Real Numbers

INTRODUCTION TO Real Numbers:-

The type of numbers, we normally use, such as

1, 15.82,-0.01, 314, etc.                

 

Positive or negative, large or small, whole numbers or decimal numbers

are all Real numbers.

 They are called "Real numbers”. because they are not

Imaginary numbers.

THE ALGEBRIC AND ORDER PROPERTIES OF |R:-

INTRODUCTION:-

We will give a short list of basic properties of addition and      

Multiplication from which all other algebraic properties

which we can be derived as theorem.

                             

 We will use the conventional notation of(a+b)and

(a.b) when discussing the properties of addition and

Multiplication.

ALGEBRAIC PROPERTIES OF |R:-

On the set of R the real number there are two binary

Operatiions,denoted

by(+)and(.) called addition and multiplication, respectively.

These operations satisfy the following properties.

AXIOMS ABOUT ADDITION OF REAL NUMBERS:-

Ø  a+b=b+a for all a,b in |R.(commutative properties of addition).

Ø  2- (a+b)+c = a+(b+c) for all a,b,c in |R (associative property of addition).

Ø  3- there exist an element 0 in |R s.t 0+a=a and a+0= a for all a in |R.(existence a zero element)

Ø  4-for each a belongs |R there exist an element -a in |R

s.t.a+(-a)=0 and (-a)+a=0 (existence of negative element).

                       MULTIPLICATION:-

                               

Ø  a.b = b.a for all a,b belongs |R (commutative property of multiplication).

                              

Ø  2- (a.b).c = a.(b.c) for all a,b,c in |R(associative property in multiplication).

                               

Ø  3- there exists an element 1in |R distinct from 0 s.t. 1.a = a and a.1 = a for all  belongs to |R.(existence of a unit element).

Ø  4- for each a is not = in |R there exists an element 1/a in |R s.t.

a.(1/a)=1 and (1/a).a =1.(existence of reciprocals)

                               

Ø  5- a.(b+c)=(a.b)+(a.c) and (b+c).a=(b.a)+(c.a) for all a,b,c belongs to |R

(Distributive property of multiplication over addition).

ROTATONAL AND IRRATIONAL NUMBERS:-

     We regard the set |N of natural numbers as a subset of |R, by

       Identifying the natural number n€|N with the n-fold sum of the unit

Element 1€|R. Similarly, we identifying “0” belongs |Z, “0” element of

“0”belongs to |R, and we identifying the n-fold of “-1” with the integer –n.

Thus, we consider |N and |Z to be subset of |R.

    

 Element of |R that can be written in the form of b/a, where a,b belongs t                                 o|Z and a is not equal to”0” are called rational numbers.

     Notation:- Q

               One consequence is that element of |R that are not in Q became known as irrational numbers, its mean that they are not rational integers.

THE ORDER PROPERTIES OF |R:-

                              There is nonempty subset real positive of |R called the set of positive real numbers, that satisfies the following properties.

Ø  If a,b belongs to real positive numbers ,then a=b belongs to real positive numbers.

Ø  If a,b belongs to real positive numbers, then a.b belongs to real positive numbers.

Ø  If a belongs to real number, then exactly on of the following holds.

·         a belongs to real positive numbers

·         a is equal to 0

·         -a is belongs to real positive numbers.

                            and is usually called the Trichotomy properties.

Ø  If a belongs to real positive numbers, a>0 & say that a is positive (or strictly positive) real number.

Ø  If a€|R+{0},a≥0 & say that a is nonnegative real number.

Ø  If -a€|R+U{0}, a≤0 & say that a is no positive real number.

Note:-

     The notation of inequality between two real numbers will how, be defined in term of then set |R+ of positive elements.

Definition:-

Ø  Let a,b be element of |R

      If a-b €|R+, then we write a<b or b<a.

      If a-b€|R+ U{0},then we write a≥b or b≤a.

ABSOLUTE VALUE AND THE REAL LINE:-

      From the trichotomy property, if a€|R, then exactly one of the    following holds.

                        a€|P,a=0,-a=|P.

    Assured that if a€|R and a ≠0, then exactly one of the number a&-a is positive.

The absolute value of a≠o is defined to be the positive one of these two numbers. The absolute value of 0 is defined to be 0.

DEFINITION:-

          The absolute value of a real number a, denoted by |a| is defined by

      |a|= {a, if a>0,

          {0, if a=0,

          {-a,if a<0.

   For example,|5|=5 and |-8|=8.

THE REAL LINE:-

    DEFINITION:-

           Let a€|R and e>0.Then the e- neighborhood of a is the set Ve(a):={x€|R |x-a|<e}.

THE PROPERTY OF SUPREMUM:

DEFINITION:-

              S be a non empty subset of r,m€|R is called a least upper bound of s if

Ø M is an upper bound of s, i.e. x≤ m¥ x€ s.

Ø Any upper bound, k€ s is > or = m ,i.e.m≤k

Ø Example:-s={1,2,3}

Therefore 3 is supremum.

GREATEST LOWER BOND (glb) (INFIMUM):-

              “S” be a nonempty subset of |R, k€|R is called infimum or glb of “s” if.

Ø “k” lower bound of “s” i.e. k≤x¥x€s.

Ø If “l” Is lower bound of “s” then l≤k.

For example.

S={1,2,3}

Therefore 1 is infimum of greatest lower bound.

COMPLETENESS AXIOM:

           If A is nonempty has infimum then it has a uniq infimum.

ARCHEMEDIAN’S PRINCIPLE:

          For any the real number” x”, there exist a natural number “n”

Such that x<n.

There is infinity many irrationals between two distinct real numbers.

DENSENESS PROPERTY:-

                Between two distinct real number there exist many irrationals.

              

                                                                                                                                - Submitted by Aman Singh

Posted in: Formulas, Maths