# Properties of Multiplication of Rational Numbers

We will learn the properties of multiplication of rational numbers i.e. closure property, commutative property, associative property, existence of multiplicative identity property, existence of multiplicative inverse property, distributive property of multiplication over addition and multiplicative property of 0.

Closure property of multiplication of rational numbers:

The product of two rational numbers is always a rational number.

If a/b and c/d are any two rational numbers then (a/b × c/d) is also a rational number.

For example:

(i) Consider the rational numbers 1/2 and 5/7. Then,

(1/2 × 5/7) = (1 × 5)/(2 × 7) = 5/14, is a rational number .

(ii) Consider the rational numbers -3/7 and 5/14. Then

(-3/7 × 5/14) = {(-3) × 5}/(7 × 14) = -15/98, is a rational number .

(iii) Consider the rational numbers -4/5 and -7/3. Then

(-4/5 × -7/3) = {(-4) × (-7)}/(5 × 3) = 28/15, is a rational number.

Commutative property of multiplication of rational numbers:

Two rational numbers can be multiplied in any order.

Thus, for any rational numbers a/b and c/d, we have:

(a/b × c/d) = (c/d × a/b)

For example:

(i) Let us consider the rational numbers 3/4 and 5/7 Then,

(3/4 × 5/7) = (3 × 5)/(4 × 7) = 15/28 and (5/7 × 3/4) = (5 × 3)/(7 × 4)

= 15/28

Therefore, (3/4 × 5/7) = (5/7 × 3/4)

(ii) Let us consider the rational numbers -2/5 and 6/7.Then,

{(-2)/5 × 6/7} = {(-2) × 6}/(5 × 7) = -12/35 and (6/7 × -2/5 )

= {6 × (-2)}/(7 × 5) = -12/35

Therefore, (-2/5 × 6/7 ) = (6/7 × (-2)/5)

(iii) Let us consider the rational numbers -2/3 and -5/7 Then,

(-2)/3 × (-5)/7 = {(-2) × (-5) }/(3 × 7) = 10/21 and (-5/7) × (-2/3)

= {(-5) × (-2)}/(7 × 3) = 10/21

Therefore, (-2/3) × (-5/7) = (-5/7) × (-2)/3

Associative property of multiplication of rational numbers:

While multiplying three or more rational numbers, they can be grouped in any order.

Thus, for any rationals a/b, c/d, and e/f we have:

(a/b × c/d) × e/f = a/b × (c/d × e/f)

For example:

Consider the rationals -5/2, -7/4 and 1/3 we have

(-5/2 × (-7)/4 ) × 1/3 = {(-5) × (-7)}/(2 × 4) ×1/3} = (35/8 × 1/3)

= (35 × 1)/(8 × 3) = 35/24

and (-5)/2 × (-7/4 × 1/3) = -5/2 × {(-7) × 1}/(4 × 3) = (-5/2 × -7/12)

= {(-5) × (-7)}/(2 × 12) = 35/24

Therefore, (-5/2 × -7/4 ) × 1/3 = (-5/2) × (-7/4 × 1/3)

Existence of multiplicative identity property:

For any rational number a/b, we have (a/b × 1) = (1 × a/b) = a/b

1 is called the multiplicative identity for rationals.

For example:

(i) Consider the rational number 3/4. Then, we have

(3/4 × 1) = (3/4 × 1/1) = (3 × 1)/(4 × 1) = 3/4 and ( 1 × 3/4 )

= (1/1 × 3/4 ) = (1 × 3)/(1 × 4) = 3/4

Therefore, (3/4 × 1) = (1 × 3/4) = 3/4.

(ii) Consider the rational -9/13. Then, we have

(-9/13 × 1) = (-9/13 × 1/1) = {(-9) × 1}/(13 × 1) = -9/13

and (1 × (-9)/13) = (1/1 × (-9)/13) = {1 × (-9)}/(1 × 13) = -9/13

Therefore, {(-9)/13 × 1} = {1 ×(-9)/13} = (-9)/13

Existence of multiplicative inverse property:

Every nonzero rational number a/b has its multiplicative inverse b/a.

Thus, (a/b × b/a) = (b/a × a/b) = 1

b/a is called the reciprocal of a/b.

Clearly, zero has no reciprocal.

Reciprocal of 1 is 1 and the reciprocal of (-1) is (-1)

For example:

(i) Reciprocal of 5/7 is 7/5, since (5/7 × 7/5) = (7/5 × 5/7) = 1

(ii) Reciprocal of -8/9 is -9/8, since (-8/9 × -9/8) = (-9/8 × -8/9 ) =1

(iii) Reciprocal of -3 is -1/3, since

(-3 × (-1)/3) = (-3/1 × (-1)/3) = {(-3) × (-1)}/(1 × 3) = 3/3 = 1

and (-1/3 × (-3)) = (-1/3 × (-3)/1) = {(-1) × (-3)}/(3 × 1) = 1

Note:

Denote the reciprocal of a/b by (a/b)-1

Clearly (a/b)-1 = b/a

Distributive property of multiplication over addition:

For any three rational numbers a/b, c/d and e/f, we have:

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

For example:

Consider the rational numbers -3/4, 2/3 and -5/6 we have

(-3)/4 × {2/3 + (-5)/6} = (-3/4) × {4 + -5/ 6} = (-3/4) × (-1)/6

= {(-3) × (-1)}/(4 × 6) = 3/24 = 1/8

again, (-3/4) × 2/3 = {(-3) × 2}/(4 × 3) = -6/12 = -1/2

and

(-3/4) ×(-5/6) = {(-3) × (-5)}/(4 × 6) = 15/24 = 5/8

Therefore, (-3/4) × 2/3 } + {(-3/4) × (-5/6)} = (-1/2 + 5/8 )

= {(-4) + 5}/8 = 1/8

Hence, (-3/4) × (2/3 + (-5)/6) = {(-3/4) × 2/3} + {(-3/4) × (-5)/6}.

Multiplicative property of 0:

Every rational number multiplied with 0 gives 0.

Thus, for any rational number a/b, we have (a/b × 0) = (0 × a/b) = 0.

For example:

(i) (5/18 × 0) = (5/18 × 0/1) = (5 × 0)/(18 × 1) = 0/18 .

Similarly, (0 × 5/8) = 0

(ii) {(-12)/17 × 0} = {(-12)/17 × 0/1} = [{(-12) × 0}/{17 × 1}] = 0/17

= 0.

Similarly, (0 × (-12)/17) = 0

Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Representation of Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

To Find Rational Numbers