Page 32 - Start Up Mathematics_8 (Non CCE)

p r
3. If q and are two rational numbers, then
s
p r
(a) q + s is also a rational number (Closure property)

p r r p
(b) q + s = s + q (Commutative property of addition)

 p r  u p  r u 
(c)  q + s  + v = q +   s + v  (Associative property of addition)



p p p
(d) q + 0 = q = 0 + q where 0 is the additive identity for addition of rational numbers.

p  − p   − p  p − p p
(e) q +   q   = 0 =   q   + q where q is the additive inverse of q .

p r p r p × r
4. If q and are two rational numbers, then q × s = q × s .
s
p r
5. If q × s are two rational numbers, then

p r
(a) q × s is also a rational number (Closure property)

p r r p
(b) q × s = s × q (Commutative property of multiplication)

 p r  u p  r u 
(c)  q × s   × v = q ×   s × v   (Associative property of multiplication)

p p p
(d) q × 1 = q = 1 × q where 1 is called the multiplicative identity for multiplication

p q q p q p
(e) q × p = =1 p × q where p is the multiplicative inverse (or reciprocal) of q

p  r u   p r   p u 
(f) q ×   s + v  =   q × s   +   q × v  (Distributive property of multiplication over addition)


 p   p 
(g)  q × 0  = 0 =   0 × q  


p r p s p r
6. If q and are two non-zero rational numbers, then q ÷ r = q × s .
s
p r p r
7. If q and are two non-zero rational numbers, then q ∏ s is also a rational number.
s
p p p p p
8. For any rational number q , q ÷=1 q and q ÷ q = 1.

p r 1  p r 
9. If q and are two non-zero rational numbers, then 2 ×  q + s  is a rational number between
s


p r .
q and s

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