Chap 1
RATIONAL NUMBERS
SOME IMPORTANT FACTS
1. Natural
numbers : The counting numbers are called Natural numbers.
1,2,3,4,5,6,7,8,…………………………………….
2. Whole
Number : All natural numbers together with zero (0) are called whole numbers.
0,1,2,3,4,5,6,7,8,…………………………………….
3. Integers
: The Whole numbers together with the negative of counting numbers are known as
integers.
……………,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,………………..
4. Fractions
: The numbers of the form a/b, where a
and b are natural numbers , are known as fractions.
5. Rational
numbers : A number of the form P/Q where
P,Q are integers and Q
≠ 0 is called rational number.
6. Every integer is a rational number
but a rational number need not be an integer.
7. Every fraction is a rational number but
a rational number numbers need not be a fraction number.
8. The operation of addition of rational
numbers has the following properties ;
a) Closure Property : The addition of
any two rational numbers is always a rational number.
b) Commutative : x + y = y + x, for any two rational numbers x
and y.
c) Associative : ( x + y ) + z = x + ( y
+ z ), for all rational numbers x,y,z .
d) Existence
of additive identity : x + 0 = 0 + x = x for all rational numbers x. the
rational number 0 is the additive identity.
e) Existence
of additive inverse : ( -x) + x = 0 = x
+ ( - x ).
9. The
multiplication of rational numbers has following properties ;
a) Closure Property : The multiplication
of any two rational numbers is always a rational number.
b) Commutative : x x y = y x x, for any two rational numbers x and
y.
c) Associative : ( x x y ) x z = x x ( y x z ), for all rational numbers x,y and z .
d) Existence
of identity : x x 1 = 1 x x = x for all rational
numbers x. the rational number 1 is the identity element for multiplication.
e) Existence
of multiplicative inverse : x x 1/x = 1 =
1/x x
x .
f) Multiplication
by 0 (zero ) : for any rational number x , x x 0 = 0 = 0 x
x.
g) Distributivity
of multiplication over addition : x x ( y + z ) = x x y + x x z, for any three
rational numbers x, y, z.
10. Subtraction
of rational numbers has following properties ;
a)
Closure property : x – y is a rational number for all rational
numbers x , y.
b)
The subtraction of rational numbers is neither
commutative nor associative.
11. Between
two rational numbers x and y , there is
a rational number x + y /2.
We can find as many rational numbers between x
and y as we want.solution
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