Recent discussions suggested the idea of associating some kind of invariant to an irrational real number. So suppose c is an irrational real number. Then it's natural to consider the subgroup S_c of the unit circle group S^1 = {z in C | |z| = 1} defined by S_c = exp(2πicZ) = {exp(2πic*n) | n in Z}. It seems hard to extract information from the group S_c that would distinguish one irrational c from another. But actually some things about S_c are closely related to the continued fraction expansion of c. I'm not even sure how you would try to distinguish S_c for an algebraic c from a S_c transcendental one, no less a quadratic irrational from a quartic one. But what if we consider the subring of C generated by S_c ??? Call it R(c). Which is just the ring Z[exp(2πic], generated over the integers by the complex number exp(2πic): R(c) = Z[exp(2πic)]. Now with this commutative ring we can form the group SL(2, R(c)) of 2 x 2 matrices over this ring with determinant = 1. Does the algebraic structure of SL(2, R(c)) distinguish different kinds of irrational reals c ??? —Dan