Getting Started

This is a PyPI package which uses only Python standard libraries, and is supported on any Linux, Mac OS or Windows system supporting Python 3.10, 3.11, or 3.12. It is CI-tested on Ubuntu Linux (22.04.4 LTS), Mac OS (13.6.6) and Windows (Windows Server 2022 21H2), but should also install on any other platform supporting these Python versions.

Installation

A standard pip install is sufficient:

pip install continuedfractions

Potential contributors may want look at the project TOML, and then the contributions overview.

Package Structure

The continuedfractions package consists of three core libraries.

• continuedfractions.lib

• continuedfractions.continuedfraction

• continuedfractions.sequences

Note

The continuedfractions.utils library contains internal utilities.

Each is summarised below. Or you can go straight to the API reference.

continuedfractions.lib

This is a library of standalone functions:

• continued_fraction_rational() - generates the (ordered) sequence of elements (coefficients) of the unique simple continued fraction of a given rational number, given as a fractions.Fraction instance.

• continued_fraction_real() - generates the sequence of elements of a simple continued fraction of a real number, given as a single int, float, str, or decimal.Decimal value; the results for float inputs may be approximate and not necessarily unique.

• fraction_from_elements() - returns a fractions.Fraction instance of the rational number represented by a simple continued fraction with the given sequence of elements.

• convergent() - returns the \(k\)-th convergent (for a positive integer \(k\)) from a sequence of elements of a (finite) simple continued fraction; the convergent is returned as a fractions.Fraction instance.

• mediant() - returns the \(k\)-th left or right mediant of two rational numbers, given as fractions.Fraction values; the mediant is returned as a fractions.Fraction instance.

Note

There are also two “wrapper” functions for computing left- and right-mediants - left_mediant() and right_mediant() - but these just call mediant().

continuedfractions.continuedfraction

This is a library containing a single main class:

• ContinuedFraction - a subclass of fractions.Fraction, designed to represent (finite) simple continued fractions as Python objects, which are fully operable as rational numbers.

continuedfractions.sequences

This is a library of functions and classes relating to ordered sequences and structures of integers and rational numbers, such as (currently) coprime integers, Farey sequences and (in the future) the Stern-Brocot tree:

• coprime_integers() - returns a sequence of integers coprime (or relatively prime) to a given positive integer.

• KSRMTree - implements an implicit/generative class for the Kanga-Saunders-Randall-Mitchell (KSRM) ternary trees for representing and generating pairs of (positive) coprime integers.

• coprime_pairs() - returns a sequence of all pairs of (positive) coprime integers less than or equal to a given positive integer. Uses the KSRM tree search() method to perform the search.

• farey_sequence() - returns a sequence of rational numbers called a Farey sequence for a given positive integer. Uses the coprime_integers() and coprime_pairs() functions.