Ostrowski numeration

In mathematics, Ostrowski numeration, named after Alexander Ostrowski, is either of two related numeration systems based on continued fractions: a non-standard positional numeral system for integers and a non-integer representation of real numbers.

Fix a positive irrational number α with continued fraction expansion [a₀; a₁, a₂, ...]. Let (q_n) be the sequence of denominators of the convergents p_n/q_n to α: so q_n = a_nq_n−1 + q_n−2. Let α_n denote Tⁿ(α) where T is the Gauss map T(x) = {1/x}, and write β_n = (−1)ⁿ⁺¹ α₀ α₁ ... α_n: we have β_n = a_nβ_n−1 + β_n−2.

Every positive real x can be written as

${\displaystyle x=\sum _{n=1}^{\infty }b_{n}\beta _{n}\ }$

where the integer coefficients 0 ≤ b_n ≤ a_n and if b_n = a_n then b_n−1 = 0.

Every positive integer N can be written uniquely as

${\displaystyle N=\sum _{n=1}^{k}b_{n}q_{n}\ }$

where the integer coefficients 0 ≤ b_n ≤ a_n and if b_n = a_n then b_n−1 = 0.

If α is the golden ratio, then all the partial quotients a_n are equal to 1, the denominators q_n are the Fibonacci numbers and we recover Zeckendorf's theorem on the Fibonacci representation of positive integers as a sum of distinct non-consecutive Fibonacci numbers.