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

Paul Alciatore
11-25-2014, 04:47 PM
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
11-25-2014, 05:21 PM
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
11-25-2014, 05:26 PM
I don't get it ? if you know his identity why don't you just try talking to him about it?

mklotz
11-25-2014, 05:32 PM
Well, maybe not all of them!

aostling
11-25-2014, 05:43 PM
This might help, and it has a video to watch while you are reading the text: http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/

boslab
11-25-2014, 06:32 PM
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
11-25-2014, 08:24 PM
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
11-25-2014, 08:38 PM
Any help?

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

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

dp
11-25-2014, 08:38 PM
This might help, and it has a video to watch while you are reading the text: http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/

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

Paul Alciatore
11-25-2014, 09:11 PM
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
11-25-2014, 09:23 PM
... 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
11-25-2014, 09:24 PM
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
11-25-2014, 09:43 PM
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?

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

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

aostling
11-25-2014, 09:48 PM
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
11-25-2014, 10:06 PM
Actually, I have an elegant proof of that, but it's too large to fit in the margin of this post...

boslab
11-26-2014, 12:41 AM
Actually, I have an elegant proof of that, but it's too large to fit in the margin of this post...
Funny, perhaps it was rulers last theorem
Mark

MrSleepy
11-26-2014, 05:57 AM
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

No Liebniz ... surely he deserves a mention. Clerk Maxwell aswell.

Rob

aostling
11-26-2014, 10:03 AM
No Liebniz ... surely he deserves a mention. Clerk Maxwell aswell.

Rob

Of course. Which reminds me, on 12 May 2003 (according to my trip diary) I was driving south in Dumfries, Scotland, on B794, a one-lane road branching off from A712. I'd seen the sign pointing to "Old Bridge of Urr" and wanted to see what that was. There were no other cars, fortunately. Several miles in I had a fleeting glimpse of some ruins, stark chimneys visible above the dense woods. I stopped, back-tracked to a long drive, and drove up it. I came to a substantial house and parked outside on the circular drive. The owner came out and greeted me with a welcome, and said "I assume you have come to see Maxwell's house." Utterly confused, I told him I had just sort of stumbled in on a whim. He explained that the ruined house, which was on his property about a hundred yards distant, was the ancestral home of James Clerk Maxwell.

And so I got a chance to see Glenlair House. I suppose it is still a ruin, but the stone walls remain and it is an inspiring place. The owner gave me tea and biscuits and we chatted about his days in the RAF.

Alan Turing should be added to the list, too.

Seastar
11-26-2014, 10:46 AM
I have been fascinated by Euler's Idenity for a long time.
I made this a few years ago.
http://imagizer.imageshack.us/v2/150x100q90/594/jv0x.jpg (https://imageshack.com/i/gijv0xj)
Bill

Iceberg86300
11-29-2014, 03:09 AM
I'll take a look at my texts, but don't hold your breath. I may have only kept my advanced engineering math text. Proofs and derivations pretty much got left in 9th grade geometry for me.

Regards,
Steve

Paul Alciatore
11-29-2014, 02:21 PM
Well, I did some poking around some math sites but still am not satisfied. Most seem to throw Taylor series at you with no explanation of how the series for the complex expression is derived. They are just shifting the mystery to another location, not throwing any real light on it.

Still looking.

dp
11-29-2014, 03:31 PM
Euler's identity is a special case of Euler's formula where n=π. The constant e is a consequence of the Taylor series that generates e. The number 1 is the multiplicative identity, i is the imaginary unit such that i^2 = -1, and pi is a consequence of geometry. Given that e and pi are irrational numbers the identity can't be calculated precisely beyond "approaches 0" but the logical analysis can describe e and pi as absolute values such that the right hand side of the identity is equal to 0.

dp
11-29-2014, 03:46 PM
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.

i^i is equivalent to e^-(π/2) = 0.207879575....

