Al6200 2008-08-22 20:13:13 (Updated 2008-10-01 22:33:06)

Hmmm... Yes. This is important for electrical engineering.

The power that comes out of ordinary wall sockets is called "AC current". In other words, the voltage that it provides is a cosine curve: cos(Bx + C), where B and C are constants (which I think depend on what country you're in).

By Euler's Identity, you can represent the voltage curve as the real part of e^(ix). So:

Real[ e^( (a +bi) x) ] = cos(Bx + C)

So you end up representing a voltage source as a complex number: a + bi, which corresponds to a certain cosine curve.

This is INCREDIBLY useful, because you can use a concept called "Impedence", which follows Ohm's law. For some impedance Z, a current I, and a voltage V:

V = I * Z

A resistor has a real positive impedance (meaning that it flattens the cosine curve, but does not shift it). A capacitor has a negative imaginary impedence, and an inductor has a positive imaginary impedence.

So if you have a voltage source V = 3 + 5i, and an inductor of 5 Henries, then the current that flows through the circuit will be I = (3 + 5i)/(5wi) where w is a constant. This is really quite awesome.

Al6200 2008-09-07 00:23:13

Well, if you write

e^(Pi * i), you get -1.

So basically if you put 3 of the weirdest and most fascinating numbers that exist into one equation, you get out...

negative one, the most dull and boring number out there...

Al6200 2008-09-07 17:35:38

No, but it is a beautiful relationship that one would not expect to exist.

Al6200 2008-09-11 18:30:30

Or is it i that is the weird number?

Al6200 2008-09-23 20:46:03

Indeed.

Al6200 2008-10-06 17:00:00

Yes, I'm in Calc 4 now.

Four dimensional volumes. Not fun.