The Real Numbers

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Introduction

       The real number system is the foundation of which the whole branch of mathematics known as ‘Real Analysis’ rests. The beauty of this mathematical system lies in the fact that for many important abstract mathematical systems the structure of the real number system serves as a model.

 

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1. Field Structure(Axoims)

 

       Let R be the set of real numbers having at least two distinct elements equipped with two fundamental algebraic operations called addition and multiplications and denoted by ‘+’ and ‘.’ respectively.

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A.1. The set R is closed with respect to addition i.e., a + b is a unique real number for any two real numbers a and b.

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A.2. (a + b) + c =a + (b + c)   ∀ a, b, c ∈ R         Associativity

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A.3. a + b = b + a        ∀ a, b ∈ R                      Commutativity

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A.4. There exists an element 0 in R : 0 + a = a    ∀ a, b ∈ R

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A.5. To each real number a there corresponds a real number, viz., -a, such that

      a + (- a) = 0

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M1.The set R is closed with respect to multiplication i.e., a.b is a unique real number for any two real numbers a and b.

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M2. a.b = b.a             ∀ a,b ∈ R      (commutativity)

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M3. (a.b).c =a.(b.c)     ∀ a,b,c ∈      (associativity)

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M4.There exists an element namely 1 ≠ 0 in R such that

    1.a = a      ∀ a ∈ R

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M5. To each element a ≠ 0 in R there exists an element 1/a in R such that

      (1/a) . a = 1        (1/a = a-1)

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AM    a(b + c) = ab + ac   ∀ a ,b, c ∈ R.     ( Distributive law)

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Because of the above properties the algebraic structure (R, +, . ) is called a field. As a matter of fact any mathematical system satisfying the above axioms is called a field. Thus we may speak of the field Q of rational numbers or the field C of complex numbers.

 

Subtraction and Division in R

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Def. The difference between two real numbers a and b is defined by a + (-b) and is denoted by a – b.

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The operation of finding the difference is called subtraction.

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Def. The quotient of a real number a by b real number b (b ≠ 0 ) is defined by a.b-1 , and is denoted by a/b.

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 The operation of finding quotient is called division.

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Some Properties of Real Numbers

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Some important consequences of the field properties of real numbers.

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1.   There exists a unique identity element for addition in R.

2.   There exists  a unique additive inverse for each element in R.

3.   a + b = a + c ⇒ b  = c.

4.   a + b = a ⇒ b = 0.

5.   a + b = 0 ⇒ b = -a.

6.   –(-a) = a.

7.   There exists a unique identity element for multiplication in R.

8.   There exists a unique multiplication inverse for each non-zero element in R.

9.   a ≠ 0, a . b = a . c ⇒ b = c.

10.   a ≠ 0, a . b = a ⇒ b = 1.

11.   a ≠ 0, a . b = 1 ⇒ b = 1/a.

12.   a ≠ 0 ⇒ 1/(1/a) = a.

13.   a . 0 = 0 ∀ a ∈ R.

14.   a ≠ 0, b ≠ 0 ⇒ a . b ≠ 0.

15.   a . b = 0 ⟺ a = 0 or b = 0.

16.   a . (-b) = -(a .b) and (-a).b = -(a.b).

17.   (-a).(-b) = a.b .

18.   (-1).a = -a.

19.  –(a + b) = -a -b.

20.   1/a.b = (1/a).(1/b), a ≠ 0, b ≠ 0.

21.   If a and b are any two real numbers, then the equation x + a = b has a unique solution  x = [b + (-a)] = b – a in R.

22.  If a,b are any real numbers and a ≠ 0, then the equation ax = b has a unique solution  x =  (1/a).b = b/a in R.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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The order axioms

The order relation ‘greater than’ (>) between pairs of real numbers satisfies the following axioms :

1. For any two real numbers a, b one any one of the following is true:                                 a > b, a = b, a < b. Its known as the law of trichotomy.

2. For a, b, c ∈ R, a > b, b > c ⇒ a > c. Its known as law of transitivity.

3. For all real numbers a, b and c, a > b and c > 0 ⇒ a + c > b + c . Its known as monotone property for addition.

4. For all real numbers a, b and c, a > b and c > 0 ⇒ ac > bc. Its known as monotone property for multiplication.

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Some more definitions

We define some other relations in the terms of the relation ‘greater than’ on the real numbers.

