Fourth Apotome/Example

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Example of Fourth 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 fourth apotome 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 apotome.