Download Rational and Irrational Numbers

pure arithmetic
1
Rational and
Irrational Numbers
introduction
You are already familiar with the system of natural numbers, whole numbers, integers, rational
numbers and their representation on the number line. You also know the four fundamental
operations of arithmetic on them — addition, subtraction, multiplication and division. In this
chapter, we shall extend our study to irrational numbers and real numbers. We shall learn
the decimal expansions of real numbers as terminating/non-terminating decimal numbers
and shall explain that every real number can be represented by a unique point on the number
line and conversely corresponding to every point on the number line there is a unique real
number. We shall also learn the surds and rationalisation of surds; simplifying expressions by
rationalising the denominator.
1.1 rational numbers
Recall the definition of a rational number.
p
Any number that can be expressed in the form , where p and q are both integers and
q
q ≠ 0 is called a rational number.
The word ‘rational’ comes from the word ‘ratio’. Thus, every rational number can be
written as the ratio of two integers.
p
Note that every integer (positive, negative or zero) can be written in the form
where
q
q = 1. For example,
5 = 5 , – 7 = −7 , 0 = 0 .
1
1
1
Hence, every integer is a rational number.
We know that the rational numbers do not have a unique representation in the form
p
1
2
3
7
53
, where p and q are both integers and q ≠ 0. For example, = = =
, etc. In fact,
=
q
2
4
6
14
106
these are equivalent rational numbers. However, if we write a rational number in the form
p
, where p and q are both integers and q ≠ 0 and p and q have no common factors except 1,
q
1
1
then among the infinitely many rational numbers equivalent to , we choose to represent
2
2
all of them on the number line.
p
Thus, a rational number can be uniquely expressed as q where p and q are both integers,
q ≠ 0 and p and q have no common factors except 1 i.e. p and q are co-prime.
It is called in the lowest terms or simplest form.
Remarks
1
•Since the division by zero is not allowed, is not a rational number i.e. the reciprocal of
0
zero is not allowed.
p
•When we write a rational number in the form , p, q ∈ I and q ≠ 0, usually we take q > 0,
q
while p may be positive, negative or zero.
•Two rational numbers
a
c
a
c
and are called equal, written as = , if and only if ad = bc.
b
d
b
d
1.1.1 Representation of rational numbers
Recall the representation of natural numbers, whole numbers, integers and rational numbers
on the number line.
Let l be a straight line which extends endlessly on both sides. Mark the positive
direction to the right by an arrowhead. Take a point O on l and label it 0 (zero). Next choose
another point, say A, on l towards the right of O and label it 1 (one). Thus the points O and
A represent the numbers 0 and 1 respectively. The length of the segment OA represents
unit length. Now mark points on l to the right of A at unit length intervals and label these
2, 3, 4, …. Similarly, mark points on l to the left of O at unit length intervals and label them
– 1, – 2, – 3, – 4, … , shown in fig. 1.1. Thus, every integer has been represented by one and only one
point on the line l.
Next, we consider the representation of rational numbers on the line l. Take one-half of the
unit length and mark points on l on both sides of O ; these points will represent the numbers
1 2 3
1
2
3
, , and − , − , − , …, shown in fig. 1.1.
2 2 2
2
2
2
−
>
– 3
6
−
2
5
− 2
– 2
4
−
2
4
3 3
− 2
−
– 1
2
−
2
2
1
−
3 3
1
− 2
O
0
1
2
3 3 A
1
2
1
2
2
4
5
3 3 3
2
7
8
3 3
2
4
2
5
2
3
6
2
7
2
>l
4
Fig. 1.1
Similarly, take one-third of the unit length and mark points on l on both sides of O ; these
points will represent the numbers
1 2 3 4
1 2 3 4
, , , , … and − , − , − , − , …. and so on. Thus, every
3 3 3 3
3 3 3 3
rational number has been represented by one and only one point on the line l.
