Geometric Representation of Real Numbers
Sets
Fundamental in mathematics is the concept of a set, class, or collection of objects having specified characteristics. For example, we speak of the set of all university professors, the set of all letters A, B, C, D, . . . , Z of the English alphabet, and so on. The individual objects of the set are called members or elements. Any part of a set is called a subset of the given set, e.g., A, B, C is a subset of A, B, C, D, . . . , Z. The set consisting of no elements is called the empty set or null set.
Real Numbers
The number system is foundational to the modern scientific and technological world. It is based on the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Thus, it is called a base ten system. The significance of the base ten terminology is enhanced by the following examples:
The collection of numbers created from the basic set is called the real number system.Significant subsets of them are listed as follows.
1. Natural numbers1, 2, 3, 4, . . . , also called positive integers, are used in counting members of a set. The sum a + b and product a ・ b or ab of any two natural numbers a and b is also a natural number. This is often expressed by saying that the set of natural numbers is closed under the operations of addition and multiplication, or satisfies the closure property with respect to these operations.
2. Negative integers and zero,denoted by –1, –2, –3, . . . , and 0, respectively, arose to permit solutions of equations such as x + b = a, where a and b are any natural numbers. This leads to the operation of subtraction, or inverse of addition, and we write x = a – b. The set of positive and negative integers and zero is called the set of integers.
3. Rational numbersor fractions such as 2 5 , 3 4 − , . . . arose to permit solutions of equations such as bx = a for all integers a and b, where b ¹_ 0. This leads to the operation of division, or inverse of multiplication, and we write x = a/b or a ÷ b, where a is the numerator and b the denominator. The set of integers is a subset of the rational numbers, since integers correspond to rational numbers where b = 1.
4. Irrational numberssuch as 
and π are numbers which are not rational; i.e., they cannot be expressed as a/b (called the quotient of a and b), where a and b are integers and b ¹ 0. The set of rational and irrational numbers is called the set of real numbers.
Geometric Representation of Real Numbers
The geometric representation of real numbers as points on a line, called the real axis, as in Figure 1.1. For each real number there corresponds one and only one point on the line, and, conversely, there is a one-to-one (see Figure 1.1) correspondence between the set of real numbers and the set of points on the line. Because of this we often use point and number interchangeably.
Figure 1.1
The set of real numbers to the right of 0 is called the set of positive numbers, the set to the left of 0 is the set of negative numbers, while 0 itself is neither positive nor negative.
Between any two rational numbers (or irrational numbers) on the line there are infinitely many rational (and irrational) numbers. This leads us to call the set of rational (or irrational) numbers an everywhere dense set.