1. Rational numbers include integers and fractions, and can be written in the form x/y where x and y are integers. 2. Addition, subtraction, multiplication, and division of rational numbers can be performed by finding equivalent fractions with a common denominator or by using algebraic expressions. 3. Two fractions are equal if their numerators divided by their denominators are equal (i.e. if aq = bp, where a/b and p/q are the fractions).

1. 1.RATIONAL NUMBERS
Many kinds of Numbers
We have seen many kinds of numbers" like natural 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.
What about the difference?
Sometimes a natural number, or a negative number, or zero.
Natural numbers, their negatives and zero are collectively called integer.
So, the sum, difference and product of two integers is again an integer, isn't it?
What about the quotient?
The result of dividing an integer by another integer may not always be an integer; it can be a fraction.
For example. 6/3=2. But what about 7/3
We can write, if we want
7
3
=
(2∗3)+1
3
= 2∗3
3
+ 1
3
= 2 1
3
What about 2/3 We cannot do anything in particular
What about negative integers?
Integers and fractions (positive or negative) are collectively
called r at i o nal numb er s.
In shod, "rational number" is the collective name for all
numbers we have seen so far.
Rational form
Every fraction has a numerator and a denominator; for instance. the numerator of is 3 for 3/4and the
denominator is 4.
What about -3/5?
We can write that the numerator is -3 and the denominator is 5 or that the numerator is 3 and the
denominator is -5

2. So any fraction can be written in the form x/y , where:r andy are integers.
Can we write an integer in this form?
So, any rational number (fraction or integer) can be written in the form x/y, where x and y are integers
Various forms
Each rational number can be written in the form x/y , where x andy are integers, in various ways.
That is, by multiplying the numerator and denominator of a rational number by the same integer, we
can get another form of the same rational number (The multiplier should not be zero, thats all).
ln the language ofalgebra
푎
푏
=
푎푥
푏푥
, x is a non zero integer
Reversing this, we can also say that if 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

2푥
2푦
=
푥
푦

푥푦+푥
푥푧+푥
=
푥(푦+1)
푥(푧+1)
=
푦+1
푧 +1
Now can't you simplify the expressions below?

푥푢+푦푢
푥푣+푦푣

3푥+6
푥+2
Addition and subtraction
Remember how we find the sum of two fractions?
For example, how so we find 2/3 +1/5
From the various forms of 2/3 and 1/5 we must choose a pair with the same denominator, right?
In all the forms of 2/3 the denominator is a multiple of 3 ; and in all the forms of 1/5 the denominator is
multiple of 5.
So, from the various forms of 2/3and 1/5 , if we want ones with the same denominator then the
denominator must be a multiple of both 3 and 5.
3 x 5 = 15 itself is a multiple of 3 and 5.

3. Thus . If we write
2/3= 10/15
1/5= 3/15
2
3
+
1
5
=
13
15
How do we write such an addition in algebra?,
퐚
퐛
+
퐩
퐪
=
퐚퐪+퐛퐩
퐛퐪
Can't you do the following problems like this?

푥
푦
−
푦
푥

1
푥
−
1
푦
 1 +
1
푥
 푥 +
푥
푥−1
Multiplication and division
Multiplication and division of fractions are a bit easier, right?
2/3*5/7= 10/21
2
35
7
=
2
3
∗
7
5
=
14
15
In the language of algebra,
풂
풃
∗
풑
풒
=
풂풑
풃풒
풂
풃
÷
풑
풒
=
풂
풃
×
풒
풑
=
풂풒
풃풑
Now try these problems on your own:

푥
푥−1
∗ 푥

4. 
푥
푥−1
/푥
Equal fractions
How do we check whether two fractions are equal?
Let's think in algebraic terms. Suppose
a/b = p/q
Here a/b and p/q are different forms of the same number
풂
풃
÷
풑
풒
= ퟏ
aq/bp = 1
If a quotient is 1, then the dividing number and the divided number should be equal. So, here we get
aq=bp
Now on the other hand, suppose four numbers q b, c ard, d are such that aq = bp
For the numbers a, b, p, q, 1f a/b = p/q , then aq = bp On the other hand if aq = bp and also b not
equal zero. Then a/b = p/q
Now try these problems on your own:
1. If x/y = 2/3 then find the value of (4x+2y)/(5x-2y)?
2. If x/y = ¾ what is the value of (5x+2y)/(5x-2y)?