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

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

    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?
    Paul A.
    SE Texas

    Make it fit.
    You can't win and there IS a penalty for trying!

  • #2
    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.
    Regards, Marv

    Home Shop Freeware - Tools for People Who Build Things
    http://www.myvirtualnetwork.com/mklotz

    Location: LA, CA, USA

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    • #3
      I don't get it ? if you know his identity why don't you just try talking to him about it?

      Comment


      • #4
        Well, maybe not all of them!
        Regards, Marv

        Home Shop Freeware - Tools for People Who Build Things
        http://www.myvirtualnetwork.com/mklotz

        Location: LA, CA, USA

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        • #5
          This might help, and it has a video to watch while you are reading the text: http://betterexplained.com/articles/...ulers-formula/
          Allan Ostling

          Phoenix, Arizona

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          • #6
            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

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            • #7
              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...

              Comment


              • #8
                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.

                Comment


                • #9
                  Originally posted by aostling View Post
                  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

                  Comment


                  • #10
                    Seance?

                    Line 2:

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


                    Originally posted by A.K. Boomer View Post
                    I don't get it ? if you know his identity why don't you just try talking to him about it?
                    Paul A.
                    SE Texas

                    Make it fit.
                    You can't win and there IS a penalty for trying!

                    Comment


                    • #11
                      Originally posted by dp View Post
                      ... 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.
                      Last edited by aostling; 11-25-2014, 10:32 PM.
                      Allan Ostling

                      Phoenix, Arizona

                      Comment


                      • #12
                        Originally posted by Paul Alciatore View Post

                        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...

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                        • #13
                          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.


                          Originally posted by oldtiffie View Post
                          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.
                          Paul A.
                          SE Texas

                          Make it fit.
                          You can't win and there IS a penalty for trying!

                          Comment


                          • #14
                            Originally posted by A.K. Boomer View Post
                            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
                            Last edited by aostling; 11-25-2014, 10:50 PM.
                            Allan Ostling

                            Phoenix, Arizona

                            Comment


                            • #15
                              Actually, I have an elegant proof of that, but it's too large to fit in the margin of this post...
                              "A machinist's (WHAP!) best friend (WHAP! WHAP!) is his hammer. (WHAP!)" - Fred Tanner, foreman, Lunenburg Foundry and Engineering machine shop, circa 1979

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