Definition:Binomial (Euclidean)/Fifth 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 fifth binomial if and only if:
- $(1): \quad b \in \Q$
- $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$
where $\Q$ denotes the set of rational numbers.
Let $a = \sqrt {13}$ and $b = 3$.
Then:
\(\ds \frac {\sqrt {a^2 - b^2} } a\) | \(=\) | \(\ds \frac {\sqrt {13 - 9} } {\sqrt {13} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\frac 4 {13} }\) | \(\ds \notin \Q\) |
Therefore $\sqrt {13} + 3$ is a fifth binomial.