Definition:Binomial (Euclidean)/First 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 first binomial if and only if:

$(1): \quad a \in \Q$

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

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

Let $a = 9$ and $b = \sqrt {17}$.

Then:

\(\ds \frac {\sqrt {a^2 - b^2} } a\)

\(=\)

\(\ds \frac {\sqrt {81 - 17} } 9\)

\(\ds \)

\(=\)

\(\ds \frac {\sqrt {64} } 9\)

\(\ds \)

\(=\)

\(\ds \frac 8 9\)

\(\ds \in \Q\)

Therefore $9 + \sqrt {17}$ is a first binomial.