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
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $1$. Scalar and Vector Quantities