\begin{definition}[Definition:Binomial (Euclidean)/Third Binomial/Example]
Let $a$ and $b$ be two positive real numbers such that $a + b$ is a binomial.
By definition, $a + b$ 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.
Let $a = \sqrt {11}$ and $b = \sqrt {\dfrac {143} {49} }$.
Then:
{{begin-eqn}}
{{eqn | l = \frac {\sqrt {a^2 - b^2} } a
      | r = \frac {\sqrt {11 - \frac {143} {49} } } {\sqrt {11} }
}}
{{eqn | r = \frac {\sqrt {\frac {396} {49} } } {\sqrt {11} }
}}
{{eqn | r = \sqrt {\frac {36} {49} }
}}
{{eqn | r = \frac 6 7
      | rr= \in \Q
}}
{{end-eqn}}
Therefore $\sqrt {11} + \sqrt {\dfrac {143} {49} }$ is a third binomial.
Category:Definitions/Euclidean Number Theory
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\end{definition}