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Understanding the most beautiful equation in Mathematics

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Re: Understanding the most beautiful equation in Mathematics

#41

Post by carstimon »

ChuckMcM wrote: post #27

And after you read this you should read this: http://symbo1ics.com/blog/?p=1089 which was kind of fun as well.

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.

Re: Understanding the most beautiful equation in Mathematics

#42

Post by thedufer »

mohas wrote: post #28

Can anyone name some of the actual uses of this equation in solving real world problems?

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.

Re: Understanding the most beautiful equation in Mathematics

#44

Post by stephencanon »

a-nikolaev wrote: post #8
cmvkk wrote: post #3

Here's my favorite explanation of this formula: http://betterexplained.com/articles/intuitive-understanding-...

My favorite: 1) Using Taylor Series, show that exp(ix)=cos(x)+i*sin(x). 2) Then the result is trivial for x=pi This image helps: https://en.wikipedia.org/wiki/File:Euler%27s_formula.svg

Taylor series are ugly; do it directly from the definition of the functions as solutions to specific differential equations =).

Re: Understanding the most beautiful equation in Mathematics

#46

Post by tome »

viraj_shah wrote: post #40
tome wrote: post #31

I think the actually remarkable equation is e^ix = cos x + i sin x The cliched "e^(i pi) + 1 = 0" is a fairly mundane consequence of the fact that pi was chosen to make this equation hold.

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…

My point is that it is only fundamental because it leads to this equation!

Re: Understanding the most beautiful equation in Mathematics

#47

Post by decasteve »

betterunix wrote: post #4

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.

Not to mention the Inverse Galois Problem is one of the long unsolved problems in Mathematics. In the league of Fermat's Last Theorem.

Re: Understanding the most beautiful equation in Mathematics

#49

Post by ChuckMcM »

carstimon wrote: post #41
ChuckMcM wrote: post #27

And after you read this you should read this: http://symbo1ics.com/blog/?p=1089 which was kind of fun as well.

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 peo…

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.

Re: Understanding the most beautiful equation in Mathematics

#50

Post by dbaupp »

dnautics wrote: post #12
lmm wrote: post #11

Earlier quoted context omitted.

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")

or you could use MacLaurin polynomial series (Taylor series at zero)

I'm probably missing something, but that is the Taylor series at 0.
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