Definition:Binomial (Euclidean)/Third Binomial

Definition
Let $a$ and $b$ be two positive real numbers, such that $a + b$ is a binomial.

Then $a + b$ is a third binomial iff $a \not \in \Q$ and $b \not \in \Q$ and $\frac {\sqrt {a^2 - b^2}} a \in \Q$, 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 binomial.