devotion & mathematics

EUCLIDEAN LINE, PLANE AND SPACES         We know about the real line or real number system R, R is one dimensional. Consider the two dimensional plane R 2 ={(a ,b)/ a ,b  Ꜫ  R}. In R 2  ,the elements of the form (a ,b) are called the ordered pairs of a & b. In R 2 , the addition and multiplication are defined as for x=( a 1 , b 1 ) , y=( a 2 , b 2 ) in R 2                                                                        x +y=(a 1+ a 2 ,b 1+ b 2  ) and x .y=(a 1. a 2  ,b 1. b 2 ) Here R 2  is called as an Euclidean  plane. Now consider the three dimensional space R 3 ={(a ,b ,c )/ a ,b ,c  Ꜫ  R}. In R 3 ,the elements of the form (a ,b ,c) are called ordered triples of a, b, c. The addition and multiplication in  R 3  are same...

  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 :       ...