Definition:Binomial (Euclidean)/Fourth 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 fourth binomial if and only if:

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

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

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

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

Then:

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

\(=\)

\(\ds \frac {\sqrt {9 - 2} } 3\)

\(\ds \)

\(=\)

\(\ds \frac {\sqrt 7} 3\)

\(\ds \notin \Q\)

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