How to approximate irrational numbers nicely?

TL;DR

This survey discusses a simple and accessible criterion for proving the irrationality of numbers through nice rational approximations, demonstrating its effectiveness on algebraic, exponential, and trigonometric irrationals.

Contribution

It highlights a straightforward irrationality criterion based on rational approximations, with explicit constructions and elementary arguments suitable for undergraduates.

Findings

Proves irrationality of various classes of numbers using the criterion

Provides explicit sequences of rational approximations

Uses elementary mathematical tools like binomial theorem

Abstract

In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of rational numbers. This criterion of irrationality is easy to prove and is of great importance. Using it, irrationality of a large class of numbers is proved. We shall apply this method to prove irrationality of algebraic, exponential and trigonometric irrational numbers. Most of the times, explicit constructions are done of the nice sequence of rational numbers. Arguments used are basic like the binomial theorem and maxima and minima of functions. Thus, this survey article is accessible to undergraduate students and has a broad appeal for anyone looking for entertaining mathematics. This article brings out classically known beautiful mathematics without using any heavyweight machinery.