Important Concepts & Solved Examples (Real Numbers) | Extra Documents, Videos & Tests for Class 10 PDF Download

 Page 1


Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 1
REAL NUMBERS
? INTRODUCTION
"God gave us the natural number, all else is the work of man". It was exclaimed by L. Kronecker (1823-1891), the
reputed German Mathematician. This statement reveals in a nut shell the significant role of the universe of numbers
played in the evolution of human thought.
N : The set of natural numbers,
W : The set of whole numbers,
Z : The set of Integers,
R
Z
Q
N
0
?2,
W
1,2,3,4,...
99,100,...
?3,
?3 + , ?5
?5,
5 3 ?
–5
6
1
3
—
3
11
2
15
e
Q : The set of rationals,
R : The set of Real Numbers.
? HISTORICAL FACTS
Dedekind was the first modern mathematician to publish in 1872 the mathematically
rigorous definition of irrational numbers. He gave explanation of their place in the
real Numbers System. He was able to demonstrate the completeness of the real number
line. He filled in the 'holes' in the system of Rational numbers with irrational Numbers.
This innovation has made Richard Dedekind an immortal figure in the history of
Mathematics.
Srinivasa Ramanujan (1887-1920) was one of the most outstanding mathematician
that India has produced. He worked on history of Numbers and discovered wonderul
properties of numbers. He stated intuitively many complicated result in mathematics.
Once a great mathematician Prof. Hardy come to India to see Ramanujan. Prof.
Hardy remarked that he has travelled in a taxi with a rather dull number viz. 1729.
Ramanujan jumped up and said, Oh! No. 1729 is very interesting number. It is the
smallest number which can be expressed as the sum of two cubes in two different
ways.
viz 1729 = 1
3
 + 12
3
,
1729 = 9
3
 + 10
3
,
? 1729 = 1
3
 + 12
3 
= 9
3
 + 10
3
? RECALL
In our day to day life, we deal with different types of numbers which can be broadly classified as follows.
Richard Dedekind
(1831-1916)
Srinivasa Ramanujan
(1887-1920)
Page 2


Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 1
REAL NUMBERS
? INTRODUCTION
"God gave us the natural number, all else is the work of man". It was exclaimed by L. Kronecker (1823-1891), the
reputed German Mathematician. This statement reveals in a nut shell the significant role of the universe of numbers
played in the evolution of human thought.
N : The set of natural numbers,
W : The set of whole numbers,
Z : The set of Integers,
R
Z
Q
N
0
?2,
W
1,2,3,4,...
99,100,...
?3,
?3 + , ?5
?5,
5 3 ?
–5
6
1
3
—
3
11
2
15
e
Q : The set of rationals,
R : The set of Real Numbers.
? HISTORICAL FACTS
Dedekind was the first modern mathematician to publish in 1872 the mathematically
rigorous definition of irrational numbers. He gave explanation of their place in the
real Numbers System. He was able to demonstrate the completeness of the real number
line. He filled in the 'holes' in the system of Rational numbers with irrational Numbers.
This innovation has made Richard Dedekind an immortal figure in the history of
Mathematics.
Srinivasa Ramanujan (1887-1920) was one of the most outstanding mathematician
that India has produced. He worked on history of Numbers and discovered wonderul
properties of numbers. He stated intuitively many complicated result in mathematics.
Once a great mathematician Prof. Hardy come to India to see Ramanujan. Prof.
Hardy remarked that he has travelled in a taxi with a rather dull number viz. 1729.
Ramanujan jumped up and said, Oh! No. 1729 is very interesting number. It is the
smallest number which can be expressed as the sum of two cubes in two different
ways.
viz 1729 = 1
3
 + 12
3
,
1729 = 9
3
 + 10
3
,
? 1729 = 1
3
 + 12
3 
= 9
3
 + 10
3
? RECALL
In our day to day life, we deal with different types of numbers which can be broadly classified as follows.
