Continued Fraction Expansion 1: Finite Expansions

This is the beginning of (hopefully) a series of questions on Continued Fraction Expansions.

This part intends to cover finite or infinite rational fraction expansions.

Knowledge of number theory is probably useful for this question.

Consider the finite sequences (a_n)_{n=0}^k and (b_n)_{n=0}^k, for some positive integer k (or infinity) where:

  • a_0 is an integer
  • The remaining terms in the sequence (a_n)_{n=0}^k are positive integers
  • b_n= image

For notational simplicity, you may denote b_n=[a_0;a_1,a_2,...,a_n]. The semicolon can be replaced by a comma, if so you wish.

If the sequence (b_n)_{n=0}^k is finite, let x=b_k. If the sequence is infinite, let x=\lim_{n \to \infty} b_n, where x is the real number represented by the continued fraction expansion. The terms b_1,b_2 etc. are called convergents and are approximations of x.

a) Deduce a method for expressing any rational number in continued fraction form. Essentially, this question is asking for an algorithm. You may want to try to express a few rational numbers into continued fraction from to get an idea of such an algorithm/method.

b) Show that every rational number can be represented by exactly two continued fractions.

The shorter representation (in number of terms in the sequence) is called the canonical representation.

c) What is the continued fraction expansion of the reciprocal of a rational number, according to the cases that arise?

This part is the most important one:

d) Show that the continued fraction expansion of a real number is finite if and only if the number is rational.

Hence, provided convergence, the continued fraction expansion of an irrational number is infinite and infinite continued fraction expansions represent an irrational number. These infinite expansions are much more interesting and I will hopefully make questions(s) on these soon. I will aim to also get the issues of convergence tackled, but for that part readers will need to know real analysis.