Alex Patin
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Front end developer / designer. Lover of technology, learning, and liberty.
Front end developer / designer. Lover of technology, learning, and liberty.

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Continued fractions

This animation by Lucas Vieira Barbosa illustrates the continued fraction representation of the golden ratio, φ (or phi), which is approximately equal to 1.618. The number φ is the larger root of the equation x^2–x–1=0, which means that φ = 1+(1/φ). The animation expresses this latter equation very well.

A continued fraction is an expression obtained by representing a number as the sum of its integer part and the reciprocal of another number, and then (recursively) writing this other number as a continued fraction.

The continued fraction expansion of a number may be written in brief as a sequence of integers, which in the case of the golden ratio would be simply be [1;1,1,1,1,...]. More generally, the continued fraction [a0;a1,a2,a3,...] corresponding to the integer sequence (a0, a1, a2, a3,...) would correspond to the number a0+1/(a1+(1/(a2+1/(a3+...)))).

In a continued fraction, the first term a0 may be negative, but all the other terms will be nonnegative integers. The continued fraction of an irrational number will give an infinite sequence, whereas the continued fraction of a rational number will terminate after a finite number of steps. This representation of a number as an integer sequence will be unique for an irrational number, whereas a rational number will correspond to two such sequences. For example, the rational number 7/3 can be written either as [2;3] = 2 + 1/3 or as [2;2,1] = 2 + 1/(2 + 1/1).

The number e, which is approximately equal to 2.71828, is the base of the natural logarithm. It is an irrational number whose decimal expansion looks more or less random. Despite this, the continued fraction of e has a pattern to it: e = [2;1,2,1,1,4,1,1,6,1,1,8,...] (http://oeis.org/A003417).

In contrast to this, neither the decimal expansion nor the continued fraction representation of the number π (approximately 3.14159) shows any obvious pattern: π = [3;7,15,1,292,1,1,1,2,1,3,1,...] (http://oeis.org/A001203). Truncating this sequence after two steps, we obtain [3;7] = 3 + 1/7, which is a familiar approximation to π. Truncating the sequence after the large entry of 292 will give a very accurate rational approximation to π, namely 103993/33102; this has the first nine decimal places correct.

I was reminded of continued fractions by my recent post about Srinivasa Ramanujan (1887-1920), who was extremely good at using them. There is a famous anecdote about Ramanujan being asked the answer to a brain-teaser type problem from a magazine. As soon as the problem was read out to him, Ramanujan said "Please take down the solution" and dictated a continued fraction. When asked how he solved the problem, Ramanujan replied:

Immediately I heard the problem, it was clear that the solution was obviously a continued fraction; I then thought, “Which continued fraction?” and the answer came to my mind. It was just as simple as this.

And in case you are wondering, this method of solving problems sounds baffling and intimidating to other professional mathematicians.

Wikipedia on continued fractions: http://en.wikipedia.org/wiki/Continued_fraction

Details of the anecdote on Ramanujan and continued fractions: http://anecdotesandallthat.blogspot.com/2012/01/mahalanobis-on-ramanujan.html

A post by me about Ramanujan and taxi cab number 1729: https://plus.google.com/101584889282878921052/posts/QtuFkDQnDQ6

Lucas Vieira Barbosa's tumblr page, where this animation came from: http://1ucasvb.tumblr.com/about

Lucas contributes many mathematical diagrams and animations to Wikipedia. If you like his work, there is a way to make a donation to it on the above page, so that he can keep producing these wonderful graphics.

#mathematics #scienceeveryday
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