A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of is The numbers di are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) β-expansion. The set of all β-expansions that have a finite representation is a subset of the ring Z[β, β−1]. There are applications of β-expansions in coding theory and models of quasicrystals .