# Condition for Incommensurability of Roots of Quadratic Equation

## Theorem

$(1): \quad a x - x^2 = \dfrac {b^2} 4$

Then $x$ and $a - x$ are incommensurable if and only if $\sqrt {a^2 - b^2}$ and $a$ are incommensurable.

In the words of Euclid:

If there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are incommensurable, the square on the greater will be greater than the square on the less by the square on a straight line incommensurable with the greater.

And, if the square on the greater be greater than the square on the less by the square on a straight line incommensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it divides it into parts which are incommensurable.

## Proof

We have that:

 $\ds x \paren {a - x} + \paren {\frac a 2 - x}^2$ $=$ $\ds a x - x^2 + \frac {a^2} 4 - 2 \frac a 2 x + x^2$ $\ds$ $=$ $\ds \frac {a^2} 4$ simplifying $\ds \leadsto \ \$ $\ds 4 x \paren {a - x} + 4 \paren {\frac a 2 - x}^2$ $=$ $\ds a^2$ $\ds \leadsto \ \$ $\ds b^2 + \paren {a - 2 x}^2$ $=$ $\ds a^2$ from $(1)$ $\ds \leadsto \ \$ $\ds a^2 - b^2$ $=$ $\ds \paren {a - 2 x}^2$ $\ds \leadsto \ \$ $\ds \sqrt {a^2 - b^2}$ $=$ $\ds a - 2 x$

Let:

$a \smile b$ denote that $a$ is incommensurable with $b$
$a \frown b$ denote that $a$ is commensurable with $b$.

### Necessary Condition

Let $\paren {a - x} \smile x$.

$a \smile x$
$x \frown 2 x$

So:

 $\ds a$ $\smile$ $\ds 2 x$ Commensurable Magnitudes are Incommensurable with Same Magnitude $\ds$ $\smile$ $\ds \paren {a - 2 x}$ Incommensurability of Sum of Incommensurable Magnitudes $\ds$ $\smile$ $\ds \sqrt {a^2 - b^2}$

$\Box$

### Sufficient Condition

Let $a \smile \sqrt {a^2 - b^2}$.

Then:

 $\ds a$ $\smile$ $\ds \sqrt {a^2 - b^2}$ $\ds$ $\smile$ $\ds \paren {a - 2 x}$ $\ds$ $\smile$ $\ds 2 x$ Incommensurability of Sum of Incommensurable Magnitudes
$x \frown 2 x$
$a \smile x$
$\paren {a - x} \smile x$

$\blacksquare$

## Historical Note

This proof is Proposition $18$ of Book $\text{X}$ of Euclid's The Elements.