Definition:Binomial (Euclidean)/Fifth Binomial/Example

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.