Quotient of Rational Numbers is Rational

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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

Euclid-X-20.png

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


Sources