Although I had originally found them in Book X of Euclid's Elements, there is a summary of the definition of apotome on the web. After sifting through the definitions there are 4 conditions Euclid checks:
only $a \in \mathbb{Q}$ or only $b \in \mathbb{Q}$ or $a,b \notin \mathbb{Q}$
$\frac{a}{b} \notin \mathbb{Q}$ but $(\frac{a}{b})^2 \in \mathbb{Q}$ (always)
$ \frac{\sqrt{a^2 - b^2}}{a} \in \mathbb{Q}$ or $ \frac{\sqrt{a^2 - b^2}}{a} \notin \mathbb{Q}$ WHY IS THIS CONDITION SO IMPORTANT??
Between the first and third items there are six possibilities. Here are examples of each kind:
- $9 - \sqrt{17}$
- $2 \sqrt{3}- 3$
- $\color{lightgray}{\sqrt{11} - \sqrt{\frac{143}{49}}}$
- $3 - \sqrt{2}$
- $\sqrt{13}-3$
- $\sqrt{7} - \sqrt{5}$
Related question Is there example of 3rd apotome of the form $\sqrt{m} - \sqrt{n}$ with $m, n \in \mathbb{Z}$ ?
$\boxed{3\sqrt{2}-10}$ any others??