periodic continued fraction

In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form

:$x = a_0 + \cfrac$

where the initial block of ''k'' + 1 partial denominators is followed by a block 'a''_''k''+1, ''a''_''k''+2,...''a''_''k''+''m''of partial denominators that repeats over and over again, ''ad infinitum''. For example, $\sqrt2$ can be expanded to a periodic continued fraction, namely as ,2,2,2,...

The partial denominators 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 ''a''_''i'' (''i'' ≥ 1) are positive integers.

Purely periodic and periodic fractions

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

:
\begin
x& = _0;a_1,a_2,\dots,a_k,a_,a_,\dots,a_,a_,a_,\dots,a_,\dots\
& = _0;a_1,a_2,\dots,a_k,\overline\end

where, in the second line, a vinculum marks the repeating block. Some textbooks use the notation

:
\begin
x& = _0;a_1,a_2,\dots,a_k,\dot a_,a_,\dots,\dot a_\end

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 = a_m and

:$x =$overline

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, ...– is purely periodic, while the regular continued fraction for the square root of two – ; 2, 2, 2, ...– is periodic, but not purely periodic.

As unimodular matrices

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
:$x =$overline

This can, in fact, be written as
:$x = \frac$
with the $\alpha,\beta,\gamma,\delta$ being integers, and satisfying $\alpha \delta-\beta \gamma=1.$ Explicit values can be obtained by writing
:$S = \begin 1 & 0\\ 1 & 1\end$
which is termed a "shift", so that
:$S^n = \begin 1 & 0\\ n & 1\end$
and similarly a reflection, given by
:$T\mapsto \begin -1 & 1\\ 0 & 1\end$
so that $T^2=I$. Both of these matrices are unimodular, arbitrary products remain unimodular. Then, given $x$ as above, the corresponding matrix is of the form
:$S^TS^T\cdots TS^ = \begin \alpha & \beta\\ \gamma & \delta\end$
and one has
:$x =$overline= \frac
as the explicit form. As all of the matrix entries are integers, this matrix belongs to the modular group $SL(2,\mathbb).$

Relation to quadratic irrationals

A quadratic irrational number is an irrational real root of the quadratic equation

:$ax^2 + bx + c = 0$

where the coefficients ''a'', ''b'', and ''c'' are integers, and the discriminant, ''b''² − 4''ac'', is greater than zero. By the quadratic formula every quadratic irrational can be written in the form

:$\zeta = \frac$

where ''P'', ''D'', and ''Q'' are integers, ''D'' > 0 is not a perfect square (but not necessarily square-free), and ''Q'' divides the quantity ''P''² − ''D'' (for example (6+)/4). Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example (3+)/2) as explained for quadratic irrationals.

By considering the complete quotients of periodic continued fractions, Euler was able to prove that if ''x'' is a regular periodic continued fraction, then ''x'' is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that ''x'' must satisfy.

Lagrange

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