Numerical Continued Fractions

  • Reza N. Jazar


In this chapter we review the method of continued fractions to show its advantage and application. Convergence and its usefulness in working with irrational numbers, as well as converting a numerical series to continued fraction back and forth are the topics of this chapter. This review makes the reader ready to derive and work with solution of differential equations in continued fractions.

We will show that all real numbers may be divided into rational and irrational. They also may be divided into algebraic and transcendental. A rational number can be expressed by a fraction of the form p/q where p and q are integers. Numbers are also either algebraic or transcendental. The method expressing rational and irrational numbers by continued fractions will be covered in this chapter to make the reader ready to solve differential equations in continued fractions.


Continued fractions Conversion of series to continued fractions Convergents Rational numbers Irrational numbers Transcendental numbers 


  1. Aydin, N., & Hammoudi, L. (2019). Al-Kāshı̄’ās Miftā h. al-Hisab, Volume I: Arithmetic. Birkhäuser, Cham: Springer Nature.CrossRefGoogle Scholar
  2. Bailey, D. H., & Borwein, J. M. (2016). Pi: The next generation: A sourcebook on the recent history of Pi and its computation. Cham: Springer.zbMATHGoogle Scholar
  3. Battin, R. H. (1999). An introduction to the mathematics and methods of astrodynamics. Reston, VA: American Institute of Aeronautics and Astronautics.zbMATHGoogle Scholar
  4. Beckmann, P. (1971). A history ofπ(PI). New York: St. Martin’s Press.zbMATHGoogle Scholar
  5. Ben-Dov, J., Horowitz, W., & Steele, J. M. (2012). Living the lunar calendar. Oxford, UK: Oxbow Books.CrossRefGoogle Scholar
  6. Borwein, J. M., & Bailey, D. (2008). Mathematics by experiment, plausible reasoning in the 21st century. New York: CRC Press.CrossRefGoogle Scholar
  7. Borwein, J. M., & Borwein, P. B. (1987). Pi and the AGM. New York: Wiley.zbMATHGoogle Scholar
  8. Clawson, C. C. (1996). Mathematieal mysterles: The beauty and magie of numbers. New York: Springer.CrossRefGoogle Scholar
  9. Dershowitz, N., & Reingold, E. M. (2008). Calendrical calculations (3rd ed.). New York, USA: Cambridge University Press.zbMATHGoogle Scholar
  10. Euler, L. (1988). Introduction to analysis of the infinite. New York: Springer. Euler’s work to 1800, Book I, Translated by J. D. Blanton.Google Scholar
  11. Feeney, D. (2007). Caesar’s calendar: Ancient time and the beginnings of history. Los Angeles, CA: University of California Press.Google Scholar
  12. Gray, L. H. (1907). On certain Persian and Armenian month-names as influenced by the Avesta calendar. Journal of the American Oriental Society, 28, 331–344.CrossRefGoogle Scholar
  13. Hannah, R. (2005). Greek and Roman calendars, constructions of time in the classical world. London, UK: Gerald Duckworth & Co. Ltd.Google Scholar
  14. Hardy, G. H., & Wright, E. M. (2008). An introduction to the theory of numbers (6th ed.). London, UK: Oxford University Press.zbMATHGoogle Scholar
  15. Herz-Fischler, R. (1987). A mathematical history of the golden number. Mineola, NY: Dover.Google Scholar
  16. Idem. (1965). The Iranian calendar, in Zoroastrian studies (2nd ed., pp. 124–131). New York: AMS Press.Google Scholar
  17. Jonathan, M. B., & Chapman, S. T. (2015). I prefer Pi: A brief history and anthology of articles in the American mathematical monthly. The American Mathematical Monthly, 122(3), 195–216.MathSciNetCrossRefGoogle Scholar
  18. Khinchin, A. Y. (1997). Continued fractions. New York: Dover.zbMATHGoogle Scholar
  19. Khrushchev, S. (2008). Orthogonal polynomials and continued fractions. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  20. Kline, M. (1972). Mathematical thought from ancient to modern times (Vol. 1). New York: Oxford University Press.zbMATHGoogle Scholar
  21. Lorentzen, L., & Waadeland, H. (2008). Numerical computation of continued fractions. In Continued fractions. Atlantis studies in mathematics for engineering and science (Vol. 1). Atlantis Press.Google Scholar
  22. Lyusternik, L. A., & Yanupolskii, A. R. (1965). Mathematical analysis, functions, series, and continued functions. London, UK: Pergamon Press. Translated by D. E. Brwn.Google Scholar
  23. Merzbach, U. C., & Boyer, C. B. (2011). A history of mathematics (3rd ed.). Hoboken, NJ: Wiley.zbMATHGoogle Scholar
  24. Morony, M. (2012). ARAB II. Arab conquest of Iran. In Encyclopaedia Iranica (Vol. II, pp. 203–210).Google Scholar
  25. Müller, J. H. (1920). On the application of continued fractions to the evaluation of certain integrals, with special reference to the incomplete Beta function. Biometrika, 22, 284–297.CrossRefGoogle Scholar
  26. Olds, C. D. (1963). Continued fractions. New York: Random House.CrossRefGoogle Scholar
  27. Pacioli, L. (1509). De divina proportione (On the Divine Proportion), Venice: Alessandro and Paganino de’ Paganini, Republic of Venice.Google Scholar
  28. Panaino, A., Abdollahy, R., & Balland, D. (1990). Calendars. Encyclopaedia Iranica (Vol. IV, pp. 658–677), Fasc. 6–7.Google Scholar
  29. Philip, A. (1921). The calendar, its history, structure and improvement. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  30. Pringsheim, A. I. (1898). Ueber die ersten Beweise der Irrationalität von e und π. Sitzungsberichte der Bayerischen Akademie der Wissenschaften Mathematisch-Physikalische Klasse, 28, 325–337.zbMATHGoogle Scholar
  31. Rogers, L. J. (1893). On the expansion of some infinite products. Proceedings of the London Mathematical Society, 24, 337–352.Google Scholar
  32. Saha, M. N., & Lahiri, N. C. (1955). History of the calendar in different countries through the ages. New Delhi: Council of Scientific & Industrial Research.Google Scholar
  33. Shidlovskii, A. B. (1989). Transcendental numbers. New York: Walter de Gruyter.CrossRefGoogle Scholar
  34. Silverman, J. H. (2011). A friendly introduction to number theory (4th ed.). London, UK: Pearson Education.Google Scholar
  35. Stolz, O. (1885). Vorlesungen über allgemeine Arithmetik: nach den Neueren Ansichten (pp. 173–175). Leipzig: Teubners.Google Scholar
  36. Tabak, J. (2004). Numbers: Computers, philosophers, and the search for meaning. New York: Facts On File.Google Scholar
  37. Yanpolskii, L. A., & Lyusternik, A. R. (1965). Mathematical analysis. New York: Pergamon Press.Google Scholar

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Authors and Affiliations

  • Reza N. Jazar
    • 1
  1. 1.School of EngineeringRMIT UniversityMelbourneAustralia

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