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Non-integer base of numeration
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<p>A non-integer representation uses non-<a href="/facts/Integer/PUwwrawx">integer</a> numbers as the <a href="/facts/Radix/J5RGWRJF">radix</a>, or base, of a <a href="/facts/Positional_notation/31DECoN4">positional numeral system</a>. For a non-integer radix <i>β</i> > 1, the value of </p> x = d n … d 2 d 1 d 0 . d − 1 d − 2 … d − m {\displaystyle x=d_{n}\dots d_{2}d_{1}d_{0}.d_{-1}d_{-2}\dots d_{-m}} <p>is </p> x = β n d n + ⋯ + β 2 d 2 + β d 1 + d 0 + β − 1 d − 1 + β − 2 d − 2 + ⋯ + β − m d − m . {\displaystyle {\begin{aligned}x&=\beta ^{n}d_{n}+\cdots +\beta ^{2}d_{2}+\beta d_{1}+d_{0}\\&\qquad +\beta ^{-1}d_{-1}+\beta ^{-2}d_{-2}+\cdots +\beta ^{-m}d_{-m}.\end{aligned}}} <p>The numbers <i>d</i><i>i</i> are non-negative integers less than <i>β</i>. This is also known as a <i>β</i>-expansion, a notion introduced by Rényi (1957) and first studied in detail by Parry (1960). Every <a href="/facts/Real_number/R02gw5Pb">real number</a> has at least one (possibly infinite) <i>β</i>-expansion. The <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of all <i>β</i>-expansions that have a finite representation is a <a href="/facts/Subset/SJGERmbA">subset</a> of the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> Z[<i>β</i>, <i>β</i>−1]. </p><p>There are applications of <i>β</i>-expansions in <a href="/facts/Coding_theory/SBqStoGL">coding theory</a> and models of <a href="/facts/Quasicrystal/evHcQeCp">quasicrystals</a>. </p>