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.