Third Apotome/Example

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.