Page 19 - ICSE Math 8

−4 −×43 −12
LCM of 3 and 1 is 3. Thus, = =
1 13 3
×
(
 5  5 −12  ( −5) − −12) −+512 7
− 
−  
−
So, x =   −−4( ) ⇒ x =     = = =
 3   3   3  3 3 3
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The other rational number is .
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EXERCISE 1.3

1. Subtract.
−3 5 −4 5 4 2
(a) from (b) from (c) from
8 −7 21 14 5 3
2. Evaluate.
15 6 2  9   − 6  1 5
−
(a) − (b) (− ) −4 (c)     (d) −
11 7 7  − 10   15  8 24
Properties of Subtraction of Rational Numbers
p
I. Closure Property: “The difference of two rational numbers is always a rational number.” Thus, if and
q
r p r
s are two rational numbers, then q − is also a rational number.
s
3  − 8  7 1
Example 12: Find: (a) −   (b) −
7  5  2 2
Solution: (a) 3  − 8  = 15 −−( 56) = 15 56+ = 71 , which is a rational number.
−


7  5  35 35 35
7 1 71− 6 3
(b) − = = = 3 = , which is a rational number.
2 2 2 2 1
Thus, rational numbers are closed under subtraction.
p
II. Commutative Property: “In subtraction, the order of numbers cannot be changed.” Thus, if q and r are
s
p r r p
two rational numbers, then q − ≠ − q .
s
s
2 5 5 2 1  − 5   − 5  1
Example 13: Find if: (a) − = − (b) −     −
=
3 4 4 3 3  6   6  3
2 5 815− − 7 5 2 15 8− 7
Solution: (a) − = = and − = =
3 4 12 12 4 3 12 12
2 5 5 2
So, −− ≠ −− (Not commutative)
3 4 4 3
− 
(b) 1 −  − 5   = 2 −− ( 5) = 25+ = 7 and  5  − 1 = −−52 = −7


3  6  6 6 6  6  3 6 6
1  − 5   5 1
− 
So, −   ≠   − (Not commutative)
3  6   6  3
Thus, commutative property does not hold true for subtraction of rational numbers.
p r u
III. Associative Property: “The order of the three numbers cannot be changed in subtraction.” So, if , and v
q s
 p r  u p r  u 
are three rational numbers, then  q − s  − ≠ q −  s −  .
  v  v 

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