Definition:Apotome/Third Apotome

Definition
Let $a$ and $b$ be two positive real numbers, such that $a - b$ is an apotome.

Then $a - b$ is a third apotome iff $a \not \in \Q$ and $b \not \in \Q$ and $\displaystyle \frac {\sqrt {a^2 - b^2}} a \in \Q$, where $\Q$ denotes the set of all rational numbers.



Example
When $a = \sqrt {11}$ and $b = \sqrt {\frac {143} {49}}$,

Therefore $\sqrt {11} - \sqrt {\frac {143} {49}}$ is a third apotome.