devotion & mathematics

contents : 1. Extended real number system 2. The symbol infinity 3. Properties of infinity THE EXTENDED REAL NUMBER SYSTEM :                                    Basically the real number system not contain + or -  ∞, it  contains all the real numbers lies between for - ∞ and + ∞ i.e.  the real number system R=(- ∞. ∞).                                     The extended real number system is a union of the real number set R and the symbols - ∞ and + ∞ i.e. Extended real system = R ⋃ {- ∞, ∞} = [- ∞, ∞]    In extended real number system,  ∞ is an upper bound   and - ∞ is a lower bound of every subset. Here we see an example to tell the main difference between real and extended real number systems.     Consider an open interval (2, ∞).  The interval (2, ∞) i...

  Some important subsets of R . We define some important subsets of R the set of real numbers , namely 1. Inductive Set 2. The set of natural numbers 3. The set of Whole numbers             4.The set of integers                                                 5.The set of rational numbers 6.The set of irrational numbers   in different manner. Inductive set :    If A is a subset of R is  such that      i) 1  Ꜫ A    ii) p  Ꜫ A implies p+1  Ꜫ A , then A is called an Inductive set. Examples and counter examples :  1.  The set R, of real numbers is an inductive set.  2. the set  Z +  of all + ve integers is an inductive set. 3. the set  Z -   of all -ve integers is not inductive since 1  ∉  Z - Natural numbers : ...

  Axioms of real numbers :                         The structure of real analysis rests on the real number system. It is a development of the set of real numbers starting from the set of natural numbers.                    We consider real numbers as undefined elements satisfying                                         I.    Field axioms                                        II.  Order axioms                                        III. Completeness axioms Field axioms :       ...