhD (psych not math) and even he was an instructor as opposed to a professor. Professor is a specific rank in academia that is not found at community colleges.

I only brought up this hijack because I was honestly confused about your statement concerning math professors that didn’t match my experience with the people I work with and about a profession that I one day hope to join.