Definition:Binomial (Euclidean)/Fourth Binomial

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

Then $a + b$ is a fourth binomial iff:
 * $(1): \quad a \in \Q$
 * $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \not \in \Q$

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



Example
Let $a = 3$ and $b = \sqrt 2$.

Then:

Therefore $3 + \sqrt 2$ is a fourth binomial.

Also see

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