Binomial is Irrational
Theorem
In the words of Euclid:
- If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be called binomial.
(The Elements: Book $\text{X}$: Proposition $36$)
Proof
Let $AB$ and $BC$ be two rational straight lines which are commensurable in square only.
By definition, $AB$ and $BC$ are incommensurable in length.
We have:
- $AB : BC = AB \cdot BC : BC^2$
Therefore from Proposition $11$ of Book $\text{X} $: Commensurability of Elements of Proportional Magnitudes:
- $AB \cdot BC$ is incommensurable with $BC^2$.
But by Proposition $6$ of Book $\text{X} $: Magnitudes with Rational Ratio are Commensurable:
- $2 AB \cdot BC$ is commensurable with $AB \cdot BC$.
We have that $AB$ and $BC$ are commensurable in square.
So from Proposition $15$ of Book $\text{X} $: Commensurability of Sum of Commensurable Magnitudes:
- $AB^2 + BC^2$ is commensurable with $BC^2$.
- $2 AB \cdot BC$ is incommensurable with $AB^2 + BC^2$.
Thus from Proposition $4$ of Book $\text{II} $: Square of Sum:
- $AC^2 = \left({AB + BC}\right)^2 = 2 AB \cdot BC + AB^2 + BC^2$ is incommensurable with $AB^2 + BC^2$.
But $AB^2 + BC^2$ is rational.
Therefore $AC^2$ is irrational.
Thus by Book $\text{X}$ Definition $4$: Rational Area:
- $AC$ is irrational.
Such a straight line is called binomial.
$\blacksquare$
Historical Note
This proof is Proposition $36$ 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