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 = \cos x + i\sin x $
Then,
$ \frac{dz}{dt} = -\sin x + i\cos x $
Also true is the statement,
$ iz = -\sin x + i\cos x $
Therefore,
$ \frac{dz}{dt} = iz $
and
$ \frac{1}{z} dz = i dt $
Integrate both sides,
$ \int \frac{1}{z} dz = \int i dt $
we get,
$ \ln z = it $
and
$ z = e^{it} + c $
since $ c = 0 $
$ z = e^{it} $
therefore,
$ e^{it} = \cos x + i\sin x $
Alternatively, proof using Taylor series is posted 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?
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