1.1.2 Properties of rational numbers
The following results hold for the system (collection) of rational numbers:
(1) If a, b are any two rational numbers, then a + b is also a rational number.
For example,
1 2 7 + 6 13
, which is a rational number.
+ =
=
3 7
21
21
(2) If a, b are any two rational numbers, then a – b is also a rational number.
For example,
1 2 7−6
1
, which is a rational number.
− =
=
3 7
21
21
(3) If a, b are any two rational numbers, then a × b is also a rational number.
1 2
2
, which is a rational number.
× =
3 7 21
a
(4) If a, b (≠ 0) are any two rational numbers, then is also a rational number.
b
12
For example,
Understanding ICSE mathematics – Ix
For example,
1 2 1 7 7
÷ = × = , which is a rational number.
3 7 3 2 6
hus, the system (collection) of rational numbers is closed under all the four fundamental
T
operations of arithmetic (except division by zero).
(5)The collection of rational numbers is ordered i.e. if a, b are any two rational numbers, then
either a < b or a > b or a = b.
(6)If a, b are any two different rational numbers, then
between them i.e. if a < b then a <
a+b
is a rational number and it lies
2
a+b
< b. Continuing this process, we find that there are
2
infinitely many rational numbers between two different rational numbers.
1 2
+
3
7 = 13 is a rational number which lies between 1 and 2 .
For example,
2
42
7
3
Illustrative Examples
Example 1. Insert one rational number between
Solution. The L.C.M. of 7 and 9 is 63.
5
4
and and arrange in ascending order.
7
9
5 5 × 9 45 4 4 × 7 28
=
=
=
, =
7 7 × 9 63 9 9 × 7 63
Since 28 < 45,
.
4 5
< .
9 7
4 5
28 + 45
+
5 9 7
73
4
63
A rational number between and =
, and the numbers in ascending
=
=
7
2
2
126
9
order are
Note
4 73 5
,
, .
9 126 7
Since infinitely many rational numbers lie between two rational numbers,
only rational number between
5
4
and .
7
9
73
is not the
126
Example 2. Insert three rational numbers between 3 and 3·5.
Solution. A rational number between 3 and 3·5 =
A rational number between 3 and 3·25 =
3 + 3·5 6·5
= 3·25.
=
2
2
3 + 3·25 6·5
= 3·125.
=
2
2
A rational number between 3 and 3·125 =
3 + 3·125 6·125
= 3·0625.
=
2
2
We note that 3 < 3·0625 < 3·125 < 3·25 < 3·5, therefore, three rational numbers between
3 and 3·5 are 3·0625, 3·125, 3·25.
Example 3. Find five rational numbers between 2 and 3.
Solution. We shall approach the problem in two ways.
Method I. A rational number between 2 and 3 =
2+ 3
5
= .
2
2
5
2+
5
2 = 9.
A rational number between 2 and =
2
2
4
rational and Irrational numbers
13
9
2+
9
4 = 17 .
A rational number between 2 and =
4
2
8
2
5
+ 3
11
5
A rational number between and 3 = 2
.
=
2
4
2
17
8
9
4
5
2
11
4
23
8
3
Fig. 1.2
11
+ 3
23
11
=
A rational number between
and 3 = 4
.
2
8
4
17 9 5 11 23
, , , ,
Thus, five rational numbers between 2 and 3 are
.
8 4 2 4 8
Method II. The other way is to find all the five rational numbers in one step.
The given numbers 2 and 3 can be written as 2 =
2
3
and 3 = .
1
1
Since we want to find five rational numbers between the given numbers, multiplying the
12
numerator and denominator of each of the above numbers by 5 + 1 i.e. by 6, we get
and
6
18 , which are equivalent to the given numbers.
6
As 12 < 13 < 14 < 15 < 16 < 17 < 18,
12
13
14
15
16
17
18
<
<
<
<
<
<
6
6
6
6
6
6
6
⇒ 2 <
13
7
5
8
17
<
<
<
<
< 3.
