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.