Richard Dedekind
(1831-1916)
Srinivasa Ramanujan
(1887-1920)
Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009
2
CLASSIFICATION OF NUMBERS
                     
N U M BE R S  
R E A L N UM BE R S 
I M A G I NA R Y N U M BE R S
R A TI ONA L N UM BE R S
I R R A TI ON A L N U M BE R S 
FR A C T I ON S 
IN T E G E R S
N E G A T I V E S
Z E R O
W H OL E N U M BE R S
N A TU R A L N U M BE R S
( i ) Natural numbers (N) : N = {1, 2, 3, 4... ?}
Remark : ( i ) The set N is infinite i.e. it has unlimited members.
( i i ) N has the smallest element namely '1'.
( i i i ) N has no largest element. i.e., give me any natural number, we can find the bigger number from
the given number.
( i v ) N does not contain '0' as a member. i.e. '0' is not a member of the set N.
( i i ) Whole numbers (W) : W = {0, 1, 2, 3, 4... ?}
Remark : ( i ) The set of whole number is infinite (unlimited elements).
( i i ) This set has the smallest members as '0'. i.e. '0' the smallest whole number. i.e., set W contain
'0' as a member.
( i i i ) The set of whole numbers has no largest member.
( i v ) Every natural number is a whole number.
( v ) Non-zero smallest whole number is '1'.
( i i i ) Integers (I or Z) : I or Z = {– ?... –3, –2, –1, 0, +1, +2, +3 ...+ ?}
Positive integers : {1, 2, 3...}, Negative integers : {.... –4, –3, –2, –1}
Remark : ( i ) This set Z is infinite.
( i i ) It has neither the greatest nor the least element.
( i i i ) Every natural number is an integer.
( i v ) Every whole number is an integer.
( i v ) The set of non-negative integer = {0, 1, 2, 3, 4,....}
( v ) The set of non-positive integer = {......–4, – 3, – 2, –1, 0}
( i v ) Rational numbers :– These are real numbers which can be expressed in the form of 
p
q
, where p and q are integers
and q ? 0.
Ex.
2 37 –17
, ,
3 15 19
, –3, 0, 10, 4.33, 7.123123123.........
Page 3


Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 1
REAL NUMBERS
? INTRODUCTION
"God gave us the natural number, all else is the work of man". It was exclaimed by L. Kronecker (1823-1891), the
reputed German Mathematician. This statement reveals in a nut shell the significant role of the universe of numbers
played in the evolution of human thought.
N : The set of natural numbers,
W : The set of whole numbers,
Z : The set of Integers,
R
Z
Q
N
0
?2,
W
1,2,3,4,...
99,100,...
?3,
?3 + , ?5
?5,
5 3 ?
–5
6
1
3
—
3
11
2
15
e
Q : The set of rationals,
R : The set of Real Numbers.
? HISTORICAL FACTS
Dedekind was the first modern mathematician to publish in 1872 the mathematically
rigorous definition of irrational numbers. He gave explanation of their place in the
real Numbers System. He was able to demonstrate the completeness of the real number
line. He filled in the 'holes' in the system of Rational numbers with irrational Numbers.
This innovation has made Richard Dedekind an immortal figure in the history of
Mathematics.
Srinivasa Ramanujan (1887-1920) was one of the most outstanding mathematician
that India has produced. He worked on history of Numbers and discovered wonderul
properties of numbers. He stated intuitively many complicated result in mathematics.
Once a great mathematician Prof. Hardy come to India to see Ramanujan. Prof.
Hardy remarked that he has travelled in a taxi with a rather dull number viz. 1729.
Ramanujan jumped up and said, Oh! No. 1729 is very interesting number. It is the
smallest number which can be expressed as the sum of two cubes in two different
ways.
viz 1729 = 1
3
 + 12
3
,
1729 = 9
3
 + 10
3
,
? 1729 = 1
3
 + 12
3 
= 9
3
 + 10
3
? RECALL
In our day to day life, we deal with different types of numbers which can be broadly classified as follows.
