Rational Multiplication Distributes over Addition

From ProofWiki
Jump to navigation Jump to search

Theorem

The operation of multiplication on the set of rational numbers $\Q$ is distributive over addition:

$\forall x, y, z \in \Q: x \times \left({y + z}\right) = \left({x \times y}\right) + \left({x \times z}\right)$
$\forall x, y, z \in \Q: \left({y + z}\right) \times x = \left({y \times x}\right) + \left({z \times x}\right)$


Proof

Follows directly from the definition of rational numbers as the quotient field of the integral domain $\left({\Z, +, \times}\right)$ of integers.

So $\left({\Q, +, \times}\right)$ is a field, and therefore a priori $\times$ is distributive over $+$ on $\Q$.

$\blacksquare$