Rational and Irrational Numbers. Approximation of Numbers by Rational Numbers (Diophantine Approximation)

Abstract

As we have already mentioned, the numbers we know the best are the integers. These occur very infrequently among the real numbers. Quotients of these, the rational numbers, occur much more often, but as we will see, in a certain sense, they make up only an insignificant part of the real numbers. It is rather surprising, however, that it can be quite difficult to decide whether or not a given number is rational.