Barrington
11-29-2014, 04:42 PM
Well, I did some poking around some math sites but still am not satisfied. Most seem to throw Taylor series at you with no explanation of how the series for the complex expression is derived. They are just shifting the mystery to another location, not throwing any real light on it. Still looking.
It's not hard to derive a Taylor Series, but a bit more tricky to prove it converges. Here's a rough guide to get you going for the first bit, in excruciating detail.

E.g. assume cos(x) can be represented by converging polynomial series, with coefficients A,B,C,D etc:-

cos(x) = A + B*(x^1) + C*(x^2) + D*(x^3) + E*(x^4) + F*(x^5) +...
for x = 0, then cos(0) = A, A = 1

Differentiate both sides wrt x:-
-sin(x) = 0 + B + 2*C*(x^1) + 3*D*(x^2) + 4*E*(x^3) + 5*F*(x^4) + 6*G*(x^5) +...
for x = 0, then -sin(0) = B, B = 0,

Differentiate again:-
-cos(x) = 0 + 0 + 2*C + 3*2*D*(x^1) + 4*3*E*(x^2) + 5*4*F*(x^3) + 6*5*G*(x^4) +...
for x = 0, then -cos(0) = 2*C, C = -1/2, (note: = -1/(2!))

and again:-
sin(x) = 0 + 0 + 0 + 3*2*D + 4*3*2*E*(x^1) + 5*4*3*(x^2) + 6*5*4*G*(x^3) +...
for x = 0, then sin(0) = 3*2*D, D = 0

and again:-
cos(x) = 0 + 0 + 0 + 0 + 4*3*2*E + 5*4*3*2*F*(x^1) + 6*5*4*3*G*(x^2) +...
for x = 0, then cos(0) = 4*3*2*E, E = 1/(4*3*2), E = 1/(4!)

and again:-
-sin(x) = 0 + 0 + 0 + 0 + 0 + 5*4*3*2*F
for x = 0, then -sin(0) = 5*4*3*2*F, F = 0

and again:-
-cos(x) = 0 + 0 + 0 + 0 + 0 + 0 +6*5*4*3*2*G +...
for x = 0, then -cos(0) = 6*5*4*3*2*G, G = -1/(6!)

etc. etc. etc., so:-

cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! +...

(Note coeffs of odd powers are zero.)

By a similar process:...

sin(x) = 0 + (x^1)/1! - (x^3)/3! + (x^5)/5! -...

(Note coeffs of even powers are zero.)

and yet again similarly, (remembering d(e^x)/dx = e^x ) :-

e^x = 1 + (x^1)/1! + (x^2)/2! + (x^3)/3! + (x^4)/4! + (x^5)/5! + (x^6)/6! +...

(Note coeffs are all non-zero.)

So, (remembering (i*x)^n = (i^n)*(x^n) ) :-

e^(i*x) = 1 +i*(x^1)/1! -1*(x^2)/2! -i*(x^3)/3! +1*(x^4)/4! +i*(x^5)/5! -1*(x^6)/6! -...

Which by inspection is clearly equal to [ cos(x) + i*sin(x) ] as previously shown.

.

Cheers

.

Daveb
11-29-2014, 04:56 PM
If Euler didn't know who he was, how on Earth is anyone going to figure it out after all this time?

Seastar
11-29-2014, 05:05 PM
I have been fascinated by Euler's Idenity for a long time.
I made this a few years ago.
http://imagizer.imageshack.us/v2/150x100q90/594/jv0x.jpg (https://imageshack.com/i/gijv0xj)
Bill
I think you have all missed the symbology of the teeth on a saw blade representing the rotation about a circle as in Eulers Identity.

Daveb
11-29-2014, 05:37 PM
Sorry Seastar, I did see it, I thought it was a Pi.
Dave

dp
11-29-2014, 05:50 PM
If Euler didn't know who he was, how on Earth is anyone going to figure it out after all this time?

That would be solved using Euler's Identity Crisis :)