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Concept
Periodic continued fraction
Summary
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 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 k = -1, a0 = am and
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.
Such periodic fractions are in one-to-one correspondence with the real quadratic irrationals. The correspondence is explicitly provided by Minkowski's question-mark function. That article also reviews tools that make it easy to work with such continued fractions. Consider first the purely periodic part
This can, in fact, be written as
with the being integers, and satisfying Explicit values can be obtained by writing
which is termed a "shift", so that
and similarly a reflection, given by
so that . Both of these matrices are unimodular, arbitrary products remain unimodular.
Official source
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