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Euler's Equation: A sign from god?

  1. Mar 26, 2004 #1
    The first time I saw Euler's equation, it blew my mind.

    [tex]e^{i\pi}+1 \equal 0[/tex]

    Here, we have three of the most important numbers in math, all related to each other in such a remarkably compact equation. Does anyone know what this means? I think you can prove this through Taylor Series, but that's not what I'm asking for. Is there some underlying, intuitive reason for why Euler's equation is true? Or is it just a big fat coincidence (hence sign from god)?

    Edit: Why is there a stupid dash over my zero? This is why I hate Latex. Stupid, unexplained stuff always happens.
     
    Last edited by a moderator: Mar 26, 2004
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  3. Mar 27, 2004 #2

    ahrkron

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    Instead of "\equal", just use "="

    [tex]e^{i\pi}+1 = 0[/tex]

    I don't see any divine influence on the equation, but it is definitely a wonderful one. I think of it as a brief, yet elegant, summary of some of our most powerful and beautiful math (algebra, complex numbers and calculus).

    In terms of the underlying reason, the proof is basically the reason you are looking for. Think of exponentiation in terms of the Taylor series (i.e., picture ("e^x" as a short notation for the series), and of complex numbers in terms of their intimate relation with rotations, which naturally bring pi to the arena. Instead of an unexplained coincidence, I see it as an inevitable result of the structure of math, lying in an unavoidable intersection of various branches of it.
     
  4. Mar 27, 2004 #3
    It's from the definition of [itex]e^z[/itex] = [itex]e^x(Cos(y) - i Sin(y))[/itex] for complex z = x + iy. And yes, this definition does come about because if you use the Taylor Series expansion for [itex]e^z[/itex] you can separate the resultant series into two which represent the trigonometric functions. However, such rearrangement of infinite series with changing signs is very dangerous, as this can change the resultant sum (as in conditionally convergent series). But Euler often played fast and loose with such unrigorous methods, and got away with it! In any case, having done that, setting x to 0 and y to [itex]\pi[/itex] gives you [itex]e^{i\pi}[/itex] = -1 which is a most ugly form. I much prefer it when the identity elements of addition and multiplication show in the equation.

    But I'm not so sure it proves there's a God. Perhaps it proves that Euler was in league with the devil! :)
     
    Last edited: Mar 27, 2004
  5. Mar 28, 2004 #4
    Hm?

    "Is there some underlying, intuitive reason for why Euler's equation is true? Or is it just a big fat coincidence (hence sign from god)?"

    Since when did coincidences imply a sign from God?
     
  6. Mar 29, 2004 #5

    HallsofIvy

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    Are you saying that you consider any "coincidence" a sign from God? You are truly blessed!
     
  7. Mar 29, 2004 #6

    selfAdjoint

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    According to the vector interpretation of complex numbers [tex]e^{i\pi} = -1 [/tex] just says that if you turn through 180o you will be facing the other way. Another way to look at it is that [tex]e^{i\theta} [/tex] generates a circle in the complex plane, and of course [tex] \pi [/tex] has its well-known relationship to a circle.

    And why does [tex]e^{i\theta} [/tex] generate a circle? Basically because if you multiply two complex numbers with magnitude and argument, the magnitude of the product will be the product of the magnitudes and the argument of the product will be the sum of th arguments, so the argument, or angle of a complex number behaves like a natural logarithm, and e is the basis of the natural logarithms.

    And why does the product of complex numbers work that way? It comes out of the distributive law of multiplication and the representation of the complex numbers in components as [tex]m cos \theta + i sin \theta [/tex] which brings us back to Euler's series.

    It hangs together because the plane has this rotation property. Whether you take that as a mystical sign or not is of course up to you.
     
  8. Mar 29, 2004 #7
    you can use taylor expansion to proof this equation
     
  9. Mar 29, 2004 #8

    Janitor

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    John Baez has a lot of interesting stuff at his website. Complex numbers, and their "higher dimensional" analog, the quaternions, are pet topics of his. I took the liberty of pasting this from there:

     
  10. Oct 22, 2005 #9
    Richard Feynmann, in his lectures at Caltech (available in book form), referred to the underlying relation that produced the referenced equation as the "crown jewel of algebra", I think.
     
  11. Oct 22, 2005 #10
    Perhaps you should be a little less humble; it is an achievement of human intellect: we did it.
     
  12. Oct 22, 2005 #11

    Pengwuino

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    since when did humans invent math? Did math not exist before us?

    I'm still rather (extremely) confused as to how this relates to God. What's so important about this equation?
     
  13. Oct 22, 2005 #12
    Since, oh, about 4000 BC.
     
  14. Oct 23, 2005 #13
    Icebreaker, math existed before we discovered it, and it will exist long after we're gone.
     
  15. Oct 23, 2005 #14

    Pengwuino

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    Unless of course, you follow that "If I believe it's true, then it's true" line of thinking.
     
  16. Oct 23, 2005 #15
    i think euler himself thought that his equation was so perfect that it proved the existence of God.
     
  17. Oct 23, 2005 #16

    Tide

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    Unless you are assuming there are/were other intelligent species in the Universe, how could that possibly be?
     
  18. Oct 23, 2005 #17

    Pengwuino

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    1 + 1 = 2 existed before humans did. We didn't invent math, we discovered it.
     
  19. Oct 23, 2005 #18

    Tide

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    I don't think so! Number is a concept and you cannot have concepts without mind. There may have been a "number" of things before the mind existed but the numbers require realization in order to exist.
     
  20. Oct 23, 2005 #19
    Euler Formula: http://mathworld.wolfram.com/EulerFormula.html

    Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician (Derbyshire 2004, p. 202).

    I doubt most of us are in that class, but, according to Gauss, so much for mystification of the formula. (It might be added that Gauss was a hard master who cared little for his students, and did absolutely nothing to help Galois or Able.)
     
    Last edited: Oct 23, 2005
  21. Oct 23, 2005 #20
    Tide: I don't think so! Number is a concept and you cannot have concepts without mind. There may have been a "number" of things before the mind existed but the numbers require realization in order to exist.

    Does a tree falling in the forest make a sound if there is no one to hear it?
     
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