If you are looking for help on Properties of Multiplication of Rational Numbers you have come the right way. Check out Closure, Commutative, Associative, Existence of Multiplicative Inverse Property, Distributive Property, Multiplicative Property of 0. Get to Know the Multiplication of Rational Numbers Properties along with few examples and get a better idea of the concept.

### Closure Property of Multiplication of Rational Numbers

Rational Numbers are closed under Multiplication. Let us assume two rational numbers a/b, c/d then their product (a/b*c/d) is also a Rational Number. Check out a few examples listed to understand the Closure Property better.

**Examples**

(i) Consider the two rational numbers 1/3 and 4/7 then

1/3*4/7

= 4/21

Therefore, 4/21 is also a Rational Number.

(ii) Consider two rational numbers -3/4 and 5/8

= -3/4*5/8

= -15/32

Thus, Product or Multiplication of Rational Numbers -3/4, 5/8 is also a Rational Number -15/32.

### Commutative Property of Multiplication of Rational Numbers

Multiplication of Rational Numbers is Commutative. Two Rational Numbers can be multiplied in any order. Let us consider two rational numbers a/b, c/d then

(a/b*c/d) = (c/d*a/b)

**Example**

(i) Consider Two Rational Numbers 3/4 and 5/2 then

(3/4*5/2) = 3/4*5/2 = 15/8

(5/2*3/4) = 5/2*3/4 = 15/8

Therefore, (3/4*5/2) = (5/2*3/4)

(ii) Consider Two Rational Numbers -4/5 and -3/7

(-4/5*-3/7) = (-4*-3)/35 = 12/35

(-3/7*-4/5) =(-3*-4)/7*5 = 12/35

Therefore, (-4/5*-3/7) = (-3/7*-4/5)

### Associative Property of Multiplication of Rational Numbers

Rational Numbers obey the Associative Property of Multiplication. While Multiplying Three or More Rational Numbers they can be grouped in any order. Consider Rational Numbers a/b, c/d, e/f then we have (a/b × c/d) × e/f = a/b × (c/d × e/f).

**Example**

Consider Rational Numbers 1/2, 4/5, 6/4 then

(1/2*4/5)*6/4 =(1*4/2*5)*6/4

= (4/10)*6/4

= 4*6/10*4

= 24/40

1/2*(4/5*6/4) = 1/2*(4*6/5*4)

= 1/2*(24/20)

=24/40

Therefore, (1/2*4/5)*6/4 = 1/2*(4/5*6/4)

### Existence of Multiplicative Identity Property

For a Rational Number a/b we get (a/b*1) = (1*a/b) = a/b. 1 is called the Multiplicative Identity of Rationals.

**Example**

(i) Consider the Rational Number 4/3 we have

(4/3*1) = (4/3*1/1) = (4*1/3*1) = 4/3

(1*4/3) = (1/1 *4/3) = (1*4/1*3) = 4/3

(4/3*1) = (1*4/3) = 4/3

### Existence of Multiplicative Inverse Property

Nonzero rational number a/b has multiplicative inverse b/a.

(a/b*b/a) = (b/a*a/b) = 1

b/a is called the Reciprocal of a/b. Zero has no reciprocal.

**Example**

(i) Reciprocal of 4/7 is 7/4 since 4/7*7/4 = 1

(ii) Reciprocal of -5/3 is -3/5 since -5/3*-3/5 = 1

### Distributive Property of Multiplication over Addition

Let us assume three rational numbers a/b, c/d, e/f . Distributive Property states that a/b*(c/d+e/f) = (a/b*c/d+a/b*e/f)

**Example**

Consider three Rational Numbers 1/3, 4/5, 7/8 then

1/3*(4/5+7/8) =1/3*((4*8+7*5)/40)

= 1/3*(67/40)

= 67/120

(1/3*4/5+1/3*7/8) = (1*4/3*5+1*7/3*8)

= (4/15+7/24)

= (4*8+7*5)/120

= 32+35/120

= 67/120

Therefore, 1/3*(4/5+7/8) = (1/3*4/5+1/3*7/8)

### Multiplicative Property of 0

Any Rational Number multiplied with 0 gives 0. For a Rational Number a/b, we have(a/b*0) = (0*a/b) = 0

**Example**

(i) (4/7*0) = (4/7*0/1) = (4*0/7*1) = 0

Similarly (0*4/7) = (0/1*4/7) =0

(ii)(-12/5*0) =(-12/5*0/1) =(-12*0/5*1) = 0

Similarly (0*-12/5) = (0/1*-12/5) = 0*-12/1*5 = 0