Al6200 Oct 1, 2008 (Updated Feb 13, 2009)

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 Oct 1, 2008

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 Oct 1, 2008

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

Al6200 Oct 1, 2008

Or is it i that is the weird number?

Al6200 Oct 1, 2008

Indeed.

Al6200 Oct 7, 2008

Yes, I'm in Calc 4 now.

Four dimensional volumes. Not fun.