We shall be using capital letters and for the sets of numbers as specified below:
, the set of natural numbers.
, the set of integers.
, the set of rational numbers.
, the set of real numbers.
Thus, means that and are natural numbers and similarly means that and are real numbers. In this way, the nature of a number in use can always be describe in brief.
At the present stage we are generally too familiar with the use of real numbers, without being attentive to the reasons for which their various properties hold. The history of systemic development of real numbers, however, does not go back very long. It was only in the last quarter of the 19th century that three proper theories of the structure of real numbers were separately proposed by three German mathematicians, viz. K. Weierstrass, R. Dedekind and G. Cantor.
A real number such that , where is said to be a rational number. It can be expressed as or , where and .
We shall restrict and (except when ) in such a manner that they may have no common divisors other than . In such a case and are called relatively prime integers. For , we have , as .
A real number which is not rational (i.e. not expressible as ) is called an irrational number. Since denotes the set of rational numbers, and stands for the set of irrational numbers.
In the present study, the set acts as the universal set in respect to sets consisting of only real numbers. Various sets dealt with in this unit are to be taken as subsets of .