Condition for Incommensurability of Roots of Quadratic Equation

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Theorem

Consider the quadratic equation:

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


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.

(The Elements: Book $\text{X}$: Proposition $18$)


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

From Incommensurability of Sum of Incommensurable Magnitudes:

$a \smile x$

From Magnitudes with Rational Ratio are Commensurable:

$x \frown 2 x$

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


From Magnitudes with Rational Ratio are Commensurable:

$x \frown 2 x$

From Commensurable Magnitudes are Incommensurable with Same Magnitude:

$a \smile x$

From Incommensurability of Sum of Incommensurable Magnitudes:

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

$\blacksquare$


Historical Note

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


Sources