Continued Fractions - Maple Programming Help

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Continued Fractions

Main Concept

A continued fraction is a unique representation of a number, obtained by recursively subtracting the integer part of that number and then computing the continued fraction of the reciprocal of the remainder, if it is non-zero. If the number is rational, this process terminates with a finite continued fraction:

 12345=2+11+12+11+13 

Otherwise, the result is called an infinite continued fraction:

  π=3+17+115+11+1...

Continued fractions can be used to find rational approximations to real numbers, by simply truncating the resulting fraction at a certain point. For example,  π 3+17.

 

The numbers appearing on the left of the expansion (the integer parts) are called coefficients.

Coefficient facts

• 

 The continued fraction coefficients of quadratic numbers (solutions of a quadratic equation with integer coefficients) eventually repeat.

• 

 For some non-quadratic numbers such as Euler's number e=2.718..., the coefficients have an obvious pattern: 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...

• 

 However, for almost all real numbers x, the geometric mean of the coefficients of the continued fraction expansion of x is the following number:

 

2.6854520010653064453097148...

 which is known as Khinchin's constant.

 

Input a Maple expression in the box below that evaluates to a real number and click Enter, or choose one from the drop-down box. Adjust the number of approximating coefficients using the slider, and see how the coefficient frequency is affected in the graph.

 

# of coefficients =

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