This is also known as a β-expansion, a notion introduced by Rényi (1957) and first studied in detail by Parry (1960).
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[1] and models of quasicrystals.
A canonical choice for the β-expansion of a given real number can be determined by the following greedy algorithm, essentially due to Rényi (1957) and formulated as given here by Frougny (1992).
Denote by ⌊x⌋ the floor function of x (that is, the greatest integer less than or equal to x) and let {x} = x − ⌊x⌋ be the fractional part of x.
With an integer base, this defines the usual radix expansion for the number x.
This construction extends the usual algorithm to possibly non-integer values of β.
First, we must define our k value (the exponent of the nearest power of β greater than n, as well as the amount of digits in
This means that every integer can be expressed in base √2 without the need of a decimal point.
In addition, the area of a regular octagon with side length 1√2 is 1100√2, the area of a regular octagon with side length 10√2 is 110000√2, the area of a regular octagon with side length 100√2 is 11000000√2, etc… In the golden base, some numbers have more than one decimal base equivalent: they are ambiguous.
The set of numbers with two different representations is dense in the reals,[6] but the question of classifying real numbers with unique β-expansions is considerably more subtle than that of integer bases.
[7] Another problem is to classify the real numbers whose β-expansions are periodic.
Then any real number in [0,1) having a periodic β-expansion must lie in Q(β).
The converse does hold if β is a Pisot number,[8] although necessary and sufficient conditions are not known.