OT and MATH: Proof or Derivation of Euler's Identity

[Paul Alciatore]

I have a bit of a background in math, but not a professional one. I have been aware of Euler's Identity since my college years, perhaps even earlier, but have never seen a derivation or proof of it that satisfied my curiosity.

Euler's Identity: e^(i*pi) + 1 = 0

or e^(i*pi) = -1

At a first glance, the explanation is simple.

http://en.wikipedia.org/wiki/Euler%27s_identity

About 1/3 of the way down the page is an explanation that only requires high school algebra to understand. But it is based on what is called Euler's Formula which is more or less just another way of expressing Euler's Identity. So, that really is not a proof or explanation.

Euler's Formula: e^ix = cos x + i sin x

http://en.wikipedia.org/wiki/Euler%27s_formula

Now it gets a lot more dicey. The derivation of this formula is apparently based on Taylor series.

I am going to go through this page and others on the subject but that is going to take some time.

Does anyone have an easier way of understanding this? Mathematically? Perhaps a text or web page that shows the derivation in understandable steps?

[mklotz]

Wow, OT and MATH, that ought to keep them away. :-)


Well, it's certainly not a proof but it does make it a bit more obvious...

In the series expansions for:

sin(x) = x - (x^3)/3! + (x^5)/5! - ...

cos(x) = 1 - (x^2)/2! + (x^4)/4! - ...

e^x = 1 + x + (x^2)/2! + (x^3)/3! + ...

plug in x = i*z [i = sqrt(-1)] and simplify all the powers of 'i',

i^2 = -1
i^3 = -i
i^4 = +1
etc,

and Euler's formula (in infinite series form) will be obvious.

[A.K. Boomer]

I don't get it ? if you know his identity why don't you just try talking to him about it?

[mklotz]

Well, maybe not all of them!

[aostling]

This might help, and it has a video to watch while you are reading the text: http://betterexplained.com/articles/...ulers-formula/

[boslab]

I'm not sure, Taylor's is often used to calculate irregular areas so why not,
Try

http://math.stackexchange.com/

At least you will be asking a mathematician !
Mark

[A.K. Boomer]

So did you ask him? if you know who he is it really should not be that hard I mean he's still alive right? so try facebook for starters and see what happens...

[oldtiffie]

Any help?

https://www.google.com.au/?gws_rd=ssl#q=euler's+theory

http://en.wikipedia.org/wiki/Euler's_theorem

Its far too deep for me - but I went looking anyway - as I do.

[dp]

This might help, and it has a video to watch while you are reading the text: http://betterexplained.com/articles/...ulers-formula/

That was quite interesting. And yes, i^i is a very puzzling concept

[Paul Alciatore]

Seance?

Line 2:

http://en.wikipedia.org/wiki/Leonhard_Euler

I don't get it ? if you know his identity why don't you just try talking to him about it?

[aostling]

... i^i is a very puzzling concept

It is indeed. I can't get a mental picture of it, but it is easily calculated on my HP48G, which returns the real value of 0.207879576.... This page explains it: https://www.math.hmc.edu/funfacts/ffiles/20013.3.shtml.

[A.K. Boomer]

Seance?

Line 2:

http://en.wikipedia.org/wiki/Leonhard_Euler

Geeze, That guy was the Tesla of math or at least one of them... some people are so advanced it's like you feel like they must be aliens or something. just outright geniuses in their fields --- might not know how to butter a piece of toast but get them in their field and watch the hell out....

just amazes me someone could take interest in stuff I slept my way through in high school...

[Paul Alciatore]

That WikiPedia article looks like it was written by a politician. It may explain it to a math PhD, but not to me.

I'm afraid I need baby steps.
Looking at some of the other responses.
The video is interesting, but hardly a rigorous proof.

Any help?

https://www.google.com.au/?gws_rd=ssl#q=euler's+theory

http://en.wikipedia.org/wiki/Euler's_theorem

Its far too deep for me - but I went looking anyway - as I do.

[aostling]

Geeze, That guy was the Tesla of math or at least one of them...

I like that analogy. Some of the other mathematical greats who are in Euler's league are:

Gauss
Newton
Archimedes
Ramanujan (whose mysterious infinite series continue to baffle the greatest minds)
Galois (who died in a duel at age 20)
Emmy Noether (who showed that the First Law of Thermodynamics is a consequence of symmetry)
Riemann (whose theories were instrumental to Einstein's General Theory of Relativity)
Cantor

[mickeyf]

Actually, I have an elegant proof of that, but it's too large to fit in the margin of this post...