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 ñ with continued fraction expansion [a<sub>0</sub>; a<sub>1</sub>, a<sub>2</sub>, ...]. Let (q<sub>n</sub>) be the sequence of denominators of the convergents p<sub>n</sub>/q<sub>n</sub> to ñ: so q<sub>n</sub> = a<sub>n</sub>q<sub>n−1</sub> + q<sub>n−2</sub>. Let ñ<sub>n</sub> denote T<sup>n</sup>(ñ) where T is the Gauss map T(x) = {1/x}, and write ò<sub>n</sub> = (−1)<sup>n+1</sup> ñ<sub>0</sub> ñ<sub>1</sub> ... ñ<sub>n</sub>: we have ò<sub>n</sub> = a<sub>n</sub>ò<sub>n−1</sub> + ò<sub>n−2</sub>.
Every positive real x can be written as
where the integer coefficients 0 ⤠b<sub>n</sub> ⤠a<sub>n</sub> and if b<sub>n</sub> = a<sub>n</sub> then b<sub>n−1</sub> = 0.
Every positive integer N can be written uniquely as
where the integer coefficients 0 ⤠b<sub>n</sub> ⤠a<sub>n</sub> and if b<sub>n</sub> = a<sub>n</sub> then b<sub>n−1</sub> = 0.
If ñ is the golden ratio, then all the partial quotients a<sub>n</sub> are equal to 1, the denominators q<sub>n</sub> 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.