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

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Properties of Multiplication of Rational Numbers

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