Richard Dedekind
(1831-1916)
Srinivasa Ramanujan
(1887-1920)
Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009
2
CLASSIFICATION OF NUMBERS
                     
N U M BE R S  
R E A L N UM BE R S 
I M A G I NA R Y N U M BE R S
R A TI ONA L N UM BE R S
I R R A TI ON A L N U M BE R S 
FR A C T I ON S 
IN T E G E R S
N E G A T I V E S
Z E R O
W H OL E N U M BE R S
N A TU R A L N U M BE R S
( i ) Natural numbers (N) : N = {1, 2, 3, 4... ?}
Remark : ( i ) The set N is infinite i.e. it has unlimited members.
( i i ) N has the smallest element namely '1'.
( i i i ) N has no largest element. i.e., give me any natural number, we can find the bigger number from
the given number.
( i v ) N does not contain '0' as a member. i.e. '0' is not a member of the set N.
( i i ) Whole numbers (W) : W = {0, 1, 2, 3, 4... ?}
Remark : ( i ) The set of whole number is infinite (unlimited elements).
( i i ) This set has the smallest members as '0'. i.e. '0' the smallest whole number. i.e., set W contain
'0' as a member.
( i i i ) The set of whole numbers has no largest member.
( i v ) Every natural number is a whole number.
( v ) Non-zero smallest whole number is '1'.
( i i i ) Integers (I or Z) : I or Z = {– ?... –3, –2, –1, 0, +1, +2, +3 ...+ ?}
Positive integers : {1, 2, 3...}, Negative integers : {.... –4, –3, –2, –1}
Remark : ( i ) This set Z is infinite.
( i i ) It has neither the greatest nor the least element.
( i i i ) Every natural number is an integer.
( i v ) Every whole number is an integer.
( i v ) The set of non-negative integer = {0, 1, 2, 3, 4,....}
( v ) The set of non-positive integer = {......–4, – 3, – 2, –1, 0}
( i v ) Rational numbers :– These are real numbers which can be expressed in the form of 
p
q
, where p and q are integers
and q ? 0.
Ex.
2 37 –17
, ,
3 15 19
, –3, 0, 10, 4.33, 7.123123123.........
Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 3
Remark : ( i ) Every integer is a rational number.
( i i ) Every terminating decimal is a rational number.
( i i i ) Every recurring decimal is a rational number.
( i v ) A non- terminating repeating decimal is called a recurring decimal.
( v ) Between any two rational numbers there are an infinite number of rational numbers. This
property is known as the density of rational numbers.
( v i ) If a and b are two rational numbers then 
1
(a b)
2
? lies between a and b.
a < 
1
(a b)
2
? < b
n rational numbers between two different rational numbers a and b are :
a + 
(b – a)
;
n 1 ?
 a + 
2(b – a)
;
n 1 ?
 a + 
3(b – a)
;
n 1 ?
a + 
4(b – a)
;
n 1 ?
.......a +
n(b – a)
;
n 1 ?
( v i i ) Every rational number can be represented either as a terminating decimal or as a non-terminating
repeating (recurring) decimals.
( v i i i ) Types of rational numbers :– (a) Terminating decimal numbers and
(b) Non-terminating repeating (recurring) decimal numbers
( v ) Irrational numbers :– A number is called irrational number, if it can not be written in the form 
p
q
, where p & q are
integers and q ? 0. All Non-terminating & Non-repeating decimal numbers are Irrational numbers.
Ex.
2, 3, 3 2, 2 3, 2 3 ? ?
, ?, e, etc...
( v i ) Real numbers :– The totality of rational numbers and irrational numbers is called the set of real numbers i.e.
rational numbers and irrational numbers taken together are called real numbers.
Every real number is either a rational number or an irrational number.
? NATURE OF THE DECIMAL EXPANSION OF RATIONAL NUMBERS
Theorem-1 : Let x be a rational number whose decimal expansion terminates. Then we can express x in the
form 
p
q
, where p and q are co-primes, and the prime factorisation of q is of the form 2
m
 ×  5
n
,
where m,n are non-negative integers.
