Definition:Binomial (Euclidean)/Sixth 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 sixth binomial if and only if:

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

$(2): \quad: b \notin \Q$

$(3): \quad: \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$

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

Let $a = \sqrt 7$ and $b = \sqrt 5$.

Then:

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

\(=\)

\(\ds \frac {\sqrt {7 - 5} } {\sqrt 7}\)

\(\ds \)

\(=\)

\(\ds \sqrt {\frac 2 7}\)

\(\ds \notin \Q\)

Therefore $\sqrt 7 + \sqrt 5$ is a sixth binomial.