Product of Rationally Expressible Numbers is Rational

Proof

 * Euclid-X-19.png

Let the rectangle $AC$ be contained by the rational straight lines $AB$ and $BC$.

Let $AB$ and $BC$ be commensurable in length.

Let the square $AD$ be described on $AB$.

From, $AD$ is rational.

Since:
 * $AB$ is commensurable in length with $BC$

and:
 * $AB = BD$

it follows that
 * $BD$ is commensurable in length with $BC$.

From Areas of Triangles and Parallelograms Proportional to Base:
 * $BD : BC = DA : AC$

Therefore $DA$ is commensurable in length with $AC$.

But $DA$ is rational.

It follows from that $AC$ is also rational.