Page 13 - Start Up Mathematics_8 (Non CCE)

p r u  p r u p r  u
So, if , and are three rational numbers, then  q + s  + v = q +   s + v  
q s
v


3
3

− 
− 
Example 8: Show that  −  1  +  +  4  =     −  1  +  +  4  (NCERT)



7
 2   7  3    2    3 
3
9
−   1
Solution:  −  1  +  +  4  =  −   +  +−28(  )  =    −1  +   −19   = ( −21) + −38( ) = −59


 2   7  3   2   21   2   21  42 42
  −  1 3   4 ( −7) +   4 −1  4 ( −3) + −56( ) −59
6
− 
− 
−  
=
and    +  +   =   +     +   = =
  2  7   3   14   3   14   3  42 42
3
− 

3
− 
So,   −  1  +  +  4  =     −  1  +  +  4 


 2   7  3    2  7   3 
IV. Existence of Additive Identity: The Role of ‘Zero’
“The sum of any rational number and 0 is the rational number itself.”
p p p p
So, if is any rational number, then += =+ , where 0 is called the additive identity for the
0
0
q q q q
rational number p .
q
3  −14 
Example 9: Find: (a) + 0 (b)   + 0
8  9 
3 3 3  −14   −14   −14 
0
0
Solution: (a) += =+ (b)   +=   =+  
0
0
8 8 8  9   9   9 
V. Existence of Additive Inverse Remember
p − p
If q is any rational number, then there exists a rational number q such that To find the additive inverse
p  − p  − p p − p p of any rational number,
change its sign.
0
q +   q   ==  q   + q where q is called the additive inverse of q .

−6 2 2
Example 10: Find the additive inverse of: (a) (b) (c) (NCERT)
−5 − 9 8
−6  6 −6
− 
Solution: (a) The additive inverse of =−   =
−5  5 5
− 
2  2   − 2  2 2 − 2
(b) The additive inverse of =−   =−   = (c) The additive inverse of =
− 9  − 9   9  9 8 8
EXERCISE 1.2
1. Verify the commutative property of addition for the following rational numbers:
5 7 −2 1 2 3 −4 10
(a) and (b) and (c) and (d) and
8 − 12 −3 5 5 10 14 21
2. Verify that (a + b) + c = a + (b + c) where
−1 3 5 2 −3 7 7 −2 −3
(a) a = , b = , c = (b) a = , b = , c = (c) a = , b = , c =
2 4 7 3 4 5 −11 5 22
5