PROPERTIES OF RATIONAL NUMBERS

There are some properties of rational numbers like closure property, commutative property and associative property. 

Let us explore these properties on the four binary operations (Addition, subtraction, multiplication and division) in mathematics.

Addition

(i) Closure Property : 

The sum of any two rational numbers is always a rational number. This is called ‘Closure property of addition’ of rational numbers. Thus, Q is closed under addition

If a/b and c/d are any two rational numbers, then

(a/b) + (c/d) is also a rational number

Example : 

2/9 + 4/9  =  6/9  =  2/3 is a rational number

(ii) Commutative Property : 

Addition of two rational numbers is commutative.

If a/b and c/d are any two rational numbers, then

(a/b) + (c/d)  =  (c/d) + (a/b)

Example : 

2/9 + 4/9  =  6/9  =  2/3 

4/9 + 2/9  =  6/9  =  2/3 

So,

2/9 + 4/9  =  4/9 + 2/9

(iii) Associative Property :

Addition of rational numbers is associative.

If a/b, c/d and e/f  are any three rational numbers, then

a/b + (c/d + e/f)  =  (a/b + c/d) + e/f

Example :

2/9 + (4/9 + 1/9)  =  2/9 + 5/9  =  7/9 

(2/9 + 4/9) + 1/9  =  6/9 + 1/9  =  7/9 

So, 

2/9 + (4/9 + 1/9)  =  (2/9 + 4/9) + 1/9

(iv) Additive Identity :

The sum of any rational number and zero is the rational number itself.

If a/b is any rational number, then

a/b + 0  =  0 + a/b  =  a/b

Zero is the additive identity for rational numbers.

Example : 

2/7 + 0  =  0 + 2/7  =  2/7

(v) Additive Inverse :

(-a/b) is the negative or additive inverse of (a/b).

If a/b is a rational number, then there exists a rational number (-a/b) such that

a/b + (-a/b)  =  (-a/b) + a/b  =  0

Example : 

Additive inverse of 3/5 is (-3/5).

Additive inverse of (-3/5) is 3/5.

Additive inverse of 0 is 0 itself. 

Subtraction

(i) Closure Property : 

The difference between any two rational numbers is always a rational number.

Hence Q is closed under subtraction.

If a/b and c/d are any two rational numbers, then

(a/b) - (c/d) is also a rational number. 

Example : 

5/9 - 2/9  =  3/9  =  1/3 is a rational number. 

(ii) Commutative Property : 

Subtraction of two rational numbers is not commutative.

If a/b and c/d are any two rational numbers, then

(a/b) - (c/d)    (c/d) - (a/b)

Example : 

5/9 - 2/9  =  3/9  =  1/3

2/9 - 5/9  =  -3/9  =  -1/3 

And,

5/9 - 2/9    2/9 - 5/9

Therefore, Commutative property is not true for subtraction.

(iii) Associative Property :

Subtraction of rational numbers is not associative.

If a/b, c/d and e/f  are any three rational numbers, then

a/b - (c/d - e/f)    (a/b - c/d) - e/f

Example :

2/9 - (4/9 - 1/9)  =  2/9 - 3/9  =  -1/9 

(2/9 - 4/9) - 1/9  =  -2/9 - 1/9  =  -3/9 

And,

2/9 - (4/9 - 1/9)    (2/9 - 4/9) - 1/9

Therefore, Associative property is not true for subtraction.

Multiplication

(i) Closure Property :

The product of two rational numbers is always a rational number. Hence Q is closed under multiplication.

If a/b and c/d are any two rational numbers, then

(a/b) x (c/d)  =  ac/bd is also a rational number. 

Example : 

5/9 x 2/9  =  10/81 is a rational number. 

(ii) Commutative Property :

Multiplication of rational numbers is commutative.

If a/b and c/d are any two rational numbers, then

(a/b) x (c/d)  =  (c/d) x (a/b)

Example : 

5/9 x 2/9  =  10/81

2/9 x 5/9  =  10/81

So, 

5/9 x 2/9  =  2/9 x 5/9

Therefore, Commutative property is true for multiplication.

(iii) Associative Property :

Multiplication of rational numbers is associative.

