1. 1
AN INTRODUCTION TO RATIONAL NUMBERS
SHYLA BEEGAM A
(M.Sc. Mathematics)
All the knowledge brings us nearer to ignorance – T.S. Eliot
2. 2
PREFACE
The book, ‘an introduction to rational numbers’ is intended for secondary
school students and teachers in kerala syllabus. In this book all the topic have been dealt
with in a simple and lucid manner. A sufficiently large no. of problems have been solved.By
studying this book ,the student is expected to understand the concept and acquire the skills
such as to any rational number(fraction or integer)in the form x/y ; where x and y are
integers; to find out various forms of the rational number ; to do more problems involving
the addition ,subtraction ,multiplication and division and to express any fraction as a
decimal.
Suggestion for the further improvement of this book will be highly
appreciated.
Shylabeegam A
3. 3
CONTENTS
Chapters pages
1. Rational numbers 4 - 7
2. Addition and subtraction of rational numbers 8 - 9
3. Multiplication and division of rational numbers 10 - 12
4. Equal fractions 13 - 17
5. Decimal forms 18 - 23
6. Reference 24
4. 4
Chapter 1
RATIONAL NUMBERS
Many kinds of numbers
We have seen many kinds of numbers ,lie natural numbers,fractions and negative numbers
,also various operations on them lie addition,subtraction,multiplication,division and
exponentiation.
The sum and product of to natural numbers is again a natural numbers.
What about the difference?
Sometimes a natural numbers,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,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
fraction.
For example,
6
3
= 2, but what about
7
3
?
We canwrite ,if we want
7
3
=
(2×3)+1
3
=
2×3
3
+
1
3
= 2 +
1
3
=2
1
3
What about
2
3
?We cannot do anything in particular.
What about negative integers? For example
−7
3
= - (2+
1
3
)=
7
3
- 2
1
3
−7
3
= -
7
3
= -2
1
3
7
−3
= -
7
3
= -2
1
3
−2
3
= -
2
3
5. 5
2
−3
= -
2
3
Integers and fraction (positive or negative ) are collectively called rational numbers.
In short , “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
3
4
is 3
and the denominator is 4.
What about -
3
5
?
We can write -
3
5
=
−3
5
,and say that the numerator is -3 and the denominator is 5.
Or we can write -
3
5
=
3
−5
,and say that the numerator is 3 and the denominator is -5.
So any fraction can be written in the form
𝑥
𝑦
where x and y are integers .
Can we write an integer in this form?
For example
2 =
2
1
We can do it in several ways as
2 =
2
1
=
4
2
=
6
3
= ……
So,any rational number (fraction or integer)can be written in the form
𝑥
𝑦
where x and y are
integers.
6. 6
Various forms
Each rational numbers can be written in the form where x and y are integers, in various
ways .
For example
1
2
=
2
4
=
3
6
=…..
3
5
=
6
10
=
9
15
= …..
2=
2
1
=
4
2
=
6
3
=………
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, that’s all).
In the language of algebra,
𝑎
𝑏
=
𝑎𝑥
𝑏𝑥
,( x, 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
2𝑥
2𝑦
=
𝑥
𝑦
𝑥𝑦+𝑥
𝑥𝑧+𝑥
=
𝑥(𝑦+1)
𝑥(𝑧+1)
=,
𝑦+1
𝑧+1
𝑥𝑦+𝑦
𝑥𝑧+𝑧
=,
(𝑥+1)𝑦
(𝑥+1)𝑧
=
𝑦
𝑧
x2 −1
𝑥−1
=
(𝑥+1)(𝑥−1)
𝑥−1
=x+1
8. 8
Chapter 2
ADDITION AND SUBTRACTION OF IRRATIONALS
Addition and subtraction
Remember how we find the sum of two fractions?
For example ,how do 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 of3 and in the all the forms of
1
5
,the
denominator is a multiple of 5.
So, from the various forms of
2
3
and
1
5
, if we want ones with the same denominator ,then the
denominator must be a multiple of both 3 and 5.
3×5=15 itself is a multiple of 3and 5.
