Definition:Apotome/Third Apotome

Definition
Let $a, b \in \R_{>0}$ be (strictly) positive real numbers such that $a - b$ is an apotome.

Then $a - b$ is a third apotome iff $a \notin \Q$ and $b \notin \Q$ and $\dfrac {\sqrt {a^2 - b^2}} a \in \Q$, where $\Q$ denotes the set of all rational numbers.



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

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

Also see

 * Definition:First Apotome
 * Definition:Second Apotome
 * Definition:Fourth Apotome
 * Definition:Fifth Apotome
 * Definition:Sixth Apotome