Theorem-2 : Let x = 
p
q
 be a rational number, such that the prime factorisation of q is of the form 2
m
 ×  5
n
, where
m,n are non-negative integers . Then, x has a decimal expansion which terminates.
Theorem-3 : Let x = 
p
q
 be a rational number, such that the prime factorisation of q is not of the form 2
m
 ×
5
n
, where m,n are non-negative integers . Then, x has a decimal expansion which is non-terminating
repeating.
Ex. ( i )
3 0 3
189 189 189
125 5 2 5
? ?
?
we observe that the prime factorisation of the denominators of these rational numbers are of the
form 2
m
 × 5
n
, where m,n are non-negative integers. Hence, 
189
125
 has terminating decimal expansion.
( i i )
17
6
= 
17
2 3 ?
we observe that the prime factorisation of the denominator of these rational numbers are not of the form 2
m
× 5
n
, where m,n are non-negative integers. Hence 
17
6
has non-terminating and repeating decimal expansion.
Page 4


Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 1
REAL NUMBERS
? INTRODUCTION
"God gave us the natural number, all else is the work of man". It was exclaimed by L. Kronecker (1823-1891), the
reputed German Mathematician. This statement reveals in a nut shell the significant role of the universe of numbers
played in the evolution of human thought.
N : The set of natural numbers,
W : The set of whole numbers,
Z : The set of Integers,
R
Z
Q
N
0
?2,
W
1,2,3,4,...
99,100,...
?3,
?3 + , ?5
?5,
5 3 ?
–5
6
1
3
—
3
11
2
15
e
Q : The set of rationals,
R : The set of Real Numbers.
? HISTORICAL FACTS
Dedekind was the first modern mathematician to publish in 1872 the mathematically
rigorous definition of irrational numbers. He gave explanation of their place in the
real Numbers System. He was able to demonstrate the completeness of the real number
line. He filled in the 'holes' in the system of Rational numbers with irrational Numbers.
This innovation has made Richard Dedekind an immortal figure in the history of
Mathematics.
Srinivasa Ramanujan (1887-1920) was one of the most outstanding mathematician
that India has produced. He worked on history of Numbers and discovered wonderul
properties of numbers. He stated intuitively many complicated result in mathematics.
Once a great mathematician Prof. Hardy come to India to see Ramanujan. Prof.
Hardy remarked that he has travelled in a taxi with a rather dull number viz. 1729.
Ramanujan jumped up and said, Oh! No. 1729 is very interesting number. It is the
smallest number which can be expressed as the sum of two cubes in two different
ways.
viz 1729 = 1
3
 + 12
3
,
1729 = 9
3
 + 10
3
,
? 1729 = 1
3
 + 12
3 
= 9
3
 + 10
3
? RECALL
In our day to day life, we deal with different types of numbers which can be broadly classified as follows.
Richard Dedekind
(1831-1916)
Srinivasa Ramanujan
(1887-1920)
Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009
2
CLASSIFICATION OF NUMBERS
                     
N U M BE R S  
R E A L N UM BE R S 
I M A G I NA R Y N U M BE R S
R A TI ONA L N UM BE R S
I R R A TI ON A L N U M BE R S 
FR A C T I ON S 
IN T E G E R S
N E G A T I V E S
Z E R O
W H OL E N U M BE R S
N A TU R A L N U M BE R S
( i ) Natural numbers (N) : N = {1, 2, 3, 4... ?}
Remark : ( i ) The set N is infinite i.e. it has unlimited members.
( i i ) N has the smallest element namely '1'.
( i i i ) N has no largest element. i.e., give me any natural number, we can find the bigger number from
the given number.
( i v ) N does not contain '0' as a member. i.e. '0' is not a member of the set N.
( i i ) Whole numbers (W) : W = {0, 1, 2, 3, 4... ?}
Remark : ( i ) The set of whole number is infinite (unlimited elements).
( i i ) This set has the smallest members as '0'. i.e. '0' the smallest whole number. i.e., set W contain
'0' as a member.
( i i i ) The set of whole numbers has no largest member.
( i v ) Every natural number is a whole number.
