Non-integer representation

A non-integer base represents a real number as a series of powers of β, typically x = Σ d_nβ⁻ⁿ, where β > 1 and the digits d_n belong to {0, 1, …, ⌊β⌋}.¹ For x in [0,1), the greedy algorithm repeatedly sets d_n = ⌊βTⁿ⁻¹(x)⌋, with T(x) = βx − ⌊βx⌋. This is the direct analogue of repeatedly taking digits in decimal or binary notation. When β is an integer, the construction reduces to the familiar positional system; when β is non-integer, not every syntactically possible digit string is valid.

A number may have more than one expansion, just as 0.999… equals 1 in decimal. The distinction between finite, eventually periodic, and aperiodic expansions becomes a central mathematical feature rather than a technical nuisance.

The expansion of 1 determines which digit sequences are legal in a given non-integer base. Parry showed that comparing sequences lexicographically with the greedy expansion of 1 gives an admissibility rule for β-expansions.² Thus the base controls not only the numerical value of each string but also the language of permitted strings, a connection formalized through β-shifts in symbolic dynamics.

For example, in the golden-ratio base φ, useful expansions can be formed with digits 0 and 1 while avoiding adjacent 1s in the standard greedy language. This restriction reflects the algebraic identity φ² = φ + 1. Different choices of β produce different combinatorial systems: some have finite descriptions by a sofic shift, while others require infinitely many forbidden patterns.¹²

Non-integer bases reveal arithmetic behavior that has no exact analogue in ordinary integer radix notation. For certain algebraic bases, especially Pisot numbers, expansions of elements of the associated number field can display strong eventual periodicity properties; these phenomena connect β-expansions with algebraic number theory and dynamical systems.³ The golden ratio is the most familiar example, but other Pisot bases are studied through their conjugates and recurrence relations.

β-expansions also provide models for numeration systems, fractal constructions, and symbolic coding of interval maps. They are mathematically useful even when they are not implemented as everyday hardware formats: the base changes the geometry of digit intervals, the entropy of the shift, and the statistical distribution of digits. General treatments place the subject between ergodic theory, number theory, and measurable dynamics.⁴

The most subtle issue is that representability and uniqueness are separate questions. Some β values admit numbers with a single expansion, while others have large families of alternative expansions; the set of numbers with unique expansions can have complicated fractal structure. A base can also be arbitrarily close to an integer without behaving like that integer, because a small change in β can alter the expansion of 1 and therefore the entire admissibility language.

Finite expansions are exceptional rather than typical. Eventually periodic expansions are closely tied to algebraic conditions on β, while generic real bases usually produce nonperiodic digit sequences. The term “base” also requires care: negative-base and complex-base numeration are neighboring subjects, but they are not the same as the real non-integer bases considered here. The standard theory assumes β > 1 and a greedy digit-selection rule.

β-expansion

A representation of a real number as a series of powers of a real base β greater than 1.

Greedy expansion

The expansion obtained by choosing the largest allowable digit at each successive position.

β-shift

The symbolic dynamical system consisting of digit sequences admissible for a particular base β.

Pisot number

An algebraic integer greater than 1 whose other algebraic conjugates all have absolute value less than 1.