**Properties of division of rational numbers :**

Students who would like to practice problems on dividing rational numbers must be aware of the properties of dividing rational numbers.

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

Let us explore these properties on the binary operation division.

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.

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

Hence, 2/3 ÷ 1/3 ≠ 1/3 ÷ 2/3

Therefore, Commutative property is not true for division.

Division 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 ÷ 4 = 1/18

(2/9 ÷ 4/9) ÷ 1/9 = 1/2 - 1/9 = 7/18

Hence, 2/9 ÷ (4/9 ÷ 1/9) ≠ (2/9 ÷ 4/9) ÷ 1/9

Therefore, Associative property is not true for division.

Apart from the stuff "Properties of division of rational numbers", now, let us look at the stuff "Properties of multiplication of rational numbers".

**(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).

5/9 x 2/9 = 10/81

2/9 x 5/9 = 10/81

Hence, 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

Hence, 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 = 1x 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 = 0x 5/7 = 0

**(vi) Multiplicative Inverse or Reciprocal :**

For every rational number a/b, a≠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 reciprocal of 2/3 is 3/2

The reciprocal of 1/3 is 3

The reciprocal of 3 is 1/3

The reciprocal of 1 is 1

The reciprocal of 0 is undefined

**(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

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

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

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

Therefore, multiplication is distributive over subtraction.

After having gone through the stuff given above, we hope that the students would have understood "Properties of division of rational numbers".

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