If a/b, c/d and e/f  are any three rational numbers, then

a/b x (c/d x e/f)  =  (a/b x c/d) x e/f

Example :

2/9 x (4/9 x 1/9)  =  2/9 x 4/81  =  8/729 

(2/9 x 4/9) x 1/9  =  8/81 x 1/9  =  8/729

So,

2/9 x (4/9 x 1/9)  =  (2/9 x 4/9) x 1/9

Therefore, Associative property is true for multiplication.

(iv) Multiplicative Identity :

The product of any rational number and 1 is the rational number itself. ‘One’ is the multiplicative identity for rational numbers.

If a/b is any rational number, then

a/b x 1 = 1 x a/b  =  a/b

Example : 

5/7 x 1  =  1 x 5/7  =  5/7

(v) Multiplication by 0 :

Every rational number multiplied with 0 gives 0.

If a/b is any rational number, then

a/b x 0  =  0 x a/b  =  0

Example : 

5/7 x 0  =  0 x 5/7  =  0

(vi) Multiplicative Inverse or Reciprocal :

For every rational number a/b, b ≠ 0, there exists a rational number c/d such that a/b x c/d = 1.

Then, c/d is the multiplicative inverse of a/b.

If b/a is a rational number, then a/b is the multiplicative inverse or reciprocal of it.

Example : 

The multiplicative inverse of 2/3 is 3/2.

The multiplicative inverse of 1/3 is 3.

The multiplicative inverse of 3 is 1/3.

The multiplicative inverse of 1 is 1.

The multiplicative inverse of 0 is undefined.

Division

(i) Closure Property :

The collection of non-zero rational numbers is closed under division.

If a/b and c/d are two rational numbers, such that c/d ≠ 0, then a/b ÷ c/d is always a rational number. 

Example : 

2/3 ÷ 1/3  =  2/3 x 3/1  =  2 is a rational number.

(ii) Commutative Property :

Division of rational numbers is not commutative.

If a/b and c/d are two rational numbers, then

a/b ÷ c/d  ≠  c/d ÷ a/b

Example :

2/3 ÷ 1/3  =  2/3 x 3/1  =  2

1/3 ÷ 2/3  =  1/3 x 3/2  =  1/2

And, 

2/3 ÷ 1/3    1/3 ÷ 2/3

Therefore, Commutative property is not true for division.

(iii) Associative Property :

Division of rational numbers is not associative.

If a/b, c/d and e/f  are any three rational numbers, then

a/b ÷ (c/÷ e/f)    (a/b ÷ c/d) ÷ e/f

Example :

2/9 ÷ (4/9 ÷ 1/9)  =  2/9 ÷ 4  =  1/18 

(2/9 ÷ 4/9) ÷ 1/9  =  1/2 - 1/9  =  7/18 

And,

2/9 ÷ (4/9 ÷ 1/9)    (2/9 ÷ 4/9) ÷ 1/9

Therefore, Associative property is not true for division.

Distributive Property

(i) Distributive Property of Multiplication over Addition :

Multiplication of rational numbers is distributive over addition.

If a/b, c/d and e/f  are any three rational numbers, then

a/b x (c/d + e/f)  =  a/b x c/d  +  a/b x e/f

Example :

1/3 x (2/5 + 1/5)  =  1/3 x 3/5  =  1/5 -----(1)

1/3 x 2/5 + 1/3 x 1/5 :

=  2/15 + 1/15

  =  (2 + 1)/15

=   3/15

=  1/5 -----(2)

From (1) and (2), 

1/3 x (2/5 + 1/5)  =  1/3 x 2/5 + 1/3 x 1/5

Therefore, Multiplication is distributive over addition.

(ii) Distributive Property of Multiplication over Subtraction :

Multiplication of rational numbers is distributive over subtraction.

If a/b, c/d and e/f  are any three rational numbers, then

a/b x (c/d - e/f)  =  a/b x c/d  -  a/b x e/f

Example :

1/3 x (2/5 - 1/5)  =  1/3 x 1/5  =  1/15

1/3 x 2/5 - 1/3 x 1/5 :

=  2/15 - 1/15

=  (2 - 1)/15

=   1/15 -----(2)

From (1) and (2), 

1/3 x (2/5 - 1/5)  =  1/3 x 2/5 - 1/3 x 1/5

Therefore, Multiplication is distributive over subtraction.

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