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
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions