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A non-integer representation uses non-integer numbers as the radix, or bases, of a positional numbering system. For a non-integer radix β > 1, the value of

x=d_n\dots d_2d_1d_0.d_{-1}d_{-2}\dots d_{-m}

x=\beta^nd_n + \cdots + \beta^2d_2 + \beta d_1 + d_0 + \beta^{-1}d_{-1} + \beta^{-2}d_{-2} + \cdots + \beta^{-m}d_{-m}.
The numbers di are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by Rényi (1957) and first studied in detail by Parry (1960). Every real number has at least one (possibly infinite) β-expansion.
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