# Arithmetic of continued fractions, does it exist?

I'm interested in the arithmmetic of continued fractions and specially in multiplication. Consider $$f(x)=\cfrac{f_{0}(x)}{1-\cfrac{f_{1}(x)}{1+f_{1}(x)-\cfrac{f_{2}(x)}{1+f_{2}(x)-\cfrac{f_{3}(x)}{1+f_{3}(x)-\cdots}}}}$$

and

$$g(x)=\cfrac{g_{0}(x)}{1-\cfrac{g_{1}(x)}{1+g_{1}(x)-\cfrac{g_{2}(x)}{1+g_{2}(x)-\cfrac{g_{3}(x)}{1+g_{3}(x)-\cdots}}}}$$

Are there arithmetic rules (algorithms) for the multiplication of continued fractions? Specifically, is it possible to obtain a continued fraction $h(x)=f(x)\cdot g(x)$ where

$$h(x)=\cfrac{h_{0}(x)}{1-\cfrac{h_{1}(x)}{1+h_{1}(x)-\cfrac{h_{2}(x)}{1+h_{2}(x)-\cfrac{h_{3}(x)}{1+h_{3}(x)-\cdots}}}}$$

I've found this, but I'd like more. Does anyone here knows of papers, algorithms, etc?

Thanks.

Update 1: Another link.

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Well, there's always Wikipedia... – shezi Oct 26 '11 at 12:51
@Neves:the answer is yes,kindly see my conjecture in this post – Nicco Sep 10 '15 at 3:22

## 1 Answer

Bill 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)

For illustration how to perform some continued fraction arithmetic see this.

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Also here. – J. M. Oct 28 '11 at 14:30
To expand on J.M.'s comment: This is a verbatim copy of the wolfram alpha page @J.M. is linking to. See also this answer where the same thing happened, a bit more shamelessly. – t.b. Oct 28 '11 at 15:15