Sixth Apotome/Example

Example of Sixth Apotome

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome.

By definition, $a - b$ is a sixth apotome 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 apotome.