Links and other sources

 

Continued Fraction Websites

  • An Introduction to Continued Fractions - Excellent all around website about all aspects of continued fractions by Dr.Ron Knott.  This is where I first saw the use of partitioning rectangles into squares into to graphically visualize rational N-fractions.  Dr. Knott does this with 45/16 in the first section of his site.  Great site!  A must see!

  • Continued Fractions - Another all around great website about continued fractions authored for an Honor's degree in Mathematics.  Lots of great links!

  • What is a Continued Fraction? - Series of discussions on and about various properties of continued fractions.

  • Exact Real Arithmetic Research - Dr. Peter Potts' website.  Dr. Potts did his doctoral thesis on Continued Fraction Arithmetic with Mobius Transformations.  Dr. Potts has many of his papers available from his website, including his doctoral thesis.  Dr. Potts has answered many emails of mine and has cleared up many areas of confusion concerning Continued Fraction Arithmetic.  Graduate level mathematics and computer science work.

  • HAKMEM Continued Fraction Arithmetic - HAKMEM concerning Continued Fraction Arithmetic written by Bill Gosper of MIT's Artificial Intelligence Laboratory.  An introductory treatment concerning Dr. Gosper's methods of doing Continued Fraction Arithmetic.  Dr. Gosper really gives a much more comprehensive treatment of the subject in...

  • Continued Fraction Arithmetic - This is, to the best of my knowledge, an unpublished (and very difficult to find...) work by Bill Gosper concerning his methods for doing Continued Fraction Arithmetic (N-Fractions).  Studying this work of Gosper's is what really what started my understanding concerning continued fraction arithmetic.  Before I found this article, I was truly lost.  Bill Gosper has a way of presenting subject matter that is really quite advanced in a manner accessible to those of us with a more modest mathematical background.  Still... I believe digesting this work would take some considerable motivation for most undergraduate mathematics and computer science majors.

 

Books

  • Numbers and Geometry, by John Stillwell.  © 1998 Springer-Verlag, New York.  Another excellent mathematics title by John Stillwell.  Particularly this is the first place I saw anyone use the partitioning of rectangles into squares to graphically illustrate the periodicity in real quadratic irrational N-fraction expansions.

  • Continued Fractions, by C.D. Olds.  © 1963 by the Mathematical Association of America.  Excellent all around source work on continued fractions.  This is where I found one of the formula I used in the pseudo-functional CF_sqrt2.  Solutions to problems included.

  • An Introduction to Continued Fractions, by Charles G. Moore.  © 1964 by The National Council of Teachers of Mathematics, Inc.  This is a great hands-on, how-to book.  This is where I learned how to take the continued fraction expansion of a real quadratic irrational and transform a periodic continued fraction back into the quadratic irrational it represents.  Solutions to problems... and lots of clearly written examples.

  • Continued Fractions, by A. Ya. Khinchin.  © 1964, 1992 by the University of Chicago.  What I found most helpful about Khinchin's work was his discussion of 'The Measure Theory of Continued Fractions' and how that relates to interval analysis of continued fraction convergents.

  • Continued Fractions, by H.S. Wall.  © 1948 D. Van Nostrand Company, Inc.  The classic advanced treatment of continued fractions.  The example I have of the continued fraction that doesn't converge is a variation of an example I saw in Wall's book.

  • A Course in Computational Number Theory, by David Bressoud and Stan Wagon.  © 2000 Key College Publishing.  Great book... not just for continued fractions, although they devote a whole chapter to discussion of continued fractions.  Especially good because the book comes with a CD-ROM of Mathematica programs for implementing numerous Number-Theoretic functions, including many for continued fractions.

  • Number Theory:  A Programmer's Guide, by Mark Herkommer.  © 1999 McGraw-Hill.  Another great computational number-theoretic book.  The author also devotes a whole chapter to continued fractions.  Includes CD-ROM which has numerous number-theoretic functions written in C, including continued fraction programs.

  • Continued Fractions with Applications, by Lisa Lorentzen and Haakon Waadeland.  © 1992 Elsevier Science Publishers.  Outstanding all around, generally quite advanced, work on continued fractions.  Exceptionally good treatment of CFRAC, the continued fraction factorization algorithm.  The authors bring you through the algorithm which really makes clear just what's going on.  Generally tough book to get, by well worth a look-see.  I don't recommend buying it, though... the publisher wants around $200!... I'm hoping it'll come down.

  • A Course in Computational Algebraic Number Theory, by Henri Cohen.  © 1993 Springer-Verlag.  Algorithms, algorithms, algorithms!... this is the source.  Spells out CFRAC, the continued fraction factorization algorithm.  Also provides numerous algorithms for implementing Euclid's algorithms, generating continued fraction expansions, and algorithms showing the use of continued fraction in the study of the properties of quadratic fields.

