Thursday, 16 July 2015

π in continued fractions

Since π is an irrational number, there's no way to express it exactly as a decimal or a simple fraction.  The next best thing is to approximate it.  The use of continued fractions is an elegant yet practical way to approximate π.  The simple continued fraction approximating π is:


\pi=3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{1+\ddots}}}}}}}}}}}

which is conveniently represented by the notation [3;7,15,1,292,1,1,1,2,1,3,1,...].

By truncating this infinite continued fraction to finite expansions, 
we get the following fractional approximations:

[3] = 3

[3;7] = 22/7
[3;7,15] =333/106
[3;7,15,1] = 355/113
[3;7,15,1,292] = 103993/33102
etc.

Unfortunately, there are no regular patterns shown in the simple continued fraction for π.

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