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What is so beautiful about Euler's Identity?

  1. Dec 11, 2008 #1
    I'm a pretty novice Physicist/Mathematician, but I've got a few offers for good universities, to show you my general level of knowledge.

    Could someone please explain in terms I will understand why this equation is considered so perfect and beautiful?
  2. jcsd
  3. Dec 11, 2008 #2


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    What is so beautiful about the Mona Lisa?

    Euler's equation, [itex]e^{i\pi}+ 1= 0[/itex], which can also be written [itex]e^{i\pi}= -1, combines five fundamental constants, 0, 1 (or -1), e, i, and [itex]\pi[/itex] into a single, simple, equation. Simplicity and depth make for beauty.
    Last edited by a moderator: Dec 13, 2008
  4. Dec 11, 2008 #3


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    Error! Should be + 1 = 0, not -1.
  5. Dec 11, 2008 #4
    'Tis true... both of the above.
  6. Dec 12, 2008 #5


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    I think that the "beauty" is in the fact that the constants are from very different branches of mathematics. 0, 1 and i are from algebra, e is from calculus/analysis, and [itex]\pi[/itex] is from geometry.
  7. Dec 12, 2008 #6


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    And in combining those fundamental constants it uses each of the 4 fundamental math operations: Addition, multiplication, exponentiation and equality.

    All to arrive at a result that seems impossible.

    How can that be anything but beautiful?
  8. Dec 12, 2008 #7
    Why does no one mention the i, is there nothing special about imaginary numbers or something?
  9. Dec 15, 2008 #8
    Am I the only one who isn't in awe of this equation?

    When I first saw it, it seemed random and just didn't make any sense, like those infinite sum formulas of Ramanujan (...one over pi equals WHAT?). But then after I studied complex analysis, and the more I learn in math, the more pedestrian and booring it becomes. It seems to just be a random consequence of much bigger ideas, and it doesn't lead to any insights by itself.

    I've thought about this a few times and tried to "see the beauty" but as far as I can tell all the awe is based purely on shock value and nothing deeper.
  10. Dec 15, 2008 #9
    It's easy to remember and makes a lot of otherwise tough math easy.
  11. Dec 15, 2008 #10


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    why so hung upon the word beautiful? try unbelievable, or wacky, or unexpected, or sexy, or what ever, but at least it ain't boring.
  12. Dec 16, 2008 #11
    I've never seen an equation that put me in "awe", but this is a pretty cool one. And it only gets better when you find uses for it.
  13. Dec 16, 2008 #12
    I've never found anything in mathematics to be beautiful. The concept of beauty in mathematics traces back to Hardy's A Mathematician's Apology and is based on a more-or-less Late 19th/Early 20th Century sense of aesthetics.

    Nevertheless, this equation has always intrigued me, since it gives us a sneak peek into the structural integrity of Mathematics as an academic discipline.
  14. Dec 17, 2008 #13
    I always don`t get it. To me, if it can solve problems, and extend new ideas, then i like it. I don` t bother with 'beauty'.
  15. Dec 17, 2008 #14
    If you ask me, something is beautiful when it's stimulating and seemingly simple (women excluded of course! eyo!!)
    Euler's identity is, to me, a 7. [tex]A\; =\; \pi r^{2}[/tex] is like a 3. [tex]e=mc^{2}[/tex] is about a 9. The Lorentz factor is a perfect 10, if you ask me. =]
    Simplified complexity.... mmmm....
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