Topology of Numbers · Problem 2.1.5

Continued Fractions — Allen Hatcher

Mathematics — Number Theory · continued-fractions-matrix-proof

Problem

This exercise is intended to illustrate the proof of Theorem 2.1 in the concrete case of the continued fraction

\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}

. (a) Write down the product

A_1 A_2 A_3 A_4 = \begin{pmatrix} 0 & 1 \\ 1 & a_1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & a_2 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & a_3 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & a_4 \end{pmatrix}

associated to

\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}

. (b) Compute the four matrices

A_1

A_1 A_2

A_1 A_2 A_3

A_1 A_2 A_3 A_4

and relate these to the edges of the zigzag path in the strip of triangles for

\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}

. (c) Compute the four matrices

A_4

A_3 A_4

A_2 A_3 A_4

A_1 A_2 A_3 A_4

and relate these to the successive fractions that one gets when one computes the value of

\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}

, namely

\frac{1}{5}

\frac{1}{4} + \frac{1}{5}

\frac{1}{3} + \frac{1}{4} + \frac{1}{5}

, and

\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}