( v ) Non-zero smallest whole number is '1'.
( i i i ) Integers (I or Z) : I or Z = {– ?... –3, –2, –1, 0, +1, +2, +3 ...+ ?}
Positive integers : {1, 2, 3...}, Negative integers : {.... –4, –3, –2, –1}
Remark : ( i ) This set Z is infinite.
( i i ) It has neither the greatest nor the least element.
( i i i ) Every natural number is an integer.
( i v ) Every whole number is an integer.
( i v ) The set of non-negative integer = {0, 1, 2, 3, 4,....}
( v ) The set of non-positive integer = {......–4, – 3, – 2, –1, 0}
( i v ) Rational numbers :– These are real numbers which can be expressed in the form of 
p
q
, where p and q are integers
and q ? 0.
Ex.
2 37 –17
, ,
3 15 19
, –3, 0, 10, 4.33, 7.123123123.........
Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 3
Remark : ( i ) Every integer is a rational number.
( i i ) Every terminating decimal is a rational number.
( i i i ) Every recurring decimal is a rational number.
( i v ) A non- terminating repeating decimal is called a recurring decimal.
( v ) Between any two rational numbers there are an infinite number of rational numbers. This
property is known as the density of rational numbers.
( v i ) If a and b are two rational numbers then 
1
(a b)
2
? lies between a and b.
a < 
1
(a b)
2
? < b
n rational numbers between two different rational numbers a and b are :
a + 
(b – a)
;
n 1 ?
 a + 
2(b – a)
;
n 1 ?
 a + 
3(b – a)
;
n 1 ?
a + 
4(b – a)
;
n 1 ?
.......a +
n(b – a)
;
n 1 ?
( v i i ) Every rational number can be represented either as a terminating decimal or as a non-terminating
repeating (recurring) decimals.
( v i i i ) Types of rational numbers :– (a) Terminating decimal numbers and
(b) Non-terminating repeating (recurring) decimal numbers
( v ) Irrational numbers :– A number is called irrational number, if it can not be written in the form 
p
q
, where p & q are
integers and q ? 0. All Non-terminating & Non-repeating decimal numbers are Irrational numbers.
Ex.
2, 3, 3 2, 2 3, 2 3 ? ?
, ?, e, etc...
( v i ) Real numbers :– The totality of rational numbers and irrational numbers is called the set of real numbers i.e.
rational numbers and irrational numbers taken together are called real numbers.
Every real number is either a rational number or an irrational number.
? NATURE OF THE DECIMAL EXPANSION OF RATIONAL NUMBERS
Theorem-1 : Let x be a rational number whose decimal expansion terminates. Then we can express x in the
form 
p
q
, where p and q are co-primes, and the prime factorisation of q is of the form 2
m
 ×  5
n
,
where m,n are non-negative integers.
Theorem-2 : Let x = 
p
q
 be a rational number, such that the prime factorisation of q is of the form 2
m
 ×  5
n
, where
m,n are non-negative integers . Then, x has a decimal expansion which terminates.
Theorem-3 : Let x = 
p
q
 be a rational number, such that the prime factorisation of q is not of the form 2
m
 ×
5
n
, where m,n are non-negative integers . Then, x has a decimal expansion which is non-terminating
repeating.
Ex. ( i )
3 0 3
189 189 189
125 5 2 5
? ?
?
we observe that the prime factorisation of the denominators of these rational numbers are of the
form 2
m
 × 5
n
, where m,n are non-negative integers. Hence, 
189
125
 has terminating decimal expansion.
( i i )
17
6
= 
17
2 3 ?
we observe that the prime factorisation of the denominator of these rational numbers are not of the form 2
m
× 5
n
, where m,n are non-negative integers. Hence 
17
6
has non-terminating and repeating decimal expansion.
Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009
4
( i i i )
17
8
 = 
3 0
17
2 5 ?
So, the denominator 8 of 
17
8
 is of the form 2
m
 × 5
n
, where m,n are non-negative integers.
Hence 
17
8
 has terminating decimal expansion.
