# 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