e^(i*pi) + 1 = 0.
where e is the base of the natural logarithms, i is the square-root of -1 (imaginary numbers), and pi is the ratio of the circumferance of a circle with its diameter.
What does it mean--what does it show--why is it remarkable? (Not sure what I'm asking here, but you probably do.)
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 :).
TheMostRemarkableFormulaInTheWorld is an example of EuleR s Theorem from Complex Analysis.
EuleR s Theorem states that:
e^(ib) = cos(b) + i * sin(b) where b is a real number.
So, if b = pi, then e^(i*pi) = cos(pi) + i * sin(pi).
Then, since cos(pi) = -1 and sin(pi) = 0,
e^(i*pi) = - 1, and e^(i*pi) + 1 = 0.
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.
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?.