6
3
2
3
6
Therefore, five rational numbers between 2 and 3 are:
13 7 5 8 17
, , , ,
.
6 3 2 3 6
Example 4. Find six rational numbers between
3
4
and .
5
5
3
4
and , multiplying the
5
5
21
and
numerator and denominator of each of the given numbers by 6 + 1 i.e. by 7, we get
35
28
, which are equivalent to the given numbers.
35
Solution. Since we want to find six rational numbers between
As 21 < 22 < 23 < 24 < 25 < 26 < 27 < 28,
21
22
23
24
25
26
27
28
<
<
<
<
<
<
<
35
35
35
35
35
35
35
35
⇒
3
22
23
24
5
26
27
4
<
<
<
<
<
<
< .
5
35
35
35
7
35
35
5
Therefore, six rational numbers between
3
4
and are:
5
5
22 23 24 5 26 27
,
,
, ,
,
.
35 35 35 7 35 35
1
3
Example 5. Insert eight rational numbers between − and
2
.
7
Solution. Writing the given numbers with same denominator 21 (L.C.M. of 3 and 7), we
get −
1
−7
2
6
=
and =
.
3
21
7
21
As – 7 < – 6 < – 5 < – 4 < – 3 < – 2 < – 1 < 0 < 1 < 6,
14
−
7
6
5
4
3
2
1
1
6
< −
< −
< −
< −
< −
< −
< 0 <
<
21
21
21
21
21
21
21
21
21
⇒ −
1
2
5
4
1
2
1
1
2
< − < −
< −
< − < −
< −
< 0 <
< .
3
7
21
21
7
21
21
21
7
Understanding ICSE mathematics – Ix
Therefore, eight rational numbers between −
− 2 , − 5 , − 4 , − 1 , − 2 , − 1 , 0, 1 .
7
21
21
7
21
21
1
2
and are:
3
7
21
3
5
Example 6. Find four rational numbers between 4 and 6 .
Solution. Writing the given numbers with same denominator 12 (L.C.M. of 4 and 6), we
get
3
9
5
10
and =
=
.
4
12
6
12
Since we want to find four rational numbers between the given numbers, multiplying the
45
50
and
numerator and denominator of the above numbers by 4 + 1 i.e. by 5, we get
, which
60
60
are equivalent to the given numbers.
As 45 < 46 < 47 < 48 < 49 < 50,
45
46
47
48
49
50
<
<
<
<
<
60
60
60
60
60
60
⇒
3
23
47
4
49
5
<
<
<
<
< .
4
30
60
5
60
6
Therefore, four rational numbers between
3
5
23 47 4 49
and are:
,
, ,
.
4
6
30 60 5 60
Exercise 1.1
2
3
and , and arrange in descending order.
9
8
1
1
2. Insert two rational numbers between and , and arrange in ascending order.
4
3
1. Insert a rational number between
3. Insert two rational numbers between –
1
1
and – and arrange in ascending order.
2
3
4. Insert three rational numbers between
1
4
and , and arrange in descending order.
3
5
5. Insert three rational numbers between 4 and 4·5.
6. Find six rational numbers between 3 and 4.
7. Find five rational numbers between
8. Find ten rational numbers between − and
9. Find six rational numbers between
3
4
and .
5
5
2
5
1
.
7
1
2
and .
2
3
1.2 irrational numbers
Look at the number line l (shown in fig. 1.1) again and think of the situation in another way.
As far as you can imagine, there are infinitely many numbers on the number line.
You may start collecting only natural numbers i.e. the numbers 1, 2, 3, 4, …. You know
that this list of natural numbers is endless i.e. there are infinitely many natural numbers. The
system of natural numbers is denoted by N.
If you put the number zero (0) in the above list, then you have the system of whole numbers
which is denoted by W.
Further, if you put all the negative integers in the set of whole numbers then you get the
system of all integers which is denoted by I (or Z).
rational and Irrational numbers
15