Definition:Binomial (Euclidean)/Fifth Binomial

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

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



Example
When $a = \sqrt {13}$ and $b = 3$,

Therefore $\sqrt {13} + 3$ is a fifth binomial.