Topology of Numbers · Problem 2.1.7

Continued Fractions — Allen Hatcher

Mathematics — Number Theory · continued-fractions-uniqueness

Problem

Show that the continued fraction for a rational number is unique except for replacing a final term

\frac{1}{a_n}

\frac{1}{a_n - 1} + \frac{1}{1}

when

a_n > 1

. For example

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