In advanced mathematics it's common to define cos and sin by these series (and pi is defined as the smallest strictly positive x with sin x = 0). (Of course that just reduces the question to "why do certain geometrical identities match this sin function")
Yeah I mean to say you can derive the sin and cosine as abstract functions from their geometrical properties, get the form of their derivatives, and then derive their Taylor series, instead of defining them as infinite series, and then showing that the function looks like their geometric equivalents.
I was also perturbed by the jump from the definition of e to the taylor expansion. I know how to get there the long way (define e first, derive properties of the exponential derivative, then construct the Taylor series), does anyone know a shortcut?
it's a little ugly, because you have to have some strong conditions to use associativity on infinite series (and I forget what they are off the top of my head). Of course, this is true for splitting up the e^x into cos(x) and sin(x) as well.
The series has to be absolutely convergent. That is, you can rearrange the terms of \sum_{i=0}^\infty a_n freely if and only if \sum_{i=0}^\infty |a_n| converges. See http://en.wikipedia.org/wiki/Riemann_series_theorem
The Taylor series is actually the expansion of the limit in the line above. There's some trickery in proving that the limit converges, but you can derive one line from the other with some straightforward combinatorics.
gohrt linked an article, and although you could derive the identity for yourself using that information, the article doesn't contain the series you're looking for.
The latter is cliched because it incorporates an additional fundamental constant, pi. Who would have thought that the ratio of the circumference of a circle to the diameter when multiplied by the imaginary number and then exponentiated by another constant e would produce such a simple equation which also includes the multiplication identity and the addition identity? Yes, pi is chosen but it certainly encompasses the trigonometry and geometry (rotations, sinx, cosx, etc.).
The advantage of the traditional format is that it not only includes four fundamental constants (1, 0, e, i and π) but it also includes the four fundamental operators (addition, multiplication, exponential and equality.)
I prefer that point of view if you want to understand Euler's identity, and I find John Baez does it even better:
http://math.ucr.edu/home/baez/trig.html
The Baez article does leave it to the reader to convince himself that exp(i*theta) is good notation for a point on a unit circle.
I have problems with the attitude of the article you linked, though. Especially
"Therefore, I’d like to complain to the thousands of people who find Euler’s identity stunning and beautiful."
followed by a snide list of reasons why someone might find it beautiful. It's very common when doing math that something amazing is obvious an hour later. I believe that we are better served by reminding ourselves that (a) nobody knows everything, and (b) the basics facts are actually very beautiful.
I totally agree on the attitude on the blog, the author is clearly working through non-mathematical issues on their own (if you read some of the other entries you can see how those challenges affect their writing). That said, I tend to see it as a counterpoint to mathematics blogs that are bit too gushing the other way. And its amusing the path that is taken as well.
And just in case you weren't perfectly satisfied with the level of mathematical rigor of the article, here is a complete, formal, machine-verified and hyperlinked version of the proof: http://us.metamath.org/mpegif/eulerid.html
There's some even more important gaps regarding analytic continuations of functions to complex numbers (and the resulting power series expansions). You can prove it this way, but it's not at all rigorous by today's standards.
I was hoping that too, but I knew it would be about Euler's identity, since to people with only incidental exposure to the concepts that underly it it seems (justifiably) inscrutable and mysterious, thus its general popularity. It's funny that cultivating the mathematician's refusal to assign meaning to results can completely change which results you find fascinating.
I would say that the Fundamental Theorem of Galois Theory is the most beautiful result of all mathematics, though Euler's identity is certainly a contender.
I'd offer that accessibility is a huge part of the beauty of Euler's identity. A few weeks into your average Calculus 2 class and it almost feels intuitive.
The field extension lattice is isomorphic to the subgroup lattice; if you really wanted to, you could write this out symbolically (but I am not sure why you would want to, since it does not really convey the meaning of the theorem any better). I suppose you might say that such an isomorphism does not qualify as an equation, but that is a bit pedantic in my opinion since such isomorphisms have all the properties of an equivalence relation.
Right, it's a beautiful theorem, not so much a beautiful equation.
Euler's identity is a beautiful equation, because it ties together several of the most fundamental objects of mathematics, with one occurrence of each, with no wasted boilerplate. The notation is part of the beauty. It looks darn good, on the surface in addition to the beyond the ideas behind the surface.
Good question. Though I suppose it's not too difficult to state in terms of the Abel-Ruffini theorem, but then again that watered down version would probably fail to mention the wide-reaching consequences of Galois...
Well, this equation is really a consequence of the more general e^ix = cos(x) + isin(x). This, Euler's Formula, enormously simplifies sinusoidal equations. Most common trigonometric identities can be proven in only 3 or 4 steps if you spend 2 of them converting to/from the exponential form, but are far more complicated in the trigonometric form. Many problems in electricity, magnetism, and basic quantum physics would be drastically less wieldy (more unwieldy?) without it.
I don't know of any cases in which it makes things possible, but there are plenty of cases where it makes things practical.
>> Euler's brilliant mathematical mind replaced the real variable x with ix
Is there any proof that the equation remains true when x -> ix transformation is made? OK, I know there is formal proof for this; can someone explain please? :-)
(Also, just a nitpick, shouldn't the addition be actually subtraction before both elippses to demonstrate the alternating sign?)