( i v )
64
455
 = 
64
5 7 13 ? ?
Clearly, 455 is not of the form 2
m
 × 5
n
. So, the decimal expansion of 
64
455
 is non-terminating repeating.
? PROOF OF IRRATIONALITY OF 2 , 3 , 5 ,......
Ex.1 Prove that 
2
 is not a rational number or there is no rational whose square is 2.        (CBSE (outside Delhi ) 2008).
Sol. Let us find the square root of 2 by long division method as shown below.
2.000000000000
1.414215
  1
+1  1
24    100
  4      96
281    400
    1    281
2824
4
11900
11296
+
+
28282
2 +
+
60400
56564
282841
1
383600
282841
2828423
3
10075900
8485269
28284265 159063100
141421325
17641775
+5
28284270
?2  = 1.414215
Clearly, the decimal representation of 
2
 is neither terminating nor repeating.
We shall prove this by the method of contradiction.
If possible, let us assume that 
2
 is a rational number.
Then 
2
 = 
a
b
 where a, b are integers having no common factor other than 1.
?
? ?
2
2
a
2
b
? ?
?
? ?
? ?
 (squaring both sides)
2 = 
2
2
a
b
a
2
 = 2b
2
? 2 divides a
2
? 2 divides a
Therefore let a = 2c for some integer c.
? a
2
 = 4c
2
Page 5


Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 1
REAL NUMBERS
? INTRODUCTION
"God gave us the natural number, all else is the work of man". It was exclaimed by L. Kronecker (1823-1891), the
reputed German Mathematician. This statement reveals in a nut shell the significant role of the universe of numbers
played in the evolution of human thought.
N : The set of natural numbers,
W : The set of whole numbers,
Z : The set of Integers,
R
Z
Q
N
0
?2,
W
1,2,3,4,...
99,100,...
?3,
?3 + , ?5
?5,
5 3 ?
–5
6
1
3
—
3
11
2
15
e
Q : The set of rationals,
R : The set of Real Numbers.
? HISTORICAL FACTS
Dedekind was the first modern mathematician to publish in 1872 the mathematically
rigorous definition of irrational numbers. He gave explanation of their place in the
real Numbers System. He was able to demonstrate the completeness of the real number
line. He filled in the 'holes' in the system of Rational numbers with irrational Numbers.
This innovation has made Richard Dedekind an immortal figure in the history of
Mathematics.
Srinivasa Ramanujan (1887-1920) was one of the most outstanding mathematician
that India has produced. He worked on history of Numbers and discovered wonderul
properties of numbers. He stated intuitively many complicated result in mathematics.
Once a great mathematician Prof. Hardy come to India to see Ramanujan. Prof.
Hardy remarked that he has travelled in a taxi with a rather dull number viz. 1729.
Ramanujan jumped up and said, Oh! No. 1729 is very interesting number. It is the
smallest number which can be expressed as the sum of two cubes in two different
ways.
viz 1729 = 1
3
 + 12
3
,
1729 = 9
3
 + 10
3
,
? 1729 = 1
3
 + 12
3 
= 9
3
 + 10
3
? RECALL
In our day to day life, we deal with different types of numbers which can be broadly classified as follows.
Richard Dedekind
(1831-1916)
Srinivasa Ramanujan
(1887-1920)
Real Numbers – Aadhar TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009
2
CLASSIFICATION OF NUMBERS
                     
N U M BE R S  
R E A L N UM BE R S 
I M A G I NA R Y N U M BE R S
R A TI ONA L N UM BE R S
I R R A TI ON A L N U M BE R S 
FR A C T I ON S 
IN T E G E R S
N E G A T I V E S
Z E R O
W H OL E N U M BE R S
N A TU R A L N U M BE R S
( i ) Natural numbers (N) : N = {1, 2, 3, 4... ?}
Remark : ( i ) The set N is infinite i.e. it has unlimited members.
( i i ) N has the smallest element namely '1'.
( i i i ) N has no largest element. i.e., give me any natural number, we can find the bigger number from
the given number.
