# PROPERTIES OF MULTIPLICATION OF RATIONAL NUMBERS

## About "Properties of multiplication of rational numbers"

Properties of multiplication of rational numbers :

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

## 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.

## 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.

## 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.

## 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

## 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

1/3 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 multiplication of rational numbers".

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