Definition:Apotome/Warning

Definition
It is a mistake to suggest that $a$ and $b$ may be any real numbers such that $\dfrac a b \notin \Q$ and $\paren {\dfrac a b}^2 \in \Q$.

For example, let $a = \pi \sqrt 2$ and $b = \pi$.

Then we see that:
 * $(1): \quad \dfrac a b = \sqrt 2 \notin \Q$
 * $(2): \quad \paren {\dfrac a b}^2 = 2 \in \Q$

But $a - b = \pi \paren {\sqrt 2 - 1}$ is not an apotome.