Second Apotome/Example

Example of Second 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 second apotome if and only if:

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

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

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

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

Then:

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

\(=\)

\(\ds \frac {\sqrt {12 - 9} } {2 \sqrt 3}\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \frac 1 2\)

\(\ds \in \Q\)

Therefore $2 \sqrt 3 - 3$ is a second apotome.