\begin{definition}[Definition:Binomial (Euclidean)/Fifth Binomial/Example]
Let $a$ and $b$ be two positive real numbers such that $a + b$ is a binomial.
By definition, $a + b$ is a fifth binomial {{iff}}:
: $(1): \quad b \in \Q$
: $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$
where $\Q$ denotes the set of rational numbers.
Let $a = \sqrt {13}$ and $b = 3$.
Then:
{{begin-eqn}}
{{eqn | l = \frac {\sqrt {a^2 - b^2} } a
      | r = \frac {\sqrt {13 - 9} } {\sqrt {13} }
}}
{{eqn | r = \sqrt {\frac 4 {13} }
      | rr= \notin \Q
}}
{{end-eqn}}
Therefore $\sqrt {13} + 3$ is a fifth binomial.
Category:Definitions/Euclidean Number Theory
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\end{definition}