Integer continued fractions for complex numbers

• Published in 2025
• Added on 2025-08-28

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.

Links

• http://arxiv.org/abs/2508.15078v1
• http://arxiv.org/pdf/2508.15078v1

Other information

BibTeX entry

@article{Integercontinuedfractionsforcomplexnumbers,
	key = {Integercontinuedfractionsforcomplexnumbers},
	type = {article},
	title = {Integer continued fractions for complex numbers},
	author = {Cormac O'Sullivan},
	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.},
	comment = {},
	date_added = {2025-08-28},
	date_published = {2025-08-28},
	urls = {http://arxiv.org/abs/2508.15078v1,http://arxiv.org/pdf/2508.15078v1},
	collections = {unusual-arithmetic},
	url = {http://arxiv.org/abs/2508.15078v1 http://arxiv.org/pdf/2508.15078v1},
	year = 2025,
	urldate = {2025-08-28},
	archivePrefix = {arXiv},
	eprint = {2508.15078},
	primaryClass = {math.NT}
}