Third Apotome/Example
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Example of Third 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 third apotome if and only if:
- $(1): \quad a \notin \Q$
- $(2): \quad b \notin \Q$
- $(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$
where $\Q$ denotes the set of rational numbers.
Let $a = \sqrt {11}$ and $b = \sqrt {\frac {143} {49} }$.
Then:
\(\ds \frac {\sqrt {a^2 - b^2} } a\) | \(=\) | \(\ds \frac {\sqrt {11 - \frac {143} {49} } } {\sqrt {11} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt {\frac {396} {49} } } {\sqrt {11} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\frac {36} {49} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 6 7\) | \(\ds \in \Q\) |
Therefore $\sqrt {11} - \sqrt {\dfrac {143} {49}}$ is a third apotome.