First Apotome/Example

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Example of First 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 first apotome 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 apotome.