Editing Positional notation (section)

=== Terminating fractions ===
The numbers which have a finite representation form the [[semiring]]
: <math>\frac{\N_0}{b^{\N_0}} := \left\{mb^{-\nu}\mid m\in \N_0 \wedge \nu\in \N_0 \right\} .</math>
More explicitly, if <math>p_1^{\nu_1} \cdot \ldots \cdot p_n^{\nu_n} := b</math> is a [[factorization]] of <math>b</math> into the primes <math>p_1, \ldots ,p_n \in \mathbb P</math> with exponents {{nowrap|<math>\nu_1, \ldots ,\nu_n \in \N</math>,<ref>The exact size of the <math>\nu_1, \ldots ,\nu_n</math> does not matter. They only have to be ≥ 1.</ref>}} then with the non-empty set of denominators <math> S := \{ p_1, \ldots, p_n \} </math>
we have
: <math> \Z_S := \left\{x \in \Q \left | \, \exists \mu_i \in \Z : x \prod_{i=1}^n {p_i}^{\mu_i} \in \Z \right . \right\} = b^{\Z} \, \Z = {\langle S\rangle}^{-1}\Z </math>
where <math>\langle S\rangle</math> is the group generated by the <math>p\in S</math> and <math> {\langle S\rangle}^{-1}\Z </math> is the so-called [[Localization (algebra)#Localization of a ring|localization]] of <math>\Z</math> with respect to {{nowrap|<math>S</math>.}}

The [[Fraction (mathematics)|denominator]] of an element of <math> \Z_S </math> contains if reduced to lowest terms only prime factors out of <math>S</math>.
This [[Ring (mathematics)|ring]] of all terminating fractions to base <math>b</math> is [[Dense set|dense]] in the field of [[rational number]]s <math>\Q</math>. Its [[Complete metric space|completion]] for the usual (Archimedean) metric is the same as for <math>\Q</math>, namely the real numbers <math>\R</math>. So, if <math> S = \{ p\} </math> then <math> \Z_{\{ p\}} </math> has not to be confused with <math>\Z_{(p)} </math>, the [[discrete valuation ring]] for the [[prime number|prime]] <math>p</math>, which is equal to <math>\Z_{T} </math> with <math> T = \mathbb P \setminus \{ p\} </math>.

If <math>b</math> divides <math>c</math>, we have <math> b^{\Z} \, \Z \subseteq c^{\Z} \, \Z.</math>