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Richard Feynman is a NobelPrizeWinner? in PhysICs? (quantum electrodynamics, 1950s?). He found this formula funny because it links all the main constants a human being is exposed to in this world. Zero and unity arise kinda naturally: one is how one starts to count, and zero comes later... when one does not want to :). pi is a constant related to our world being Euclidean (otherwise, the ratio of the length of a circumference to its diameter would not be a universal constant, i.e. the same for all circumferences). The e constant is related to the speed of change, or growth, or whatever like that, as the solution to the simplest growth equation dy/dx=y is y=e^x. Finally, i is the concept introduced mathematically to have a nice property that all polynomials of degree n have exactly n roots in the complex plane. So, quite a lot of rather deep concepts are interrelated within this formula. Of course, there is a number of other ways to arrive to any of those numbers... which only underlines their fundamentality :). |
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Richard Feynman is a NobelPrize winner in PhysICs? (QuantumElectrodynamics?, 1950s?). He found this formula funny because it links all the main constants a human being is exposed to in this world. Zero and unity arise kinda naturally: one is how one starts to count, and zero comes later... when one does not want to :). pi is a constant related to our world being Euclidean (otherwise, the ratio of the length of a circumference to its diameter would not be a universal constant, i.e. the same for all circumferences). The e constant is related to the speed of change, or growth, or whatever like that, as the solution to the simplest growth equation dy/dx=y is y=e^x. Finally, i is the concept introduced mathematically to have a nice property that all polynomials of degree n have exactly n roots in the complex plane. So, quite a lot of rather deep concepts are interrelated within this formula. Of course, there is a number of other ways to arrive to any of those numbers... which only underlines their fundamentality :). |
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TheMostRemarkableFormulaInTheWorld is an example of Euler’s Theorem from Complex Analysis. |
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TheMostRemarkableFormulaInTheWorld is an example of EuleR s Theorem from Complex Analysis. |
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Euler’s Theorem states that: |
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EuleR s Theorem states that: |
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Then, since cos(pi) = 1 and sin(pi) = 0, |
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Then, since cos(pi) = -1 and sin(pi) = 0, |
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The proof of Euler’s Theorem involves the definition of "e," by a Taylor’s Series Expansion of e^z, where z is a complex number, DeMoivre?’s Formula, and the Taylor’s Series Expansion of the sine and cosine functions. |
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The proof of Euler's Theorem involves the definition of "e," by a TaylorsSeriesExpansion? of e^z, where z is a complex number, DeMoivresFormula?, and the Taylor's Series Expansion of the sine and cosine functions. |
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Despite, this last remark, Euler's Theorem is considered a direct consequence of the formualtion of e^z, where z is a complex number. |
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Despite, this last remark, Euler's Theorem is considered a direct consequence of the extension of the definition of the function e over the ComplexNumbers?. |
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