Two Answers to a Common Question on Diagonalization

TL;DR

The paper explores two explanations—using decimal expansions and continued fractions—for why diagonalization proves the uncountability of real numbers but not rationals, clarifying a common student question.

Contribution

It introduces two distinct answers, one traditional and one unconventional, to explain the limitations of diagonalization in counting rational numbers.

Findings

Decimal expansion approach clarifies diagonalization limits.

Continued fractions offer an alternative perspective.

Both explanations enhance understanding of uncountability proofs.

Abstract

A common question from students on the usual diagonalization proof for the uncountability of the set of real numbers is: when a representation of real numbers, such as the decimal expansions of real numbers, allows us to use the diagonalization argument to prove that the set of real numbers is uncountable, why can't we similarly apply the diagonalization argument to rational numbers in the same representation? why doesn't the argument similarly prove that the set of rational numbers is uncountable too? We consider two answers to this question. We first discuss an answer that is based on the familiar decimal expansions. We then present an unconventional answer that is based on continued fractions.