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Revision as of 10:51, 15 December 2011
On The Most Beautiful Equatin
$ e^{ix} = \cos x + i * \sin x $
I am happy to have met Euler’s Identity outside of my college education. I was reading William Dunham’s The Mathematical Universe – a good read for anyone - and in its concluding chapter Z, Professor Dunham presented the Euler’s Identity. The effect was strong- I had read how mathematicians had labored to estimate the value of pi, the unnaturally frequent occurrence of the natural number e in our universe, the great baffle and nervous air among mathematical circle in formulating the notion of i, and here this superstar cast is thrown into a simple but complete scene with no other than the brilliant Euler as its playwright.
It was pleasant to learn of the equation in this manner, I was even more pleased to learn that Euler’s Identity is a strong candidate for the crown of most beautiful mathematical equation (the competition is Maxwell’s Equation). By this time, I was informed that Euler’s Identity also casts two fundamental constant 0 (additive identity) and 1 (multiplicative identity) and also position the cast with three basic arithmetic operations: addition, multiplication, and exponentiation.
I tried sharing the same enthusiasm I had with my mother, but she was not too impressed I think. Perhaps the beauty of the Euler’s Identity is best capture in a form of poetry (haiku, specifically):
e to the i pi
Add one and you get zero
Is that weird or what? –[spdqbr]
Yes, it is the strangest connection among the immensely important mathematical constants, peaceful coexistence of three branches of mathematics - calculus (e), algebra (0, 1, i), and geometry ($ \pi $). I’ve found very simple derivation of Euler’s equation from a blog I cannot locate anymore. It only requires elementary calculus skills:
$ Suppose, z Alternatively, proof using Taylor series is posted [[HW1.3_Chris_Cadwallader_-_Euler's_forumla_ECE301Fall2008mboutin|here]] This is certainly not the first time in which mathematics was able to draw unforeseen conclusion. For example, the number of primes up to a postivie number n approaches n/ln(n) as n approaches infinity, area under f(t), derivative of F(t), can be calculated by F(t) (well, this connection may seem rather unconnected to me for I haven’t given rigorous thought about it). Sometimes, I do wonder what the utility of that beautiful equation is any way, like an engineer. But if that isn’t interesting, what is? [[Honors_Project|Back to Daniel's Honor Project]] [[2011_Fall_MA_265_Walther|Back to MA265 Fall 2011 Prof. Walther]] [[MA265|Back to MA265]] $
