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