Zero Product of Numbers implies Factors are Zero

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Theorem

On all the number systems:

natural numbers $\N$
integers $\Z$
rational numbers $\Q$
real numbers $\R$
complex numbers $\C$

the following holds.

Let $a \times b = 0$.

Then either $a = 0$ or $b = 0$.


Proof

From Natural Numbers have No Proper Zero Divisors

$\forall a, b \in \N: a \times b = 0 \implies a = 0 \text { or } b = 0$

We have:

Integers form Integral Domain
Rational Numbers form Integral Domain
Real Numbers form Integral Domain
Complex Numbers form Integral Domain

Hence by definition of integral domain:

$a \times b = 0 \implies a = 0 \text { or } b = 0$

where $a, b \in \Z, \Q, \R, \C$.

$\blacksquare$


Sources