Hatcher-Topology-of-Numbers · Problem 2.1.2

Continued Fractions

Mathematics — Algebraic Topology · euclidean-algorithm-continued-fractions

Problem

(a) Compute the continued fraction for

38/83

and display the steps of the Euclidean algorithm for 38 and 83 as a sequence of equations involving only integers. (b) For the same number

38/83

compute the associated strip of triangles grouped into fans, including the labeling of the vertices of all the triangles. (c) Take the continued fraction

\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}

you got in part (a) and reverse the order of the numbers

a_i

to get a continued fraction

\frac{1}{a_n} + \frac{1}{a_{n-1}} + \cdots + \frac{1}{a_1}

. Compute the value

p/q

of this continued fraction, and also compute the strip of triangles for this fraction

p/q

. What is the relationship between

p/q

and

38/83

Related theory

Continued Fractions

Browse the problem library