Periodic continued fraction
In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form
where the initial block of k + 1 partial denominators is followed by a block [ak+1, ak+2,…ak+m] of partial denominators that repeats over and over again, ad infinitum. For example, can be expanded to a periodic continued fraction, namely as [1,2,2,2,...].
The partial denominators {ai} can in general be any real or complex numbers. That general case is treated in the article convergence problem. The remainder of this article is devoted to the subject of simple continued fractions that are also periodic. In other words, the remainder of this article assumes that all the partial denominators ai (i ≥ 1) are positive integers.
Since all the partial numerators in a regular continued fraction are equal to unity we can adopt a shorthand notation in which the continued fraction shown above is written as
where, in the second line, a vinculum marks the repeating block. Some textbooks use the notation
where the repeating block is indicated by dots over its first and last terms.
If the initial non-repeating block is not present – that is, if
the regular continued fraction x is said to be purely periodic. For example, the regular continued fraction for the golden ratio φ – given by [1; 1, 1, 1, …] – is purely periodic, while the regular continued fraction for the square root of two – [1; 2, 2, 2, …] – is periodic, but not purely periodic.
A quadratic irrational number is an irrational real root of the quadratic equation
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