Definition:Binomial (Euclidean)/Third Binomial

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

Then $a + b$ is a third binomial iff:
 * $(1): \quad a \notin \Q$
 * $(2): \quad b \notin \Q$
 * $(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$

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



Example
Let $a = \sqrt {11}$ and $b = \sqrt {\dfrac {143} {49} }$.

Then:

Therefore $\sqrt {11} + \sqrt {\dfrac {143} {49} }$ is a third binomial.

Also see

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