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NUMBER SYSTEMS
1
CHAPTER 1
NUMBER SYSTEMS
1.1 Introduction
In your earlier classes, you have learnt about the number line and how to represent
various types of numbers on it (see Fig. 1.1).
Fig. 1.1 : The number line
Just imagine you start from zero and go on walking along this number line in the
positive direction. As far as your eyes can see, there are numbers, numbers and
numbers!
Fig. 1.2
Now suppose you start walking along the number line, and collecting some of the
numbers. Get a bag ready to store them!
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2
MATHEMATICS
You might begin with picking up only natural
numbers like 1, 2, 3, and so on. You know that this list
goes on for ever. (Why is this true?) So, now your
bag contains infinitely many natural numbers! Recall
that we denote this collection by the symbol N.
58 1 N
16
0 5
71 31 652
10 9 4 2
601 7 40
4
Now turn and walk all the way back, pick up
zero and put it into the bag. You now have the
collection of whole numbers which is denoted by
the symbol W.
0
16 3
57440 2
9
601582
W
57
-7
-4
0
Now, stretching in front of you are many, many negative integers. Put all the
negative integers into your bag. What is your new collection? Recall that it is the
collection of all integers, and it is denoted by the symbol Z.
-66-21
Z comes from the
Why Z ?
German word
-3
16 71 58
0 53
31 2
42 2
6017 40
4
“zahlen”, which means
“to count”.
Z
0
166 3
-75 2 -40
22 1 9
Are there some numbers still left on the line? Of course! There are numbers like
1, 3 ,
−2005
or even
. If you put all such numbers also into the bag, it will now be the
2 4
2006
17
981
–
05
20 006
2
–12
13
9
58
Q
5 19 6
–6620
–
7
7
2 19
9
12 -65 99 14
1 –
9
3 81 13–672 60
1
16 1 –1 12
9
4
05
20 006
2
58
3 7 14 –6
7
1
16 2
9
99
3
89 0
4 6625
27 – –5
–860
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NUMBER SYSTEMS
3
collection of rational numbers. The collection of rational numbers is denoted by Q.
‘Rational’ comes from the word ‘ratio’, and Q comes from the word ‘quotient’.
You may recall the definition of rational numbers:
A number ‘r’ is called a rational number, if it can be written in the form
p
,
q
where p and q are integers and q ≠ 0. (Why do we insist that q ≠ 0?)
Notice that all the numbers now in the bag can be written in the form
p
, where p
q
−25
; here p = –25
1
and q = 1. Therefore, the rational numbers also include the natural numbers, whole
numbers and integers.
You also know that the rational numbers do not have a unique representation in
and q are integers and q ≠ 0. For example, –25 can be written as
the form
=
1
2
10
25
p
, where p and q are integers and q ≠ 0. For example, =
=
=
q
2
4
20
50
47
, and so on. These are equivalent rational numbers (or fractions). However,
94
p
p
is a rational number, or when we represent
on the number
q
q
line, we assume that q ≠ 0 and that p and q have no common factors other than 1
(that is, p and q are co-prime). So, on the number line, among the infinitely many
1
1
fractions equivalent to , we will choose to represent all of them.
2
2
Now, let us solve some examples about the different types of numbers, which you
have studied in earlier classes.
when we say that
Example 1 : Are the following statements true or false? Give reasons for your answers.
(i) Every whole number is a natural number.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
Solution : (i) False, because zero is a whole number but not a natural number.
m
(ii) True, because every integer m can be expressed in the form
, and so it is a
1
rational number.
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4
MATHEMATICS
(iii) False, because
3
is not an integer.
5
Example 2 : Find five rational numbers between 1 and 2.
We can approach this problem in at least two ways.
Solution 1 : Recall that to find a rational number between r and s, you can add r and
r+s
3
lies between r and s. So,
is a number
2
2
between 1 and 2. You can proceed in this manner to find four more rational numbers
s and divide the sum by 2, that is
between 1 and 2. These four numbers are
5 11 13
7
, ,
and .
4 8 8
4
Solution 2 : The other option is to find all the five rational numbers in one step. Since
we want five numbers, we write 1 and 2 as rational numbers with denominator 5 + 1,
i.e., 1 =
6
12
7 8 9 10
11
and 2 =
. Then you can check that , , ,
and
are all rational
6
6
6 6 6 6
6
numbers between 1 and 2. So, the five numbers are
7 4 3 5
11
, , , and .
6 3 2 3
6
Remark : Notice that in Example 2, you were asked to find five rational numbers
between 1 and 2. But, you must have realised that in fact there are infinitely many
rational numbers between 1 and 2. In general, there are infinitely many rational
numbers between any two given rational numbers.
Let us take a look at the number line again. Have you picked up all the numbers?
Not, yet. The fact is that there are infinitely many more numbers left on the number
line! There are gaps in between the places of the numbers you picked up, and not just
one or two but infinitely many. The amazing thing is that there are infinitely many
numbers lying between any two of these gaps too!
So we are left with the following questions:
1. What are the numbers, that are left on the number
line, called?
2. How do we recognise them? That is, how do we
distinguish them from the rationals (rational
numbers)?
These questions will be answered in the next section.
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