eiπ + 1 = 0
Not a definition at all, but rather a proof of the fundamental interconnectedness of all things and the unreasonable effectiveness of mathematics, this is something that still blows my mind. After I read it in a book, I told it to my high school math teacher, but she didn't believe me. It blew my father's mind when I told it to him, but of course he demanded proof. I couldn't pull that off until college, but when I could, this is the proof that I gave him, and will now give to you. Keep in mind that this uses calculus, so if you haven't had any exposure to Taylor Series, this won't mean much to you. Alright, let's begin...
First, some notation: ! means "factorial".
x! = "x factorial" = 1 * 2 * 3 * 4 * ... * (x-1) * x
4! = "4 factorial"
For the real proof, you have to remember that notation, and you have to remember Taylor Expansions. It's all about Taylor Expansions. (Actually, these are special-case Taylor series, and are called Maclaurin Series
, but only hard-core math geek
s care about the difference) Taylor expansions use derivative
s around f(0) and polynomial
s to approximate a function. As they use more derivatives ("as n goes to infinity
") they become exact approximations. When I put ... below, it means "this series continues on to infinity
Here is the Taylor expansion of sin(x):
sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - ...
And here is cos
cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - ...
And here is the Taylor expansion of ex
ex = 1 + x/1! + x^2/2! + x^3/3! + x^4/4! + ...
Now let's look at the taylor expansion of eix
, while remembering the fact that i2
e^(ix) = 1 + ix/1! + (ix)^2/2! + (ix)^3/3! + (ix)^4/4! + ...
= 1 + ix/1! + i^2*x^2/2! + i^3*x^3/3! + i^4*x^4/4! + ...
= 1 + ix + -1*x^2/2! + -1*i*x^3/3! + x^4/4! + ...
= 1 + ix - x^2/2! - ix^3/3! + x^4/4! + ix^5/5! - x^6/6! + ...
Now, notice that this is just the Taylor expansion of cos(x) plus i times the Taylor expansion of sin(x) (this fact is the amazing one!). Therefore:
e^(ix) = cos(x) + i*sin(x)
From there it is a short step to:
e^(i*pi) = cos(pi) + i*sin(pi)
= -1 + i*0
Proving, for once and for all, that:
e^(i*pi) = -1
e^(i*pi) + 1 = 0
Whoah. This last equation is definitely the preferred form, because it includes all of the really important numbers: 1, e, π, and 0. The fact that this is even true at all has led to people referring to it as god's equation
, and has led many a good mathematician astray into fields of philosophical wonder.
vruba says Re Euler's identity: he was a Swiss math guy named "Euler", you big silly.