Editing Quadratic irrational number (section)

==Real quadratic irrational numbers and indefinite binary quadratic forms==
We may rewrite a quadratic irrationality as follows:

:<math>\frac{a+b\sqrt{c}} d = \frac{a+\sqrt{b^2c}} d.</math>

It follows that every quadratic irrational number can be written in the form

:<math>\frac{a+\sqrt{c}} d.</math>

This expression is not unique.

Fix a non-square, positive integer <math>c</math> [[Modular arithmetic|congruent]] to <math>0</math> or <math>1</math> modulo <math>4</math>, and define a set <math>S_c</math> as

: <math>S_c = \left\{ \frac{a+\sqrt{c}} d \colon a, d \text{ integers, } \, d \text{ even}, \, a^2 \equiv c \pmod{2d} \right\}.</math>

Every quadratic irrationality is in some set <math>S_c</math>, since the congruence conditions can be met by scaling the numerator and denominator by an appropriate factor.

A [[matrix (mathematics)|matrix]]

:<math>\begin{pmatrix} \alpha & \beta\\ \gamma & \delta\end{pmatrix}</math>

with integer entries and <math>\alpha \delta-\beta \gamma=1</math> can be used to transform a number <math>y</math> in <math>S_c</math>.  The transformed number is

:<math>z = \frac{\alpha y+\beta}{\gamma y+\delta}</math>

If <math>y</math> is in <math>S_c</math>, then <math>z</math> is too.

The relation between <math>y</math> and <math>z</math> above is an [[equivalence relation]].  (This follows, for instance, because the above transformation gives a [[Group action (mathematics)|group action]] of the [[Group (mathematics)|group]] of integer matrices with [[determinant]] 1 on the set <math>S_c</math>.) Thus, <math>S_c</math> partitions into [[equivalence class]]es.  Each equivalence class comprises a collection of quadratic irrationalities with each pair equivalent through the action of some matrix.  Serret's theorem implies that the regular continued fraction expansions of equivalent quadratic irrationalities are eventually the same, that is, their sequences of partial quotients have the same tail.  Thus, all numbers in an equivalence class have continued fraction expansions that are eventually periodic with the same tail.

There are finitely many equivalence classes of quadratic irrationalities in <math>S_c</math>. The standard [[mathematical proof|proof]] of this involves considering the map <math>\varphi</math> from [[binary quadratic form]]s of discriminant <math>c</math> to <math>S_c</math> given by

:<math> \varphi (tx^2 + uxy + vy^2) = \frac{-u + \sqrt{c}}{2t}</math>

A computation shows that <math>\varphi</math> is a [[bijection]] that respects the matrix action on each set.  The equivalence classes of quadratic irrationalities are then in bijection with the equivalence classes of binary quadratic forms, and Lagrange showed that there are finitely many equivalence classes of binary quadratic forms of given discriminant.

Through the bijection <math>\varphi</math>, expanding a number in <math>S_c</math> in a continued fraction corresponds to reducing the quadratic form.  The eventually periodic nature of the continued fraction is then reflected in the eventually periodic nature of the orbit of a quadratic form under reduction, with reduced quadratic irrationalities (those with a purely periodic continued fraction) corresponding to reduced quadratic forms.