Thus ,if we write
2
3
=
2×5
3×5
=
10
15
1
3
=
1×3
5×3
=
3
15
Then we can find
2
3
+
1
5
=
10
15
+
3
5
=
13
15
How do we write such an addition in algebra?
To find
𝑎
𝑏
+
𝑝
𝑞
,first mae the denominator of each equal tobq.
𝑎
𝑏
=
𝑎×𝑞
𝑏×𝑞
=
𝑎𝑞
𝑏𝑞
9. 9
𝑝
𝑞
=
𝑝×𝑏
𝑞×𝑏
From these, we see that
𝑎
𝑏
+
𝑝
𝑞
=
𝑎𝑞
𝑏𝑞
+
𝑏𝑝
𝑏𝑞
=
𝑎𝑞+𝑏𝑝
𝑏𝑞
Lie this ,we can also see that
𝑎
𝑏
−
𝑝
𝑞
=
𝑎𝑞 − 𝑏𝑝
𝑏𝑞
Let’s look at some examples:
𝑥
𝑦
+
𝑦
𝑥
=
𝑥
𝑥𝑦
+
𝑦
𝑥𝑦
=
𝑥+𝑦
𝑥𝑦
1
𝑥
+
1
𝑦
=
𝑦
𝑥𝑦
+
𝑥
𝑥𝑦
=
𝑥+𝑦
𝑥𝑦
1
𝑥
-
1
𝑥+1
=
𝑥+1
𝑥(𝑥+1)
-
𝑥
𝑥(𝑥+1)
=
1
𝑥(𝑥+1)
x+
1
𝑦
=
𝑥𝑦
𝑦
+
1
𝑦
=
𝑥𝑦+1
𝑦
Can’t you do the following problems like this?
𝑥
𝑦
-
𝑦
𝑥
1
𝑥−1
+
𝑥
𝑥+1
1
𝑥
-
1
𝑦
1+
1
𝑥
x+
1
𝑥
2
2𝑥+1
+
3
2𝑥−1
x+
𝑥
𝑥−1
1
𝑥+2
-
1
𝑥−2
1-
2
𝑥+1
10. 10
Chapter 3
MULTIPLICATION AND DIVISIONOF IRRATIONALS
Multiplication and division
Multiplication and division of fractions are a bit easier ,right?
2
3
×
5
7
=
2×5
3×7
=
10
21
2
3
÷
5
7
=
2
3
×
7
5
=
2×7
3×5
=
14
15
In the language of algebra,
𝑎
𝑏
×
𝑝
𝑞
= =
𝑎𝑝
𝑏𝑞
𝑎
𝑏
÷
𝑝
𝑞
=
𝑎
𝑏
×
𝑞
𝑝
=
𝑎𝑞
𝑏𝑝
Let’s look at some examples also.
𝑥
𝑦
×
𝑥+1
𝑦+1
=
𝑥(𝑥+1)
𝑦(𝑦+1)
=
𝑥2+𝑥
𝑦2+𝑦
13. 13
Chapter 4
EQUAL FRACTIONS
Equal fractions
How do we check whether two fractions are equal?
For example, how do we check whether
36
48
and
42
56
are different forms of the same fraction?
One method is to reduce each to the lowest terms .For that,the common factors in the
numerator and denominator should be removed in each.
36
48
=
218
224
18
24
=
29
212
=
9
12
=
33
34
=
3
4
42
56
=
221
228
=
21
28
=
73
74
=
3
4
What do we see from this?
36
48
=
42
56
This method is not easy to apply if the numerator and denominator are large. For example
in
187
209
and
221
247
the common factors of the numerator and denominator are not easily got as in
the first example .
14. 14
Let’s thin in algebraic terms.
Suppose
𝑎
𝑏
=
𝑝
𝑞
.
Here
𝑎
𝑏
and
𝑝
𝑞
are different forms of the same number (like
2
3
and
4
6
) .A number divided by itself
is 1,right?
So,
𝑎
𝑏
divided by
𝑝
𝑞
should be 1.
𝑎
𝑏
÷
𝑝
𝑞
= 1
That is ,
𝑎
𝑏
×
𝑞
𝑝
= 1
This means
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 a,b ,c and d are such that aq=bp
If we also have b ≠ 0 and q≠ 0 ,then by reversing the arguments used just now, we would
get
𝑎
𝑏
=
𝑝
𝑞
. so, what do we see here?
