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:
 * $(1): \quad: a \notin \Q$
 * $(2): \quad: b \notin \Q$
 * $(3): \quad: \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$

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



Example
Let $a = \sqrt 7$ and $b = \sqrt 5$.

Then:

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

Also see

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