Definition:Binomial (Euclidean)/Second Binomial/Example

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Example

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


By definition, $a + b$ is a second binomial if and only if:

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

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


Let $a = 2 \sqrt 3$ and $b = 3$.

Then:

\(\ds \frac {\sqrt {a^2 - b^2} } a\) \(=\) \(\ds \frac {\sqrt {12 - 9} } {2 \sqrt 3}\)
\(\ds \) \(=\) \(\ds \frac {\sqrt 3} {2 \sqrt 3}\)
\(\ds \) \(=\) \(\ds \frac 1 2\) \(\ds \in \Q\)

Therefore $2 \sqrt 3 + 3$ is a second binomial.