  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.
