 
Summary: Rationals and the Modular Group
Roger C. Alperin
The modular group k is the quotient group PSL2(Z) = SL,(Z)/{&I} of SL2(Z),
the group of 2 x 2 integer matrices of determinant 1. In [I]we gave an elementary
proof that .A' has the structure of a free product of a cyclic group of order 2
generated by the image of A = (::,) and a cyclic group of order 3 generated by
theimageof B = (i :I.The free product structure provides a description of the nontrivial elements of
k as unique strings of A's and B's with the property that there are no two
consecutive A's and no three consecutive B7s;we refer to these as reduced strings.
We explained this free product structure in terms of the action of the modular
group on the irrationals. In this note we describe the action on the rationals; this
can be viewed as a way of describing the inverse of the Euclidean algorithm.
The group SL2(Z)acts via linear transformations on R2 as column vectors and
this gives an action of .A' via linear fractional transformations on the projective
line P'(R), the real numbers together with m. We may also view P'(R) as the
slopes of nonzero vectors, that is, the equivalence classes of R2  (:)induced by
nonzero scalar multiplication; the equivalence class of the vector e = (i) is
, ,
denoted x, the equivalence class of the vector , q # 0 is the same as that of
(':I and corresponds to the real number z For the matrix (::)in
