Integer continued fractions for complex numbers

TL;DR

This paper extends the concept of continued fractions to complex numbers, introducing a unique representation with useful properties and a geometric interpretation, based on classical algorithms by Lagrange and Gauss.

Contribution

It presents a novel complex continued fraction framework that is unique and geometrically interpretable, expanding classical real continued fractions to the complex domain.

Findings

Unique complex continued fraction representations.

Geometric interpretation via cutting sequences.

Extension of classical algorithms to complex numbers.

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.