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File:ExpIPi.gif

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    Original file (360 × 323 pixels, file size: 11 KB, MIME type: image/gif, looped, 9 frames, 4.5 s)

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    Summary

    Description This is a demonstration that Exp(i*Pi)=-1 (called Euler's formula, or Euler's identity). It uses the formula (1+z/N)^N --> Exp(z) (as N increases). The Nth power is displayed as a repeated multiplication in the complex plane. As N increases, you can see that the final result (the last point) approaches -1, the actual value of Exp(i*pi).
    Date
    Source Own work
     This diagram was created with Mathematica by n.
    Author Sbyrnes321

    Licensing

    Public domain I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
    In some countries this may not be legally possible; if so:
    I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. 
    (* Source code written in Mathematica 6.0, by Steve Byrnes, 2008. I release this code into the public domain. *)

plot1 = Table[
  ListPlot[Table[{Re[(1 + (\[ImaginaryI] \[Pi])/n)^m], 
     Im[(1 + (\[ImaginaryI] \[Pi])/n)^m]}, {m, 0, n}], 
   PlotJoined -> True, PlotMarkers -> Automatic, 
   PlotRange -> {{-2.5, 1.1}, {0, \[Pi] + .05}}, AxesOrigin -> {0, 0},
    AxesLabel -> {"Real part", "Imaginary part"}, 
   PlotLabel -> "N = " <> ToString[n], 
   AspectRatio -> Automatic], {n, {1, 2, 3, 4, 5, 10, 20, 50, 100}}];

Export["ExpIPi.gif", plot1, "DisplayDurations" -> {2}, 
 "AnimationRepititions" -> Infinity ]

    Captions

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    Items portrayed in this file

    depicts

    inception

    5 May 2008

    data size

    11,049 byte

    height

    323 pixel

    width

    360 pixel

    media type

    image/gif

    checksum

    bc4adc1aeb42ee540abae5d8536f11ef312132be

    determination method or standard: SHA-1

    File history

    Click on a date/time to view the file as it appeared at that time.

    Date/TimeThumbnailDimensionsUserComment
    current19:46, 25 March 2010Thumbnail for version as of 19:46, 25 March 2010360 × 323 (11 KB)Aiyizooptimized animation, converted to 16 color mode
    17:19, 5 May 2008Thumbnail for version as of 17:19, 5 May 2008360 × 323 (20 KB)Sbyrnes321{{Information |Description=This is a demonstration that Exp(I*Pi)=-1 (called Euler's formula, or Euler's identity). It uses the formula (1+z/N)^N --> Exp(z) (as N increases). The Nth power is displayed as a repeated multiplication in the complex plane. As
    16:58, 5 May 2008Thumbnail for version as of 16:58, 5 May 2008360 × 308 (18 KB)Sbyrnes321{{Information |Description=This is a demonstration that Exp(I*Pi)=-1 (called Euler's formula, or Euler's identity). It uses the formula (1+z/N)^N --> Exp(z) (as N increases). The Nth power is displayed as a repeated multiplication in the complex plane. As

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