quote:

Originally posted by Mister_Michel Now my question is: What are you in life by knowing this?

a knob.

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quote:

Originally posted by DigiNut Anyway, complex numbers have very real meaning, you just don't learn it in high school. They're as fundamental to signal processing as arithmetic is to your high school courses.

from my understanding, they are often used only to avoid having to solve differential equations though.

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quote:

Originally posted by Mister_Michel Now my question is: What are you in life by knowing this? (been asking myself that question for many years now, still no answer )

you are someone who knows the answer to this. If you know and understand the answer then it shows a little bit more of your reasoning ability.

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I see your 4 Crushs and raise you 3 As The Rush Comes. - Yan from PvD's first summerstage event in '03

Actually, Zarathustra, just a little technicality...

You're wrong.

The square root of any complex number (including the real set) is a one-to-one mapping. For example, the square root of 4 is only 2, not {2,-2}. Otherwise you wouldn't be able to distinguish -(4^0.5) from +(4^0.5). If this confuses you, let me know....I'll try to explain it differently :P

--HyprLogik

quote:

Originally posted by HyprLogik Actually, Zarathustra, just a little technicality... You're wrong. The square root of any complex number (including the real set) is a one-to-one mapping. For example, the square root of 4 is only 2, not {2,-2}. Otherwise you wouldn't be able to distinguish -(4^0.5) from +(4^0.5). If this confuses you, let me know....I'll try to explain it differently :P --HyprLogik

What are you talking about? This is absolutely wrong. If my memory of my complex numbers course serves me correctly, it can actually be shown that the nth root of any complex number actually has n values, not 1.

When you write a complex number Z as Ae^ix, which is actually Ay corresponds to (A^y)e^(ixy). The value of A (magnitude) is raised to the power y as with real arithmetic, and the angle is multiplied by y.

But neither the original angle x nor the resulting angle xy are unique. Since we're dealing with polar coordinates, x = arg(z) = arg(z) + 2*pi*k, where k is any integer value including zero. So when you multiply your angle by 2 - say it's 45° (pi/4) in the context of the original question - you end up with 90° (pi/2). Simple right? But if your original angle was 225° (5*pi/4), you end up with 450° (5*pi/2). 5*pi/2 is the same angle as pi/2, and your angle is only defined from (-pi,pi] so it must be written as pi/2.

And like I said, you can extend it to any nth root. If you take a 60° (pi/3) angle and take its cube to get a 180° (pi) value, you don't know whether the original angle was 60°, 180°, or -60°.

What you've stated here:

quote:

Otherwise you wouldn't be able to distinguish -(4^0.5) from +(4^0.5).

is simply not correct. When you raise a complex number to the power 0.5, you are by definition multiplying the angle by 0.5. To put it another way, when you use this operation on a real number, you are actually by definition taking the positive square root.

Mathematically, x^0.5 = |sqrt(x)|. Therefore, -(x^0.5) = -|sqrt(x)|. The former must be positive, the latter must be negative, assuming x is positive and real. There is no discrepancy here, it's simply a misunderstanding on your part of the difference between sqrt(x) and x^0.5.

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