Sets of Numbers

We shall be using capital letters $$\mathbb{N},\mathbb{Z},\mathbb{Q}$$ and $$\mathbb{R}$$ for the sets of numbers as specified below:

$$\mathbb{N} = \left\{ {x:x = 1,2,3 \ldots } \right\}$$, the set of natural numbers.
$$\mathbb{Z} = \left\{ {x:x = \ldots , – 2, – 1,0,1,2 \ldots } \right\}$$, the set of integers.
$$\mathbb{Q} = \left\{ {x:x{\text{ is a rational numbers}}} \right\}$$, the set of rational numbers.
$$\mathbb{R} = \left\{ {x:x{\text{ is a real numbers}}} \right\}$$, the set of real numbers.

Thus, $$a,b \in \mathbb{N}$$ means that $$a$$ and $$b$$ are natural numbers and similarly $$c,d \in \mathbb{R}$$ means that $$c$$ and $$d$$ 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.

Rational Numbers
A real number $$x$$ such that $$nx = m$$, where $$n \in \mathbb{N} \wedge m \in \mathbb{Z}$$ is said to be a rational number. It can be expressed as $$m{n^{ – 1}}$$ or $$\frac{m}{n}$$, where $$n \ne 0$$ and $$m,n \in \mathbb{Z}$$.

We shall restrict $$m$$ and $$n$$ (except when $$m = 0$$) in such a manner that they may have no common divisors other than $$\mathbb{Z}$$. In such a case $$m$$ and $$n$$ are called relatively prime integers. For $$m = 0$$, we have $$x = \frac{m}{n} = 0$$, as $$n \ne 0$$.

Irrational Numbers
A real number which is not rational (i.e. not expressible as $$m/n$$) is called an irrational number. Since $$\mathbb{Q}$$ denotes the set of rational numbers, and $$\mathbb{R} – \mathbb{Q}$$ stands for the set of irrational numbers.

In the present study, the set $$\mathbb{R}$$ acts as the universal set $$U$$ in respect to sets consisting of only real numbers. Various sets dealt with in this unit are to be taken as subsets of $$\mathbb{R}$$.