Chapter 1 – Rational Numbers
Key Notes
1. The numbers of the form 
, where a and b are integers and
b ≠ 0, are called rational numbers.
2. All rational numbers can be represented on a number line.
3. (i) A rational number is said to be positive if its numerator and denominator are either both positive or both negative.
(ii) A rational number is said to be negative if its numerator and denominator are of opposite signs.
4. (i) if 
is a rational number and m is a non-zero integer then 
.
(ii) if is a rational number and m is a common divisor of both a and
b then 
.
5. A rational number is said to be in standard form if a
and b are integers having no common divisor other than 1 and b is
positive.
6. 
only when (a × d) = (b × c).
7. To compare two or more rational numbers, express each of them as rational number with positive denominator. Take the LCM of these positive denominators and express each rational number with this LCM as denominator. Then, the number having the greater numerator is greater.
8. Rational numbers are closed under the operations of addition, subtraction and multiplication.
9. If and 
are any two rational numbers then
(i) 
is also a rational number [closure property]
(ii) = 
[commutative law of addition]
(iii) + 
= 
+ 
[associative law of addition]
(iv) 
.
0 is called the identity element for addition (additive identity) of rational numbers.
(v) 
is called the additive inverse of
10. If and are any two rational numbers then 
.
11. If and are any two rational numbers then
(i) 
is also a rational
number. [closure property]
(ii) 
· [commutative law of multiplication]
(iii) 
· [associative law of multiplication]
(iv) 
· 1 is called the multiplicative identity
for rational numbers.
(v) 
. 
is called the multiplicative inverse or
reciprocal of ·
(vi) 
(vii) 
12. If and are two rational numbers such that 
then 
.
13. (i) if and are two rational numbers and then 
is also a rational number.
(ii) For every rational number , we have 
and 
.
14. If x and y be two rational numbers such that x
< y then 
(x + y) is
a rational number between x and y.
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