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−1 2 5
Example 8: Verify the associative property of addition for , and .
4 5 7
Solution: By grouping the first two numbers, we have
3
5
−1
−+ 
+
   4 +  2  + 7 =  58   + 5 = 20 + 5 = 21 100 = 121

7
 5
 20
140
7
140
By grouping the last two numbers, we have
−1  2  5 −1  14 + 25  −1 39 −35 156 121
+
+  +  = +   = + = =
4  5 7  4  35  4 35 140 140
5
−1
Thus,    4 +  2  + = −1 +  2 + 7  5  . Hence, associative property is verified.


4
 5
 5
7
Existence of Additive Identity
We know that, zero is the rational number which when added to any rational number, gives the
same number as a result.
3 3 3 3 −3 −3  −  3 −3
For example, += , 0 + = . Also, += 0 , +   =
0
0
4 4 4 4 4 4  4  4
Thus, 0 is called the additive identity of rational numbers.
In general,
p p p p
if is a rational number, then +=+0 0 = , where 0 is the additive identity of rational numbers.
q q q q
Example 9: Find the additive identity of rational number −3 .
5
−3  −  3 −3
Solution: Since, +=+   =
0
0
5  5  5
Hence, 0 is the additive identity for the rational number −3 .
5
Existence of Additive Inverse

We know that for every integer, ‘a’ there is an additive inverse ‘−a’ such that a + (−a) = (–a) + a = 0.
−3 3 3  −  3
Similarly, every rational number has its additive inverse. For example, + = +    = 0 . So,
3 is the additive inverse of −3 . 4 4 4  4
4 4

5  − 5  − 5 5 −5 5
Similarly, +   =   + = 0 . So, is the additive inverse of .
7  7   7  7 7 7

In general,
p −p
if is a rational number, then there exists a rational number such that
q q
p
p +  − p =  − p + p = 0 . Thus, −p is called the additive inverse of and vice versa.


q  q   q  q q q


17 Rational Numbers