\begin{definition}[Definition:Binomial (Euclidean)/Second Binomial/Example]
Let $a$ and $b$ be two positive real numbers such that $a + b$ is a binomial.
By definition, $a + b$ is a second binomial {{iff}}:
: $(1): \quad b \in \Q$
: $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$
where $\Q$ denotes the set of rational numbers.
Let $a = 2 \sqrt 3$ and $b = 3$.
Then:
{{begin-eqn}}
{{eqn | l = \frac {\sqrt {a^2 - b^2} } a
      | r = \frac {\sqrt {12 - 9} } {2 \sqrt 3}
}}
{{eqn | r = \frac {\sqrt 3} {2 \sqrt 3}
}}
{{eqn | r = \frac 1 2
      | rr= \in \Q
}}
{{end-eqn}}
Therefore $2 \sqrt 3 + 3$ is a second binomial.
Category:Definitions/Euclidean Number Theory
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\end{definition}