1. The order relation ‘less than’(<) between the real numbers a and b is defined as            a < b if b > a.

2. A real number a is said to be greater than or equal to b ( a ≥ b ) if either                         a > b or a = b.

3. A real number a is said to be less than or equal to  b ( a ≤ b ) if either                              a < b or a = b.

4. A real number a is said to be positive if a > 0.

5. A real number a is said to be negative if a < 0.

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Some properties of order relation

1.  For each real number a, one and only one of the following holds:                                   a > 0, a = 0, -a > 0.

2.  For each real number a, one and only one of the following holds:                                         a < 0, a = 0, -a < 0.

3.  (i)   a ∈ R^+ ⟺ a > 0 and a ∈ R^- ⟺ a < 0.  (ii)  a ∈ R^+, b ∈ R^- ⇒ a > b                          i.e.  every positive number is greater than every negative number.

4. a, b ∈ R^+ ⇒ a + b ∈ R^+ and ab ∈ R^+ i.e., a > 0, b > 0 ⇒ a + b > 0 and ab > 0.

5. a,b ∈ R^- ⇒ a + b ∈ R^- and ab ∈ R^+.

6. a < b and b < c ⇒ a < c.

7. a < b ⟺ a + c < b + c,        a < b and c < ⇒ ac > bc.

8. a < 0 ⟺ -a > 0, a > 0 ⟺ -a < 0

9. a > b ⟺ a – b > 0, a < b ⟺ a – b< 0.

10. a > b ⟺ -a < -b.

11. a > 0 ⟺ 1/a > 0.

12. (i) a > b > 0 ⇒ 1/b > 1/a > 0.     (ii) 0 < a < b ⇒ 1/a > 1/b.

13. a ≠ 0 ⇒ a² > 0. In particular 1 > 0.

14. a > b > 0 ⇒ a² > b², a < b < 0 ⇒ a² > b².

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The relations ‘≥’ and ‘≤’ are known as the weak inequalities while the relations ‘>’ and ‘<’ are known as the strict inequalities.

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Questions

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1.      Show that there is no rational number whose square is 2.

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Solution. Let us assume that there exists a rational number whose square is 2. Let p, q be two integers without a common factor.

                   (p/q) ² = 2.

            ⇒ p² = 2q².

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Now, q is a integer, so is q² and 2q². Thus p² is an integer divisible by 2. A such p must itself be divisible by 2, for otherwise, its square would not be divisible by 2. Let p = 2n where n is an integer.

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    ⇒ q² = 2n²

So that the integer q is also divisible by 2.

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p, q have a common factor 2. Thus our assumption is wrong.

So that, there exists no rational number whose square is 2.

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2.      Show that there is no rational number whose square is 3.

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Solution. Let us assume that there exists a rational number whose square is 3. Let p, q be two integers without a common factor.

                   (p/q) ² = 3.

            ⇒ p² = 3q².

Now, q is an integer, so is q² and 3q². Thus p² is an integer divisible by 3. A such p must itself be divisible by 3, for otherwise, its square would not be divisible by 3. Let p = 3n where n is an integer.

    ⇒ q² = 3n²

So that the integer q is also divisible by 3.

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p, q have a common factor 2. Thus our assumption is wrong.

 

3.      Show that there exists no rational number whose square is 8.

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Solution. Let us assume that there exists a rational number, whose square is 8. Let q be the smallest positive integer such that for some integer p, (p/q) ² = 8.

 ⇒          2 < p/q < 3

⇒           2q < p < 3

⇒              0 < p- 2q < q.

Thus, p-2q is a positive integer smaller than q and as such

                        (p/q)(p-2q) is not is an integer.

Again, (p/q)(p-2q) = (p²/q) – 2p

                              = (p²/q²).q – 2p = 8q – 2p = is an integer.

So (p²/q²)(p – 2q) is an integer.

Thus, we arrive at a contradiction, Hence, √8 is not a rational number.

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