This document discusses rational numbers and their properties. It defines rational numbers as numbers that can be expressed as fractions where the numerator and denominator are integers. It describes how to add, subtract, multiply and divide rational numbers. It also discusses how any rational number can be expressed in different equivalent forms by multiplying the numerator and denominator by the same integer or reducing common factors. The document also introduces decimal representations of rational numbers.

2. Jana- gana - mana – Adhinayaka , Jaya He
Bharata- Bhagya- Vidhata
Punjab- Sindhu -Gujarata-Maratha
Dravida-Utkala-Banga
Vindhya- Himachala-Yamuna- Ganga
Uchchhala- JaladthaTaranga
Tava Subha Name Jage
Tava Subha Ashisa Mage
Gahe Tava Jaya Gatha.
Jana-Gana- Mangala Dayaka, Jaya He
Bharata-Bhagya-Vidhata,
Jaya He, Jaya He, Jaya He,
Jaya, Jaya, Jaya, Jaya He

3.  DEFINITION
 RATIONAL FORM
 VARIOUS FORMS
 ADDITION & SUBTRACTION
 MULTIPLICATION & DIVISION
 DECIMAL FORMS

4. we have seen many kinds of numbers, fractions and negative numbers ;also various
operations on them like addition ,subtraction ,multiplication ,division and
exponentiation
The sum and product of two natural numbers is again a natural number
The difference of two natural number is a natural number, or a negative number ,
or zero natural numbers, their negatives and zero are collectively called integers .
so the sum difference and product of two integers is again an integer the result of
dividing and integer by another integer may not always be an integer, it can be
fraction. For example , 6 = 2
3
7 = (2 × 3) +1 = 2 × 3 + 1 = 2 + 1 = 2 1
3 3 3 3 3 3
Integers and fractions ( positive or negative) are collectively called rational numbers

5. Every fraction has a numerator and a denominator ; for instance, the
numerator of 3 is 3 and the denominator is 4
4
-3 we can write -3 = 3, and say that numerator is -3
5 5 5
and the denominator is 5.
Or we can write -3 = 3, and say that numerator is 3
5 5
and the denominator is -5
So any fraction can be written in the form x where x and y
y
are integers
We can write integer in the form for example
2= 2
1

6. We can do it in several ways as
2= 2= 4= 6 = …..
1 2 3
So any rational number (fraction or integer )can be written in form x
where x and y are integers y

7.  VARIOUS FORMS
multiplying the numerator and denominator of a rational
number by the same integer, we can get another form of the
same rational number
In the language of algebra
a = a x (a x , a non –zero integer example
b b x
1 = 2=3
2 4 6
The numerator and denominator of a rational number has any
common factor ,then by removing this factor, we get a simpler
form of the same rational number
for example
2x = x
2y y

8. ADDITION & SUBTRACTION
If a , c be two rational number then , i ) a + c = a +c , ii) a – c = a - c
b b b b b b b b
If a , p be two rational number then a + p first make the
b q b q
denominator of each equal to b q
a = a× q = a q
b b ×q b q
p = p× b = b p
q q ×b b q
From these , we see that
a + p = a q + b p = a q+ b p
b q b q b q b q
Like this, we can also see that
a - p = aq- bp
b q b q

9.  Example
2 2 2 2
 x + y = x + y = x + y
y x x y x y xy
2 2 2 2
 x - y = x - y = x - y
y x x y x y x y

10. a × p = a p
b q b q
a ÷ p = a × q = a q
b q b p b p
Examples
 x × u = x × u = x × u = x u
v 1 v 1× v v
 x ÷ u = x × v = x v
v u u

11. EQUAL FRACTIONS
a and p are different forms of the same number . A number
b q
divided by itself is 1.
So, a divided by p should be 1
b q
a ÷ p = 1
b q
That is a × q = 1
b p
This means
a q = 1
b p
If a quotient is 1, then the dividing number and the divided number
should be equal. So , here we get
a q = b p

12.  For the numbers a , b , p , q, if a = p ,then a q = b p . On the
b q
Other hand if a q = b p and also b ≠0, q ≠ 0 then a = p
b q

13.  DECIMAL FORMS
1 = 0.1
10
39 = 0.39
100
Some fractions with denominator not a power of 10 can be
expressed as decimals by converting the denominator to a power
of 10.
For example,
1 = 5 = 0.5
2 10
3 = 75 = 0.75
4 100

14. Decimal form
1
8 0.125
8 1.0
8
20
16
40
40
0
1 = 0.125
8

15. 1
3 0.333..
3 1.0
9
10
9
10
9
………
1 = 0.333..
3

16.  We have also decimal forms like
1 = 0.1666...
6
5 = 0.41666…
12