write closure , commutative ,associative property of rational number for addition ,subtraction , multiplication , division with one example of each
Answers
Answer:
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)