Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Finite difference approximations are finite difference quotients in the terminology employed above. These algebraic approximations can be "exactly" expressed in terms of Dedekind eta quotients. The quotients by congruence subgroups are of significant interest. The canonical homomorphisms from "R" to the quotients formula 204 induce a homomorphism formula 205. Then use linear algebra to determine a maximal independent set among these eta quotients. First, the remainders are real numbers, although the quotients are integers as before. For Dedekind domains with all quotients by maximal ideals finite, SK is a torsion group. DM stacks are limited to quotients by finite group actions. More precisely, it relates the difference quotients to the Riemann sums. The badly approximable numbers are precisely those with bounded partial quotients. Assuming that we have not found formula 77 many linearly independent eta quotients. Rules for finding the elasticity of products and quotients are simpler than those for derivatives. Reduce the number of eta quotients to consider. For the extended algorithm, the successive quotients are used.