For the numbers a, b, p , q ,if
𝑎
𝑏
=
𝑝
𝑞
then aq=bp .o the other hand if aq =bp and also b≠ 0,q ≠0
then
𝑎
𝑏
=
𝑝
𝑞
So to know whether
187
209
and
221
247
are equal ,we need only
Check whether the products 187×247 and209×221 are equal. We can do this either by
hand or use a calculator.
187×247=46189
15. 15
209×221=46189
Thus we see that
187
209
=
221
247
We can also note another thing .suppose
𝑎
𝑏
=
𝑝
𝑞
From this ,we get
aq=bp
And so
𝑎𝑞
𝑏𝑝
=1
This can be written
𝑎
𝑝
×
𝑞
𝑏
=1
This gives
𝑏
𝑞
÷
𝑏
𝑞
=1
Which in turn means
𝑎
𝑝
=
𝑏
𝑞
Thus we see that ,
For the numbers a,b,p,q,if
𝑎
𝑏
=
𝑝
𝑞
then
𝑎
𝑝
=
𝑏
𝑞
For example, from
187
209
=
221
247
,
seen earlier, we also get
187
221
=
209
247
16. 16
Let’s do some problems using these ideas .
𝑥
𝑦
=
2
3
What is
4𝑥+2𝑦
5𝑥−2𝑦
?
From
𝑥
𝑦
=
2
3
we get 3x=2y . We also have 2y in the numerator of the fraction to be evaluated.
These we can replace by3x.
4𝑥 + 2𝑦
5𝑥 − 2𝑦
=
4𝑥 + 3𝑥
5𝑥 − 3𝑥
=
7𝑥
2𝑥
=
7
2
If
𝑥
𝑦
=
3
5
, whatis
2𝑥+4𝑦
6𝑥−𝑦
?
Here also ,we can start by writing 5x=3y but then there is no 3y in the fraction to be
evaluated ,and so we cannot use 5x =3y directly ,as in the first example .Let’s look at
another method.
From
𝑥
𝑦
=
3
5
, we get
𝑥
3
=
𝑦
5
.This means the fraction
𝑥
3
and
𝑦
5
are different forms of the same
number .So,we can denote them both by the same letter .If we write z for both, then
x=3×
𝑥
3
= 3𝑧
And 𝑦 = 5 ×
36
48
=5z
So,we get
2𝑥+4𝑦
6𝑥−𝑦
=
(2×3𝑧)+(4×5𝑧)
(6×3𝑧)−5𝑧
=
26𝑧
13𝑧
=
26
13
=2
Prove that if
𝑎
𝑏
=
𝑝
𝑞
, then
𝑎+𝑏
𝑎−𝑏
=
𝑝+𝑞
𝑝−𝑞
From
𝑎
𝑏
=
𝑝
𝑞
we get
𝑎
𝑝
=
𝑏
𝑞
17. 17
As in the last problem ,if we denote the different forms
𝑎
𝑝
and
𝑏
𝑞
of the same number by the
single letterk , then we get
a=
𝑎
𝑝
× =kp
b =
𝑏
𝑞
×q=kq
And so
𝑎 + 𝑏
𝑎 − 𝑏
=
k𝑝 + k𝑞
k𝑝 − k𝑞
=
k(𝑝 + 𝑞)
k(𝑝 − 𝑞)
=
𝑝 + 𝑞
𝑝 − 𝑞
Now try these problems on your own:
If
𝑥
𝑦
=
3
4
then what is
5𝑥+2𝑦
5𝑥−2𝑦
?
Prove that if
𝑥
𝑦
=
𝑢
𝑣
then
2𝑥+5𝑦
4𝑥+6𝑦
=
2𝑢+5𝑣
4𝑢+6𝑣
. Will this be true if we use some other numbers
instead of 2,5,4 and 6?
Prove that if
x
y
=
u
v
then each is equal to
2𝑥+5𝑢
2𝑦+5𝑣
. Is this true for other numbers instead
of 2and 5?
