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
Introduction of Rational Numbers
Is Every Rational Number a Natural Number?
Is Every Rational Number an Integer?
Is Every Rational Number a Fraction?
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
Rational Numbers in Descending Order
Representation of Rational Numbers on the Number Line
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 Same Denominator
Subtraction of Rational Number with Different Denominator
Subtraction of Rational Numbers
Properties of Subtraction of Rational Numbers
Rational Expressions Involving Addition and Subtraction
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
Reciprocal of a Rational Number
Rational Expressions Involving Division
Properties of Division of Rational Numbers
Rational Numbers between Two Rational Numbers
8th Grade Math Practice
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Worksheet on Properties of Division of Rational Numbers
Worksheet on Finding Rational Numbers between Two Rational Numbers
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