eiπ + 1 = 0
Euler's Identity. 'tis a thing of pure beauty.
Three very suspicious numbers in a menage-trois, creating something real. How can two irrational numbers and an imaginary number work together to make a very real integer? It boggles the mind entirely. Somewhere in a past left behind, this was the first equation to make me sit up and consider imaginary numbers as something more than a trick.
The standard wuss way of explaining this (as happened to me) was that of pre-cooked trigonometry.
eiϑ = cos(ϑ) + i sin(ϑ)
But that is just a completely arbitrary equation, when you really think about it. And I'm an incorrigible skeptic. But that's where that lesson ended and Math is not taught as much as lectured on. But somewhere during my engineering, I learnt about the Taylor series, for approximating sine and cosine values. Except, it's not really an approximation, but an infinite series and the partial sum, is used for approximation.
ϑ3 ϑ5 ϑ7
sin(ϑ) = ϑ - — + — - — + ...
3! 5! 7!
ϑ2 ϑ4 ϑ6
cos(ϑ) = 1 - — + — - — + ...
2! 4! 6!
Remember, that works on radians, not regular 'ol degrees. So ironically, when you throw the magic number in there and spend an eternity calculating it, the sin(π) works out to be one huge zero. And it has to, because looking at it from pure geometry and sine as a pure fraction.
Now, I never understood how an infinite number of operations could ever result in a finite number. Well, it's the ghost of something familiar - Zeno's Paradox. And well, Archimedes debunked it, way before I could even attempt it.
Now if you shift a little from geometry of lengths into the world of co-ordinate geometry, you suddenly realize that imaginary math is literally co-ordinate geometry in disguise, except with imaginary numbers (woooo ...). Pull that very Eucledian right triangle into a unit circle on the imaginary plane, the boundaries between the disciplines start to disappear.
The imaginary pixie dust sprinkled on e results in another taylor series expansion, which ironically just shows to go how you can really go mad learning mathematics. Now, the taylor series expansion for just plain 'ol ex goes like this.
x2 x3 x4
ex = 1 + x + — + — + — + ...
2! 3! 4!
Now, here's the clincher. If x just happened to be i, the alternate coefficents would be negative. Oh, yes ... that's pure imaginary pixie dust, but once you get hooked on it, there's no getting off it :)
Now, we get to the final and crucial equation all over again.
eiπ = cos(π) + i sin(&pi); eiπ = -1 + 0 i eiπ + 1 = 0
Time to run out on the streets and yell out that ... "they were right, eiπ is REAL!".
--That's not right! Heck, that's not even wrong!
-- Wolfgang Pauli