PROPERTIES OF RATIONAL NUMBER - 2

Associativity

1. Addition: For any three rational numbers, say, 3/4, 5/8, and – 2/3, we have to prove

                                    3/4 + {5/8 + (– 2/3)} = {3/4 + 5/8} + (– 2/3)

LHS = 3/4 + {5/8 + (–2/3)}

        = 3/4 + {15 – 16/24}

        = 3/4 – 1/24

        = 18 – 1/24

        = 17/24

RHS = {3/4 +5/8} + (–2/3)

         = {6 + 5/8} + (–2/3)

         = 11/8 – 2/3

         = 33 – 16/24

         = 17/24

 LSH = RHS

Addition is associative for rational numbers. In general, for any three rational numbers a, b and c,

                                    a + (b + c) = (a + b) + c

2. Multiplication: For any three rational numbers, say, 8/9, 5/11 and 7/3, we have to prove

                        8/9 ×(5/11 × 7/3) = (8/9 × 5/11) × 7/3

                

 LHS = 8/9 × (5/11 × 7/3)

          = 8/9 × 35/33

          = 280/297

                        RHS = (8/9 × 5/11) × 7/3

                                   = 40/99 × 7/3

                                   = 280/297

               

 LHS = RHS

Therefore, multiplication is associative for rational numbers. In general, for any three rational numbers a, b and c,

                                    a × (b × c) = (a × b) × c

3. Subtraction: For any three rational numbers, say, 3/4, 5/6 and 2/3, we have to prove

                        3/4 – (5/6 – 2/3) = (3/4 – 5/6) – 2/3

                        LHS = 3/4 – (5/6 – 2/3)

                                   = 3/4 – 1/6 = 9 – 2/12

                                   = 7/12

                        RHS = (3/4 – 5/6) – 2/3

                                   = (9 – 10/12) – 2/3

                                   = – 1/12 – 2/3

                                 = – 1– 8/12

                                    = –9/12 = –3/4

i.e.,                    LHS ≠ RHS

Therefore, subtraction is not associated for rational numbers. In general, for any three rational numbers a, b and c,

                                                a – (b – c) ≠ (a – b) – c

4. Division: For any three rational number, say, 3/5, 2/7 and 4/9, we have to prove

                                    3/5 ÷ (2/7 ÷ 4/9) = (3/5 ÷ 2/7) ÷ 4/9

                         LHS = 3/5 ÷ (2/7 ÷ 4/9)

         = 3/5 ÷ (2/7 × 9/4)

         = 3/5 × 14/9

         = 14/15

                        RHS = (3/5 ÷2/7) ÷ 4/9

                                    = ( 3/5 × 7/2) ÷ 4/9

                                    = 21/10 ÷ 4/9

                                    = 21/10 × 9/4

                                     =189/40

i.e.,                    LHS ≠ RHS

Therefore, division is not associative for rational numbers. In general, for any three rational numbers a, b and c,    

                                                a  ÷ (b  ÷ c) ≠ (a  ÷ b) ÷ c

 Distributivity

For any three rational numbers, say, 4/5, 4/3 and 6/5, we have to prove

                        4/5 ×{4/3 + 6/5} = {4/5 × 4/3} + {4/5 × 6/5}

            LHS = 4/5 ×{4/3 + 6/5}

                      = 4/5 × {20 +18/15}

                      = 4/5 × 38/15

                      = 152/75

            RHS = {4/5 × 4/3} + {4/5 × 6/5}

                      = 16/15 + 24/25

                      = 80 + 72/75

                      = 152/75

i.e.,        LHS = RHS

Therefore, multiplication is distributive over addition for rational numbers. In general, for any three rational numbers a, b and c,

                        a ×( b + c) = (a ×b) + (a × c)                                                                 i

Similarly, we can pore that multiplication is distributive over subtraction for rational numbers, that is

                        a × (b – c) = (a × b) – (a × c)                                                                ii

It may be noted that in equation (i) if a and b are positive and c is negative, then equation (i) becomes equation (ii).

Additive Identity

Consider the following

                                    5/9 + 0 = 0 + 5/9

                                                    = 5/9

                                 23/25 + 0 = 0 + 23/25

                                                    = 23/25

As is clear for the above examples, when 0 is added to any rational number, it leaves the rational number unaltered. Therefore, 0 is called the identity of addition or additive identity of rational numbers.

In general, for any rational number p,

                        p + 0 = 0 +p = p

Additive Inverse

For every rational number p/q, there is a rational number – p/q such that

                                    p/q + – p/q = – p/q + p/q = 0

Here – p/q is termed as the additive inverse of p/q. Also p/q is the additive inverse of – p/q.

Note: Since 0 + 0 = 0, the additive inverse of zero is zero itself.

Multiplicative Identity

Consider the following expression

                                                4/9 × 1 = 1 × 4/9 = 4/9

                                                17/25 × 1 = 1 × 17/25 = 17/25

When 1 is multiplied with any rational number, it leaves the rational number unaltered. One (1) is called the identity of multiplication or multiplicative identity of rational numbers.

In general, for any rational number p,

                                                            p × 1 = 1× p = p

Multiplicative Inverse

For a given rational number, a/b, if there exists another rational number c/d such that

a/b ×c/d = 1

then c/d is called the multiplicative inverse or reciprocal of a/b.

If c/d is the reciprocal of a/b, then a/b is the reciprocal of c/d.

In other words, they are both inverse or reciprocals of each other.

Note: 1. Since – 1× –1 = + 1 and + 1 × + 1 = + 1, – 1 and + 1 are their own reciprocals.

          2. There is no rational number which when multiplied by 0 gives 1 as product. Hence zero has no reciprocal or multiplicative inverse.

For example:                11/15 × 15/11 = 1,  –11/17 × – 17/11 = 1.