THE REAL NUMBER SYSTEM (BSC 2ND SEM)

• Field Properties: A field is an algebraic structure (F,+,.) consisting of a non-empty set F and two binary operations '+' and '.' (called addition and multiplication respectively) defined on it and satisfying the following axioms for any element a,b,c of the set F

     (A) a+b=b+a and ab=ba (commutative laws)
     (B) (a+b)+c=a+(b+c) and (ab)c=a(bc) (associative laws)
     (C) a(b+c)=ab+ac (distributive law)
     (D) There are distinct real numbers 0 and 1 such that a+0=a and a.1=a for each a
     (E) For each 'a' there is a real number '-a' such that a+(-a)=0 and if a≠0 there is a real number 1/a  such that a(1/a)=1
     A set on which two operations are defined so as to have properties (A) to (E) is called field.
 

• Ordered Field: The real number system is ordered by the relation <, which has the following properties--

     (F) For each pair of real numbers a and b, exactly one of the following is true
          a=b or a<b or b<a
     (G) If a<b and b<c then a<c (the relation < is transitive)
     (H) If a<b then a+c<b+c for any c and if 0<c, then ac<bc.
     A field with an order relation satisfy (F) to (H) is an ordered field.