Loading AI tools
From Wikipedia, the free encyclopedia
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.
This article needs additional citations for verification. (May 2022) |
Fix a positive irrational number α with continued fraction expansion [a0; a1, a2, ...]. Let (qn) be the sequence of denominators of the convergents pn/qn to α: so qn = anqn−1 + qn−2. Let αn denote Tn(α) where T is the Gauss map T(x) = {1/x}, and write βn = (−1)n+1 α0 α1 ... αn: we have βn = anβn−1 + βn−2.
Every positive real x can be written as
where the integer coefficients 0 ≤ bn ≤ an and if bn = an then bn−1 = 0.
Every positive integer N can be written uniquely as
where the integer coefficients 0 ≤ bn ≤ an and if bn = an then bn−1 = 0.
If α is the golden ratio, then all the partial quotients an are equal to 1, the denominators qn 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.
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.