Editing Modular group (section)

==Number-theoretic properties==
The unit determinant of

:<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math>

implies that the fractions <sub><big>{{math|{{sfrac|''a''|''b''}}}}</big></sub>, <sub><big>{{math|{{sfrac|''a''|''c''}}}}</big></sub>, <sub><big>{{math|{{sfrac|''c''|''d''}}}}</big></sub>, <sub><big>{{math|{{sfrac|''b''|''d''}}}}</big></sub> are all irreducible, that is having no common factors (provided the denominators are non-zero, of course).  More generally, if {{math|{{sfrac|''p''|''q''}}}} is an irreducible fraction, then

:<math>\frac{ap+bq}{cp+dq}</math>

is also irreducible (again, provided the denominator be non-zero). Any pair of irreducible fractions can be connected in this way; that is, for any pair {{math|{{sfrac|''p''|''q''}}}} and {{math|{{sfrac|''r''|''s''}}}} of irreducible fractions, there exist elements

:<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\operatorname{SL}(2, \mathbb Z)</math>

such that

:<math>r = ap+bq \quad \mbox{ and } \quad s=cp+dq.</math>

Elements of the modular group provide a symmetry on the two-dimensional [[period lattice|lattice]]. Let {{math|''ω''<sub>1</sub>}} and {{math|''ω''<sub>2</sub>}} be two [[complex number]]s whose ratio is not real. Then the set of points

:<math>\Lambda (\omega_1, \omega_2)=\{ m\omega_1 +n\omega_2 : m,n\in \mathbb Z \}</math>

is a lattice of parallelograms on the plane.  A different pair of vectors {{math|''α''<sub>1</sub>}} and {{math|''α''<sub>2</sub>}} will generate exactly the same lattice if and only if

:<math>\begin{pmatrix}\alpha_1 \\ \alpha_2 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix}</math>

for some matrix in {{math|GL(2, '''Z''')}}. It is for this reason that [[doubly periodic function]]s, such as [[elliptic functions]], possess a modular group symmetry.

The action of the modular group on the rational numbers can most easily be understood by envisioning a square grid, with grid point {{math|(''p'', ''q'')}} corresponding to the fraction {{math|{{sfrac|''p''|''q''}}}} (see [[Euclid's orchard]]).  An irreducible fraction is one that is ''visible'' from the origin; the action of the modular group on a fraction never takes a ''visible'' (irreducible) to a ''hidden'' (reducible) one, and vice versa.

Note that any member of the modular group maps the [[projectively extended real line]] one-to-one to itself, and furthermore bijectively maps the projectively extended rational line (the rationals with infinity) to itself, the [[irrational number|irrational]]s to the irrationals, the [[transcendental number]]s to the transcendental numbers, the non-real numbers to the non-real numbers, the upper half-plane to the upper half-plane, et cetera.

If {{math|{{sfrac|''p''<sub>''n''−1</sub>|''q''<sub>''n''−1</sub>}}}} and {{math|{{sfrac|''p''<sub>''n''</sub>|''q''<sub>''n''</sub>}}}} are two successive  convergents of a [[continued fraction]], then the matrix

:<math>\begin{pmatrix} p_{n-1} & p_{n} \\ q_{n-1} & q_{n} \end{pmatrix}</math>

belongs to {{math|GL(2, '''Z''')}}.  In particular, if {{math|''bc'' − ''ad'' {{=}} 1}} for positive integers {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} with {{math|''a'' < ''b''}} and {{math|''c'' < ''d''}} then {{math|{{sfrac|''a''|''b''}}}} and {{math|{{sfrac|''c''|''d''}}}} will be neighbours in the [[Farey sequence]] of order {{math|max(''b'', ''d'')}}. Important special cases of continued fraction convergents include the [[Fibonacci number]]s and solutions to [[Pell's equation]].  In both cases, the numbers can be arranged to form a [[semigroup]] subset of the modular group.