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Is it possible for the product of 2 non-zero complex numbers to be 0?

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    $0$ is complex. – 2017-02-20
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    If neither of the numbers is $0$, then no. The first thought that comes to mind is that the magnitude of the product is the product of the magnitudes, which is nonzero. – 2017-02-20
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    @ElliotG thanks for pointing that out. I revised my question – 2017-02-20

3 Answers 3

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No, it is not possible.

For complex numbers $z_1,z_2 \in \mathbb{C}$ we have that $$z_1\cdot z_2 = 0 \quad\implies\quad z_1 = 0\quad\text{ or }\quad z_2 = 0.$$

As a hint: non-zero complex numbers have multiplicative inverses.

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Let $z_1 = r_1e^{i\theta_1}$ and $z_2 = r_2e^{i\theta_2}$ then $z_1z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$. Now $\exp(.)$ is never $0$ so for $z_1z_2$ to be $0$ we need one of $r_1, r_2$ to be $0$. But this immediately implies that one of $z_1,z_2$ have to be $0$. So no.

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It's not possible. the product of 2 complex number can equal 0 if and only if one of the numbers is zero. i.e a.b = 0 where a,b are complex numbers Then either a or b is 0

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    The question is why – 2017-03-02