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:
Otherwise, the result is called an infinite continued fraction:
Continued fractions can be used to find rational approximations to real numbers, by simply truncating the resulting fraction at a certain point. For example, .
The numbers appearing on the left of the expansion (the integer parts) are called coefficients.
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Coefficient facts
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The continued fraction coefficients of quadratic numbers (solutions of a quadratic equation with integer coefficients) eventually repeat.
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For some non-quadratic numbers such as Euler's number , the coefficients have an obvious pattern: 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...
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However, for almost all real numbers , the geometric mean of the coefficients of the continued fraction expansion of is the following number:
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2.6854520010653064453097148...
which is known as Khinchin's constant.
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