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:
 * $(1): \quad b \in \Q$
 * $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \not \in \Q$

where $\Q$ denotes the set of rational numbers.



Example
Let $a = \sqrt {13}$ and $b = 3$.

Then:

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

Also see

 * Definition:First Binomial
 * Definition:Second Binomial
 * Definition:Third Binomial
 * Definition:Fourth Binomial
 * Definition:Sixth Binomial