Category:Examples of Apotome

This category contains examples of Apotome.

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a > b$.

Then $a - b$ is an apotome if and only if:

$(1): \quad \dfrac a b \notin \Q$

$(2): \quad \paren {\dfrac a b}^2 \in \Q$

where $\Q$ denotes the set of rational numbers.

In the words of Euclid:

If from a rational straight line there be subtracted a rational straight line commensurable with the whole in square only, the remainder is irrational; and let it be called an apotome.

(The Elements: Book $\text{X}$: Proposition $73$)