Definition:Binomial (Euclidean)/Sixth Binomial

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

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



Example
When $a = \sqrt 7$ and $b = \sqrt 5$,

Therefore $\sqrt 7 + \sqrt 5$ is a sixth binomial.