Product of two complex numbers = 0

[vesu]

Actually, it isn't, which is Whovian's point. Since we're trying to prove that there are no zero divisors, so we can't assume that just because e^iθ≠0, then ze^iθ=0 implies z=0 because e^iθ might be a zero divisor. You should show how to get rid of the complex exponential, e.g. take the modulus or multiply by e^-iθ.

Now I agree that the OP was likely using circular reasoning while the rest of us glossed over it, thinking "Obviously you can get rid of the exponential." It was a good catch by Whovian.

so

$zw = 0$
$re^{i\theta} \times qe^{i\phi} = 0$
$rqe^{i(\theta + \phi)} = 0$
$e^{i(\theta + \phi)} \not= 0$
$\frac{rqe^{i(\theta + \phi)}}{e^{i(\theta + \phi)}} = \frac{0}{e^{i(\theta + \phi)}}$
$rq = 0$
$\therefore r=0$ or $q=0$
$\therefore z = 0$ or $w = 0$

Um, I can't find anything at all in our course material about taking the modulus of a complex exponential. According to WolframAlpha $|e^{i\theta}| = 1$ assuming $\theta$ is real? So, uh...

$zw = 0$
$re^{i\theta} \times qe^{i\phi} = 0$
$rqe^{i(\theta + \phi)} = 0$
$|rqe^{i(\theta + \phi)}| = |0|$
$rq = 0$
$\therefore r=0$ or $q=0$
$\therefore z = 0$ or $w = 0$

Both of these seem okay I think?

 

[clamtrox]

According to WolframAlpha $|e^{i\theta}| = 1$ assuming $\theta$ is real?

True, but you can see fairly easily again that this claim is equivalent with claiming that |zw|= |z| |w|. Maybe you can somehow convince yourself (without resorting to wolframalpha) that this is true?

 

[cepheid]

Actually, it isn't, which is Whovian's point. Since we're trying to prove that there are no zero divisors, so we can't assume that just because e^iθ≠0, then ze^iθ=0 implies z=0 because e^iθ might be a zero divisor. You should show how to get rid of the complex exponential, e.g. take the modulus or multiply by e^-iθ.

Now I agree that the OP was likely using circular reasoning while the rest of us glossed over it, thinking "Obviously you can get rid of the exponential." It was a good catch by Whovian.

What is a "zero divisor?"

 

[clamtrox]

What is a "zero divisor?"

http://en.wikipedia.org/wiki/Zero_divisor

 

[cepheid]

http://en.wikipedia.org/wiki/Zero_divisor

Ah ok. So we can't assume a priori that the set of complex numbers possesses the property that the product of two of its non-zero elements must be non-zero.

I had no idea, to be honest.