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Suppose that $z_{1}$ and $z_{2}$ are complex numbers. What can be said about $z_{1}$ or $z_{2}$ if $z_{1}z_{2}=0$?

3 Answers 3

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Assume that $z_1, z_2 \in \mathbb{C}$ and that $z_1 z_2 = 0$. Now if $z_1 \neq 0$, then $\frac{1}{z_1} \in \mathbb{C}$. Thus $0 = \frac{1}{z_1}\cdot 0 = \frac{1}{z_1}(z_1 z_2) = (\frac{1}{z_1}z_1) z_2 = 1 \cdot z_2 = z_2$ and hence $z_2 = 0$. So either $z_1 = 0$ or $z_2 = 0$ (or both).

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Notice that $0=|0|=|z_1 z_2 | = |z_1| |z_2|,$ and because $|z_1|$ and $|z_2|$ are both real numbers, either $|z_1|=0$ or $|z_2|=0$.

But then either $z_1=0$ or $z_2=0.$

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    You reduced a problem concerning $\mathbb C$ to a problem concerning $\mathbb R$. It might still be a problem for the OP. – 2017-02-23
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If $z_{1}z_{2}=0$ then we have $z_1=0$ or $z_2=0$.

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    @Havalhassan It's a property of rings. – 2017-02-23
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    $\mathbb C$ is a field and fields are zero-divisor free. – 2017-02-23
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    @rubik it is a property if integral domains, or fields, not of rings. – 2017-02-23
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    @Improve You're right. It would have been better to say "of some rings", i.e. integral domains. – 2017-02-23