Extended Integer Angles and Their Summation

Abstract

Let us start with the following question. Suppose that we have arbitrary numbers a, b, and c satisfying a+b=c. How do we calculate the continued fraction for c knowing the continued fractions for a and b? It turns out that this question is not a natural question within the theory of continued fractions. One can hardly imagine any law to write the continued fraction for the sum directly. The main obstacle here is that the summation of rational numbers does not have a geometric explanation in terms of the integer lattice. In this chapter we propose to consider a “geometric summation” of continued fractions, which we consider a summation of integer angles.

We start with the notion of extended integer angles that are the integer analogues of Euclidean angles of the type kπ+φ for arbitrary integers k. We classify extended angles by writing normal forms representing all of them. Then we define the M-sums of extended angles and integer angles. Further, we express the continued fractions of extended angles in terms of the corresponding normal forms. Finally, we conclude the proof of theorem on the sum of integer angles in integer triangles, which is based on the techniques introduced in this chapter.