  • An Introduction to the Theory of Numbers, by G.H. Hardy and E.M. Wright.  © 1979 Oxford University Press, Fifth Edition.  Quadratic fields, continued fractions... probably much more.  This classic is a real goldmine for the number theorist.

  • The Art of Computer Programming:  Volume 2 Seminumerical Algorithms, by Donald Knuth.  © 1998 Addison-Wesley.  If I have to say much about the utility of one of Knuth's books, then I'm definitely speaking to someone who has never read Knuth.  Knuth definitely has the talent of being able to present difficult material in a manner easily accessible to virtually anyone... given at least some motivation.  For this project, Knuth taught me about radix arithmetic, GCDs, continued fractions (even a bit of continued fraction arithmetic), and more...  Can't say enough about Knuth and this books.  Must have!  And... of course, Knuth is famous for his extensive solutions to problems.

  • Factorization and Primality Testing, by David M. Bressoud.  © 1989 Springer-Verlag, New York.  GCD, Continued Fractions, Bhascara-Brouncker Algorithm, and CFRAC.  Mostly for if, and when, I explain CFRAC, but the method here to find the square root of a real quadratic irrational is essentially the Bhascara-Brouncker Algorithm of sorts, but I didn't realize it's utility in this area until I read Moore's book (listed above).

  • Visual Complex Analysis, by Tristan Needham.  © 1997 Tristan Needham.  Excellent book on complex analysis... especially if you're visually-challenged as I am.  Extensive treatment of Mobius Transformations.  Mostly used for background on Mobius Transformations and their properties.  I can't say enough good things about this book... if you have any interest in complex numbers and analysis, I'd highly recommend you get this book.  I'd get the hardcover if you still can... last I knew, FatBrain.com still had it in stock.

  • Advanced Engineering Mathematics, by Erwin Kreyszig, third and fifth editions.  © 1962, 1967, 1972 by John Wiley and Sons, Inc. (Third Edition).  © 1983 by John Wiley and Sons, Inc. (Fifth Edition).  Mostly used for background on Linear Fractional, or Mobius Transformations.  Great book!  If you are interested in math, and you don't have one of the editions of this classic, I'd definitely seek to rectify that situation sooner rather than later.

 

Papers and articles

  • Of course, many of Dr. Pott's papers, including his doctoral thesis.  It's a lot easier on my wrist if I just direct you again to his website where most of the papers of his that I've read are available... Exact Real Arithmetic Research  There may be a couple papers of his I've read that aren't listed on his website.  I'll look into to that and list them here in the future.  Graduate level mathematics and computer science.

  • "Exact Real Arithmetic with Continued Fractions", by Jean E Vuillemin.  IEEE Transactions on Computers, Vol. 39, No. 8, August 1990.  Excellent treatment of various algorithms for Continued Fraction Arithmetic on redundant representations.  Graduate level mathematics and computer science.

  • "Exact Real Arithmetic with Continued Fractions", by Jean E. Vuillemin.  Lisp and Functional Programming, April 1988.  Earlier version of the above paper.  It has proven valuable to have both versions, as there are some significant differences in the two treatments.  Graduate level mathematics and computer science.

  • "An Algorithm for Redundant Binary Bit-Pipelined Rational Arithmetic", by Peter Kornerup and David W. Matula.  IEEE Transactions in Computers, Vol. 39, No. 8, August 1990.  Presentation of an algorithm for doing rational arithmetic with redundant continued fractions.  You'll notice that I default to Kornerup's and Matula's assignment of the vertices on the coefficient cube, even when I implement one of Gosper's algorithms.  It doesn't effect computation and I feel it just reduces confusion to stick with one standard, although I do show how Gosper assigns the vertices to the coefficient cube.  Graduate level mathematics and computer science.

  • "Continued Fractions and the Euclidean Algorithm", by William F. Hammond.  Lecture notes prepared for MATH 326, Spring 1997, Department of Mathematics and Statistics, University of Albany.  Excellent paper illuminating the connections between continued fractions and the Euclidean algorithm.  Also gives a good treatment of the use of matrices with continued fractions.

 

Software used in constructing this website

  • Most of the mathematical notation was generated in MSWord with MathType which links directly into Word.  Then I captured this output into PaintShop Pro and exported a transparent GIF or JPEG for inclusion into a FrontPage webpage.  I experimented with Scientific Workplace and LATEX, but finally resolved that the above procedure gave greater flexibility, and the output GIFs and JPEGs looked better.

  • Some of the graphs and Illustrative diagrams were produced in Geometer's SketchPad and some in Euklid.  I also used F(z) to generate the spherical plots used on the page illustrating the continued fraction that doesn't converge.  Some of the diagrams, particularly for continued fraction arithmetic, were produced with PaintShop Pro as the base.

  • I also used my TI-89 extensively in the generation of the continued fractions, transformations of continued fractions, and continued fraction arithmetic.  I also used Mathematica, and Maple and Matlab to a lesser extent.