Sunday, May 10, 2009

My own cranky beliefs

I've been enjoying and the wacky people profiled within. Except that yesterday I noticed that I have a cranky belief myself. Ooh, the hypocrisy, it burns.
I reject Euler's equation, which I will explain below.
e^xi=cos x + i sin x
So if you take Euler's own constant, e, and raise it to a power involving "i," which is the square root of negative one, you get a complex number based on the variable "x." This of course gives you every mathematician's favorite equation ever:
e^pi*i + 0 =1
Since this equation connects five important mathematical constants, so many mathematicians proclaim this the most beautiful equation ever invented.
Mathematical beauty is another idea that I reject. To be rude about it, it's pretty much jerking off over numbers. Oh god, yes, pi and e, so beautiful! Wank, wank, wank. My interest in math is mostly about correct measurements. Poets may claim that beauty is truth, but I expect my numbers to represent something in the physical world, and base my assumptions of their correctness on the correspondence to fact. And besides, the truth is often hideous. People are inhumanly cruel to each other, the universe doesn't care if you live or die, people have done monstrous things to each other for no good reason, and everything in the universe, you included, will eventually die. The universe itself will eventually die.
The "Beautiful equation" revolves around the cosine and sine functions. Cosine and sine are trigonometric ratios first found in triangles. In a right triangle, cosine is the ratio of the side adjacent to the angle to the hypotenuse (the long side). Sine is the ratio of the side opposite to the angle to the hypotenuse. The ratios are the same for any particular angle. Later, the metaphor was extended to circles, allowing for angles greater than 90 degrees.
In high end math, radians are used instead of degrees. In radians, the circle is divided into 2*pi sections. So 2*pi is the entire circle, pi is half the circle, and 1/2 pi is one quarter of the circle. Radians are dimensionless.
Remember Euler's equation?
e^xi=cos x + i sin x
Plugging in pi would result in the complex number -1 + 0i. Anything times 0 is 0, removing the imaginary part.
However, from goofing off with a calculator, I know that without the imaginary part, e to the power pi is slightly greater than 23. (It's irrational.) Throwing in imaginary numbers seems to be causing an abrupt, inexplicable change.
In fact, raising numbers to an imaginary exponent is so poorly defined that neither calculator nor amateur mathematician can explain how to do it. Natural numbered exponents are "Multiplying a number by itself that many times." a^2 = a*a, a^3 = a*a*a, and so on. Fractional exponents involve roots. a^1.5 is a times the square root of a.
So if "i" is the square root of -1, what would a^i indicate? Would it's result be real, like a, imaginary like i, complex, or something completely different?
Also, I would expect "e^xi" to be linear as x. As x increases, larger and larger results should return. If not larger along the real axis, than in total distance from the origin. However, Euler's equation is cyclical. Sine and Cosine go around in a circle, first increasing, then decreasing, then increasing again, back and forth forever. There should be a reason for this discrepancy.
Mathematicians often have a strange relationship with their numbers, and grow quite attached. When a Greek mathematician discovered irrational numbers, he was thrown off a boat for ruining their sense of mathematical order. That the other mathematicians could not disprove the existence of irrational numbers drove them absolutely crazy. It was proof that mathematics was not the pure orderly truth that they thought it was, and that it also involved messy generalities, just like real life.
I likewise suspect that Euler put together his equation on the grounds that he liked its structure, not because it actually corresponded to anything. Moreso, modern mathematicians love the idea that many of their fundamental constants are closely related. I kind of hope that I'm the one who's wrong about this, because I could take being wrong way more than they could.
I would like any commentators to describe what I got wrong, or describe a proof. A graphical proof would be the easiest to understand, but those are the hardest to construct in the first place. If someone does prove me wrong, I will add the "Stupid" tag to this post and describe their proof.
EDIT: Dr. Phillip Spencer of the university of Toronto's math department proves Euler's equation based on the Taylor series, and therefore this entire post is wrong. Having proved Euler's equation, all else follows. Well done, Dr. Spencer.

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