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+ 4 % 1 ¼ 3 ÷ ¾
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Chapter4 6 ½ –

Rational Numbers

Introduction If a rational number having its numerator and
In previous class, we have learnt that the denominator are both positive or both
difference of two counting numbers is not negative, then it is said to be positive
always a natural number. For example, 9 - 9 2 - 2 - 3 76
, ,
,
,
2 4 = - 2 ...(i) For ex am ple : 11 - 11 3 3 , - 5 81 etc. are
-
-
1 5 = - 4 ...(ii) pos i tive ra tio nal num bers. A ra tio nal num ber
In (i) and (ii), we can observe that -2 and -4 is said to be negative, if its numerator and
are not natural numbers. This problem gave denominator are such that one of them is a
us the idea of integers. Now in integers, when positive integer and another is a negative
we multiply an integer by another integer, the
result is also an integer. For example, integer.
-1 3 -72 5
- ´ 2 = -2 ... (iii) For ex am ple : , , , etc. are
1
2 -7 5 -12
2
- ´ -3) = 6 ...(iv)
(
neg a tive ra tio nal num bers.
From the above (iii) and (iv), we can notice that
-2 and 6 are also integers. But in case of The set of rational numbers is the set whose
division of integers, we do not always get the elements are natural numbers, negative
same result, numbers, zero and all positive and negative
)
3 4 (- 2 1 fractions.
.
i e. , , , ………are not integers. So, it
2 7 5 6 Representation of Rational Numbers on
means the division of integers also demands Number Line
for another numbers system consisting of
fractions, as well as integers that is fulfilled by We already know the method of constructing a
the Ra tio nal num bers. number line to represent the integers. Now we
use the same number line to represent the
Defining Rational Numbers
rational numbers. For this purpose, we draw a
A number that can be expressed in the form of
p number line as given below.
where p and q are integers and q ¹ 0, is
q
3 4 2 1 –3 –2 –1 0 +1 +2 +3
called the rational number, e g. . , , , - ,
2 7 5 6 Fig. : 1
are examples of rational numbers.

Now we divide each segment of the above number line into two equal parts, as given in the
following diagram

6 5 4 3 2 1 1 2 3 4 5 6
– – – – – – + + + + + +
2 2 2 2 2 2 2 2 2 2 2 2
–3 –3 –1 0 +1 +2 +3
Fig. : 2

In the figure 2 the number line represents the rational numbers which are given below.
6 5 4 3 2 1 1 2 3 4 5 6
…, - , - , - - , - , - , 0 , + , + , + + + + , ……
2 2 2 2 2 2 2 2 2 2 2 2

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Mathematics In Focus - 7