18. 18
Chapter 7
DECIMAL FORMS
Decimal forms
Do you remember how fraction with denominator a power of 10,such as 10,100,1000 and
so on,can be expressed as a decimal?
For example,
1
10
=0.1
23
10
=2.3
39
100
= 0.39
Some fractions with denominator not apower of 10 can be expressed as decimals by
converting the denominator to a power of 10.
For example,
1
2
=
5
10
=0.5
4
5
=
8
10
=0.8
3
4
=
75
100
=0.75
1
8
=
125
100
=0.125
Decimal form express fraction as sums of powers of
1
10
just as we write natural numbers as
sums of power of 10.
19. 19
For example,
243=(2×100)+(4×10)+3
1
8
= 0.125 =
1
10
+
2
100
+
5
1000
Can we write every fraction likethis ?
First, let’s look at a fraction which can be so expressed:
1
8
=
1
10
+
2
100
+
5
1000
How do we get this?
First,let’s write
1
8
using
1
10
1
8
=
1
10
×
10
8
Now we write the
10
8
in this as
10
8
= 1 +
2
8
Then we get
1
8
=
1
10
× (1 +
2
8
) =
1
10
+
2
80
Next we write
2
80
=
1
100
×
200
80
=
1
100
×
20
8
And the
20
8
in this as
20
8
= 2 +
4
8
20. 20
Then we get
1
8
=
1
10
+
1
100
(2+
4
8
)
=
1
10
+
2
100
+
4
800
Next we write
4
800
=
1
1000
×
4000
800
=
1
1000
×
40
8
And
40
8
=5
So that we get
1
8
=
1
10
+
2
100
+
5
1000
=0.125
We must note another point in this .the number
1
10
=0.1 that we get in the first step of this process
is
2
80
less than
1
8
.The number
1
10
+
2
100
= 0.12 is only
4
800
less than
1
8
. Thus the numbers got at every
stage comes closer and closer to
1
8
.Finally we get the number
1
10
+
2
100
+
5
1000
= 0.125 ,which is
the decimal number actually equal to
1
8
Now try writing
1
16
as a decimal .
Not all fractions can be expressed as a decimal like this .For example,consider
1
3
.No multiple of 3
is a power of 10. (why?)So,in all the various forms of
1
3
,none will have a power of 10as
denominator .
Still,we can try on
1
3
,the tric we used to write
1
8
as a decimal and see what happens .First we write
1
3
=
1
10
×
10
3
And then write the
10
3
in it as
21. 21
10
3
= 3 +
1
3
1
3
=
1
10
(3 +
1
3
) =
3
10
+
1
30
Thus
3
10
= .3is
1
30
less than
1
3
1
3
− 0.3 =
1
30
What if we continue? We can write
1
30
=
1
100
×
100
30
=
1
100
×
10
3
and use
10
3
= 3 +
1
3
again, to get
1
30
=
1
100
(3 +
1
3
) =
3
100
+
1
300
So, we can write
1
3
=
3
10
+
1
30
=
3
10
+
3
100
+
1
300
From this we find
1
3
− (
3
10
+
3
100
) =
1
300
That is,
1
3
− 0.33 =
1
300
Continuing this ,we get
22. 22
1
3
− 0.333 =
1
3000
1
3
− 0.3333 =
1
30000
1
3
− 0.33333 =
1
300000
and so on .
In other words ,we get decimals closer and closer to
1
3
. But the process will never end;and we
will never get a decimal actually equal to
1
3
This fact we write in shortened form as
1
3
= 0.333….
5
11
= 0.4545….
What does this mean ?
The fraction whose decimal forms are 0.4, 0.45, 0.454, 0.4545, ….get closer and closer to
5
11
.
Here the repetition is in pairs of 4 and 5. Like this , we also see that
4
27
=0.148148148…….
Where the triple 1,4,8 repeat.(try it)
We have also decimal forms lie
1
6
= 0.1666….
5
12
= 0.41666….
23. 23
5
24
= 0.2083333……
Where the repetition stars after some non-repeating digits.
Now try writing the following fractions as decimals .
1
9
2
9
1
7
1
11
2
11
1
12
