Which of these properties hold false for Multiplication of rational numbers?
Answers
Step-by-step explanation:
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