Periodic Continued Fraction

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

:$x = a_0 + \cfrac$

where the initial block _0; a_1, \dots, a_k/math> of ''k''+1 partial denominators is followed by a block _, a_, \dots, a_/math> of ''m'' partial denominators that repeats ''ad infinitum''. For example, $\sqrt2$ can be expanded to the periodic continued fraction ; 2, 2, 2, .../math>.

This article considers only the case of periodic regular continued fractions. In other words, the remainder of this article assumes that all the partial denominators ''a''_''i'' (''i'' ≥ 1) are positive integers. The general case, where the partial denominators ''a''_''i'' are arbitrary real or complex numbers, is treated in the article convergence problem.

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 ; 1, 1, 1, \dots/math> of the golden ratio φ is purely periodic, while the regular continued fraction ; 2, 2, 2, \dots/math> of $\sqrt2$ is periodic, but not purely periodic. However, the regular continued fraction ; 2, 2, 2, \dots/math> of the silver ratio ⁠$\sigma = \sqrt2 + 1$ is purely periodic.

As unimodular matrices

Periodic continued 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, $b^2 - 4ac$, 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 $(6 + \sqrt)/4$). Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example $(3 + \sqrt)/2$) 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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