Dedekind's theory of real number in hindi |BSC MATHEMATICS | DEDEKIND CUT |

In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind, are а method of construction of the real numbers from the rational numbers Intro Let Q be the set of rational numbers. It is well known that Q is an ordered field and also the set Q is equipped with a relation called “less than” which is an order relation. Between two rational numbers there exists an infinite number of elements of Q. 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., 2–√,3–√… etc.) which are not rational. For example, let we have to prove that 2–√ 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 2–√ is a rational number. Then according to the definition of rational numbers 2–√=pq , where p & q are relatively prime integers. Hence, (2–√)2=p2/q2 or p2=2q2 . This implies that p is even. Let p=2m , then (2m)2=2q2 or q2=2m2 . Thus q is also even if 2 is rational. But since both are even, they are not relatively prime, which is a contradiction. Hence 2–√ 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 Q 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 to work on the following definitions: Rational number A number which can be represented as pq where p is an integer and q is a non-zero integer i.e., p∈Z and q∈Z∖{0} and p and q are relatively prime as their greatest common divisor is 1, i.e., (p,q)=1 . Ordered Field Here, Q is, an algebraic structure on which the operations of addition, subtraction, multiplication & division by a non-zero number can be carried out. Least or Smallest Element Let A⊆Q and a∈Q . Then a is said to be a least element of A if (i) a∈A and (ii) a≤x for every x∈A . Greatest or Largest Element Let A⊆Q and b∈Q . Then b is said to be a least element of A if (i) b∈A and (ii) x≤b for every x∈A . 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 P , then the set of rational numbers is divided into two classes L and U . The rational numbers on the left, i.e. the rational numbers less than the number corresponding to the point of cut P are all in L and the rational numbers on the right, i.e. The rational number greater than the point are all in U . If the point P is not a rational number then every rational number either belongs to L or U . But if P is a rational number, then it may be considered as an element of U .