Gosper has invented an algorithm for performing analytic addition, subtraction, multiplication, and division using continued fractions. It requires keeping track of eight integers which are conceptually arranged at the polyhedron vertices of a cube. Although this algorithm has not appeared in print, similar algorithms have been constructed by Vuillemin (1987) and Liardet and Stambul (1998).
Gosper's algorithm for computing the continued fraction for (ax+b)/(cx+d) from the continued fraction for x is described by Gosper (1972), Knuth (1998, Exercise 220.127.116.11, pp. 360 and 601), and Fowler (1999). (In line 9 of Knuth's solution, X_k<-|_A/C_| should be replaced by X_k<-min(|_A/C_|,|_(A+B)/(C+D)_|).) Gosper (1972) and Knuth (1981) also mention the bivariate case (axy+bx+cy+d)/(Axy+Bx+Cy+D).
My description follows part of Gosper reproduced on perl.plover.com. This document is text and unnumbered, you may wish to start by searching for "Addition, Multiplication, etc. of Two Continued Fractions" prior to reading the whole thing.
So I can define arithmetic as operations on this matrix which make , , , , , zero and read the answer from and . This is more interesting when and are continued fractions, which is the subject of the following tasks.