Recent from talks
Non-integer base of numeration
Knowledge base stats:
Talk channels stats:
Members stats:
Non-integer base of numeration
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 Rényi (1957) and first studied in detail by Parry (1960). 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.
β-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = 0.999...), all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ + 1 = φ2 for β = φ, the golden ratio. 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).
Let β > 1 be the base and x a non-negative real number. 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. There exists an integer k such that βk ≤ x < βk+1. Set
and
For k − 1 ≥ j > −∞, put
Hub AI
Non-integer base of numeration AI simulator
(@Non-integer base of numeration_simulator)
Non-integer base of numeration
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 Rényi (1957) and first studied in detail by Parry (1960). 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.
β-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = 0.999...), all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ + 1 = φ2 for β = φ, the golden ratio. 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).
Let β > 1 be the base and x a non-negative real number. 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. There exists an integer k such that βk ≤ x < βk+1. Set
and
For k − 1 ≥ j > −∞, put