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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Continued fractions for complex numbers and values of binary quadratic forms
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by S. G. Dani and Arnaldo Nogueira PDF
Trans. Amer. Math. Soc. 366 (2014), 3553-3583 Request permission


We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Such numerous distinct expansions are possible for a complex number. They can be arrived at through various algorithms, as also in a more general way than what we call “iteration sequences”. We consider in this broader context the analogues of the Lagrange theorem characterizing quadratic surds, the growth properties of the denominators of the convergents, and the overall relation between sequences satisfying certain conditions, in terms of non-occurrence of certain finite blocks, and the sequences involved in continued fraction expansions. The results are also applied to describe a class of binary quadratic forms with complex coefficients whose values over the set of pairs of Gaussian integers form a dense set of complex numbers.
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Additional Information
  • S. G. Dani
  • Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India
  • Address at time of publication: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
  • MR Author ID: 54445
  • Email:
  • Arnaldo Nogueira
  • Affiliation: Aix-Marseille Université, Institut de Mathématiques de Luminy, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
  • Email:
  • Received by editor(s): March 4, 2011
  • Received by editor(s) in revised form: August 8, 2012
  • Published electronically: March 4, 2014
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 3553-3583
  • MSC (2010): Primary 11A55, 11H55; Secondary 22Fxx
  • DOI:
  • MathSciNet review: 3192607