You might find the Continued Fraction Calculator useful in this section.
- What is the pattern under the square root signs in the table above: that is, what is the nth term in the series 5,8,13,29,29,40,... ?
- What is the next line in the table above for T(11)?
- Express the n-th line, that is T(n) as a formula involving square-roots.
- T(1) is Phi = ( 1 + √5 )/2.
- T(4) = 2 + √5 and also involves √5. Using the Table of Properties of Phi express T(4) as a power of Phi.
- T(11) also involves √5. What is T(11)?
Is it a power of Phi too?
- What is the pattern here? Which powers of Phi are also silver means and which silver means are they?
[Hint: the answer involves the Lucas numbers.]
- What powers of Phi are missing in the answer to the previous question? What are their continued fractions?
- Express all the powers of Phi in the form (X+Y√5)/2. Find a formula for Phin in terms of the Lucas and Fibonacci numbers.
Continued Fractions and the Fibonacci NumbersIn this section we will take a closer look at the links between continued fractions and the Fibonacci Numbers.
Squared Fibonacci Number RatiosWhat is the period of the continued fractions of the following numbers?
The fractions above are the squares of the ratio of successive Fibonacci numbers:
- 25/9 = (5/3)2 = (Fib(5)/Fib(4))2
- 64/25 = (8/5)2 = (Fib(6)/Fib(5))2
- 169/64 = (13/8)2 = (Fib(7)/Fib(6))2
What other continued fraction patterns in fractions formed from Fibonacci numbers (and the Lucas Numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, ... ) can you find?
Continued Fractions of Quadratic Fibonacci Ratios Brother Alfred Brousseau in The Fibonacci Quarterly vol 9 (1971) pages 427 - 435.
Continued Fractions of Fibonacci and Lucas Ratios Brother Alfred Brousseau in The Fibonacci Quarterly vol 2 (1964) pages 269 - 276.
= 1x2–1 + 0x2–2 + 1x2–3 + 1x2–4 + 0x2–5 + 1x2–6 + ...
Expressed as a normal decimal fraction, it is
The surprise in store is what happens if we express this number as a continued fraction. It is
Perhaps even more remarkably, a discussion on sci.math newsgroup proves a result that Robert Sawyer posted:- that we can replace the base 2 by any real number bigger than 1 and the result is still true!
A Series and Its Associated Continued Fraction J L Davison, Fibonacci Quarterly vol 63, 1977, pages 29-32.
A Simple Proof of a Remarkable Continued Fraction Identity P G Anderson, T C Brown, P J-S Shiue Proceeding American Mathematical Society vol 123 (1995), pgs 2005-2009
has a proof that the Rabbit constant is indeed the continued fraction given above.
|The continued fraction for √5 φ =||5 – √5||= 1.3819660112501051518... is [1;2,1,1,1,1,1,1,1,...]|
|and its convergents are: 1,||3||,||4||,||7||,||11||,||18||,||29||, ...|
Taking the reciprocal of this value, i.e.
|Φ||=||2||= 0.72360679774997896964... = [0;1,2,1,1,1,1,1,1,1,1,...]|
5 – √5
The Strong Law of Small Numbers Richard K Guy in The American Mathematical Monthly, Vol 95, 1988, pages 697-712, Example 14.
Eric Weisstein's page on The Rabbit Constant
Chaos in Numberland: The secret life of continued fractions Prof John D Barrow, a +Plus magazine online article which, if you have enjoyed this page, will extend your knowledge with more on applications of continued fractions. Some of it is at undergrauate mathematics level.
References to articles and books
- C. Kimberling, A visual Euclidean algorithm in Mathematics Teacher, vol 76 (1983) pages 108-109.
- is an early reference to the excellent Rectangle Jigsaw approach to continued fractions that we explored at the top of this page. An even earlier description of this method is found in chapter IV Fibonacci Numbers and Geometry of:
- Fibonacci Numbers, N N Vorob'ev, Birkhauser (2003 translation of the 1951 Russian original).
- This slim classic is a translation from the Russian Chisla fibonachchi, Gostekhteoretizdat (1951). This classic contains many of the fundamental Fibonacci and Golden section results and proofs as well as a chapter on continued fractions and their properties.
- Introduction to Number Theory with Computing by R B J T Allenby and E Redfern
- 1989, Edward Arnold publishers, ISBN: 0713136618
is an excellent book on continued fractions and lots of other related and interesting things to do with numbers and suggestions for programming exercises and explorations using your computer.
- The Higher Arithmetic by Harold Davenport,
- Cambridge University Press, (7th edition) 1999, ISBN: 0521422272
is an enjoyable and readable book about Number Theory which has an excellent chapter on Continued Fractions and proves some of the results we have found above. (More information and you can order it online via the title-link.)
Beware though! We have used [a,b,c,d,...]=X/Y as our concise notation for a continued fraction but Davenport uses [a,b,c,d,..] to mean the numerator only, that is, just the X part of the (ordinary) fraction!
- Introduction to the Theory of numbers by G H Hardy and E M Wright
- Oxford University Press, (6th edition, 2008), ISBN: 0199219869
is a classic but definitely at mathematics undergraduate level. It takes the reader through some of the fundamental results on continued fractions. Earlier editions do not have an Index, but there is a Web page Index to editions 4 and 5 that you may find useful. This latest edition, the 6th, is revised and has some new material on Elliptic functions too.
- Continued Fractions by A Y Khinchin,
- This is a Dover book (1997), ISBN: 0 486 69630 8, well produced, slim and cheap, but it is quite formal and abstract, so probably only of interest to serious mathematicians!
- Continued fractions (A Popular Survey) Roger F Wheeler, published by The Mathematical Association, 2002.
- It is available via the MA website.
This book is a more gentle and systematic introduction, with plenty of illustrative examples and tables.
- A Limited Arithmetic on Simple Continued Fractions, C T Long and J H Jordan,
- Fibonacci Quarterly, Vol 5, 1967, pp 113-128;
- A Limited Arithmetic on Simple Continued Fractions - II, C T Long and J H Jordan,
- Fibonacci Quarterly, Vol 8, 1970, pp 135-157;
- A Limited Arithmetic on Simple Continued Fractions - III, C T Long,
- Fibonacci Quarterly, Vol 19, 1981, pp 163-175;