# Quotient of Rationally Expressible Numbers is Rational

## Theorem

In the words of Euclid:

*If a rational area be applied to a rational straight line, it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied.*

(*The Elements*: Book $\text{X}$: Proposition $20$)

## Proof

Let the rational area $AC$ be applied to the rational straight line $AB$ such that $BC$ is its breadth.

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

From Book $\text{X}$ Definition $4$: Rational Area, $AD$ is rational.

But $AC$ is also rational.

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

From Areas of Triangles and Parallelograms Proportional to Base:

- $BD : BC = DA : AC$

Therefore from Commensurability of Elements of Proportional Magnitudes $DB$ is also commensurable in length with $BC$.

As $DB = BA$ it follows that $AB$ is also commensurable in length with $BC$.

But $AB$ is rational.

Therefore $BC$ is rational and commensurable in length with $AB$.

$\blacksquare$

## Historical Note

This proof is Proposition $20$ of Book $\text{X}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions