Integer continued fractions for complex numbers

2025/08/20 by Cormac O'Sullivan, Cormac O’Sullivan, O'Sullivan, Cormac · 2 voices
Computer Science · Mathematics · #Advanced Mathematical Identities #Extension (predicate logic) #Integer (computer science) #Mathematical Dynamics and Fractals #Multiple #Natural number #Prime number #Real number #Scheme (mathematics) #Sequence (biology) #math.NT #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2508.15078

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new representations are shown to be unique, and to have useful properties. They also admit a geometric cutting sequence interpretation.

Discussions

• Integer continued fractions for complex numbers [hn, 53 points, 2 comments]
• Integer continued fractions for complex numbers https://arxiv.org/abs/2508.15078 [bsky, 0 points, 0 comments]

Related

• Continued fractions for complex numbers and values of binary quadratic forms
• Continued fraction expansions of complex numbers, Lagrange's theorem, and badly approximable numbers
• Continued fraction expansions for complex numbers - a general approach
• Periodic points in complex continued fractions by J. Hurwitz
• Generalized continued fraction expansions of complex numbers, and applications to quadratic and badly approximable numbers