Let be the set of rational numbers. It is well known that is an ordered field and also the set is equipped with a relation called “less than” which is an order relation. Between two rational numbers there exists infinite number of elements of . Thus the system of rational numbers seems to be dense and so apparently complete. But it is quite easy to show that there exist some numbers (?) (e.g., etc.) which are not rational. For example, let we have to prove that is not a rational number or in other words, there exist no rational number whose square is 2. To do that if possible, purpose that is a rational number. Then according to the definition of rational numbers , where p & q are relatively prime integers. Hence, or . This implies that p is even. Let , then or . Thus is also even if 2 is rational. But since both are even, they are not relatively prime, which is a contradiction. Hence is not a rational number and the proof is complete. Similarly we can prove that why other irrational numbers are not rational. From this proof, it is clear that the set is not complete and dense and that there are some gaps between the rational numbers in form of irrational numbers. This remark shows the necessity of forming a more comprehensive system of numbers other that the system of rational number. The elements of this extended set will be called a real number. The following three approaches have been made for defining a real number.
- Dedekind’s Theory
- Cantor’s Theory
- Method of Decimal Representation
The method known as Dedekind’s Theory will be discussed in this not, which is due to R. Dedekind (1831-1916). To discuss this theory we need the following definitions:
Ordered Field: Here, is, an algebraic structure on which the operations of addition, subtraction, multiplication & division by a non-zero number can be carried out.
Dedekind’s Section (Cut) of the Set of All the Rational Numbers
Since the set of rational numbers is an ordered field, we may consider the rational numbers to be arranged in order on straight line from left to right. Now if we cut this line by some point , then the set of rational numbers is divided into two classes and . The rational numbers on the left, i.e. the rational numbers less than the number corresponding to the point of cut are all in and the rational numbers on the right, i.e. The rational number greater than the point are all in . If the point is not a rational number then every rational number either belongs to or . But if is a rational number, then it may be considered as an element of .
Let satisfying the following conditions:
Let . Then the ordered pair is called a section or a cut of the set of rational numbers. This section of the set of rational numbers is called a real number.
Notation: The set of real numbers is denote by .
Let then and are called Lower and Upper Class of respectively. These classes will be denoted by and respectively.
From the definition of a section of rational numbers, it is clear that and . Thus a real number is uniquely determined iff its lower class is known.
Let Then prove that is a lower class of a real number.
Proof: Since and is non-empty proper subset of .
Let and . If then . If so .
Let . If then . If then .
Let for and , for any .
And similarly, .
Thus, and Hence has no greatest element.
Since, satisfies all the conditions of a section of rational numbers, it is a lower class of a real number. [Proved]
Remark: In the given problem, is an upper class of a real number given by the set , since it has no smallest element.
Real Rational Number: The real number is said to be a real rational number if its upper class has a smallest element. If is the smallest element of , then we write .
Irrational Number: The real number is said to be an irrational number if does not have a smallest element.
Important Theorems & Results
If is a section of rational numbers, then
- is a non-empty proper subset of .
- if is a positive rational number, then there exists such that
- if contains some positive rational numbers and then there exists and such that .