commensurable numbersCommensurableNumbers2008-07-08 16:01:552009-08-24 15:11:01Definitionreal numbercommensurableincommensurablecommensurability% this is the default PlanetMath preamble. as your knowledge
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Two positive real numbers $a$ and $b$ are {\em commensurable}, iff there exists a positive real number $u$ such that
\begin{align}
a = mu, \quad b = nu
\end{align}
with some positive integers $m$ and $n$.
If the positive numbers $a$ and $b$ are not commensurable, they are {\em incommensurable}.\\
\textbf{Theorem.}\, The positive numbers $a$ and $b$ are commensurable if and only if their ratio is a rational number
$\displaystyle\frac{m}{n}$\, ($m,\,n \in \mathbb{Z}$).\\
{\em Proof.}\, The equations (1) imply the \PMlinkname{proportion}{ProportionEquation}
\begin{align}
\frac{a}{b} = \frac{m}{n}.
\end{align}
Conversely, if (2) is valid with\, $m,\,n \in \mathbb{Z}$,\, then we can write
$$a = m\!\cdot\!\frac{b}{n}, \quad b = n\!\cdot\!\frac{b}{n},$$
which means that $a$ and $b$ are multiples of $\displaystyle\frac{b}{n}$ and thus commensurable.\, Q.E.D.\\
\textbf{Example.}\, The lengths of the side and the diagonal of \PMlinkid{square}{1086} are always incommensurable.
\subsection{Commensurability as relation}
\begin{itemize}
\item The commensurability is an equivalence relation in the set $\mathbb{R}_+$ of the positive reals:\, the reflexivity and the symmetry are trivial;\, if\, $a\!:\!b = r$\, and\, $b\!:\!c = s$,\, then\, $a\!:\!c = (a\!:\!b)(b\!:\!c) = rs$,\, whence one obtains the transitivity.\\
\item The equivalence classes of the commensurability are of the form
$$[\varrho] \,:=\, \{r\varrho\,\vdots\;\; r \in \mathbb{Q}_+\}.$$
\item One of the equivalence classes is the set\, $[1] = \mathbb{Q}_+$\, of the positive rationals, all others consist of positive irrational numbers.
\item If one sets\; $[\varrho]\!\cdot\![\sigma] := [\varrho\sigma]$,\, the equivalence classes form with respect to this binary operation an Abelian group.
\end{itemize}