( i v ) N does not contain '0' as a member. i.e. '0' is not a member of the set N.
( i i ) Whole numbers (W) : W = {0, 1, 2, 3, 4... ?}
Remark : ( i ) The set of whole number is infinite (unlimited elements).
( i i ) This set has the smallest members as '0'. i.e. '0' the smallest whole number. i.e., set W contain
'0' as a member.
( i i i ) The set of whole numbers has no largest member.
( i v ) Every natural number is a whole number.
( v ) Non-zero smallest whole number is '1'.
( i i i ) Integers (I or Z) : I or Z = {– ?... –3, –2, –1, 0, +1, +2, +3 ...+ ?}
Positive integers : {1, 2, 3...}, Negative integers : {.... –4, –3, –2, –1}
Remark : ( i ) This set Z is infinite.
( i i ) It has neither the greatest nor the least element.
( i i i ) Every natural number is an integer.
( i v ) Every whole number is an integer.
( i v ) The set of non-negative integer = {0, 1, 2, 3, 4,....}
( v ) The set of non-positive integer = {......–4, – 3, – 2, –1, 0}
( i v ) Rational numbers :– These are real numbers which can be expressed in the form of 
p
q
, where p and q are integers
and q ? 0.
Ex.
2 37 –17
, ,
3 15 19
, –3, 0, 10, 4.33, 7.123123123.........
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Remark : ( i ) Every integer is a rational number.
( i i ) Every terminating decimal is a rational number.
( i i i ) Every recurring decimal is a rational number.
( i v ) A non- terminating repeating decimal is called a recurring decimal.
( v ) Between any two rational numbers there are an infinite number of rational numbers. This
property is known as the density of rational numbers.
( v i ) If a and b are two rational numbers then 
1
(a b)
2
? lies between a and b.
a < 
1
(a b)
2
? < b
n rational numbers between two different rational numbers a and b are :
a + 
(b – a)
;
n 1 ?
 a + 
2(b – a)
;
n 1 ?
 a + 
3(b – a)
;
n 1 ?
a + 
4(b – a)
;
n 1 ?
.......a +
n(b – a)
;
n 1 ?
( v i i ) Every rational number can be represented either as a terminating decimal or as a non-terminating
repeating (recurring) decimals.
( v i i i ) Types of rational numbers :– (a) Terminating decimal numbers and
(b) Non-terminating repeating (recurring) decimal numbers
( v ) Irrational numbers :– A number is called irrational number, if it can not be written in the form 
p
q
, where p & q are
integers and q ? 0. All Non-terminating & Non-repeating decimal numbers are Irrational numbers.
Ex.
2, 3, 3 2, 2 3, 2 3 ? ?
, ?, e, etc...
( v i ) Real numbers :– The totality of rational numbers and irrational numbers is called the set of real numbers i.e.
rational numbers and irrational numbers taken together are called real numbers.
Every real number is either a rational number or an irrational number.
? NATURE OF THE DECIMAL EXPANSION OF RATIONAL NUMBERS
Theorem-1 : Let x be a rational number whose decimal expansion terminates. Then we can express x in the
form 
p
q
, where p and q are co-primes, and the prime factorisation of q is of the form 2
m
 ×  5
n
,
where m,n are non-negative integers.
Theorem-2 : Let x = 
p
q
 be a rational number, such that the prime factorisation of q is of the form 2
m
 ×  5
n
, where
m,n are non-negative integers . Then, x has a decimal expansion which terminates.
Theorem-3 : Let x = 
p
q
 be a rational number, such that the prime factorisation of q is not of the form 2
m
 ×
5
n
, where m,n are non-negative integers . Then, x has a decimal expansion which is non-terminating
repeating.
Ex. ( i )
3 0 3
189 189 189
125 5 2 5
? ?
?
we observe that the prime factorisation of the denominators of these rational numbers are of the
form 2
m
 × 5
n
, where m,n are non-negative integers. Hence, 
189
125
 has terminating decimal expansion.
( i i )
17
6
= 
17
2 3 ?
we observe that the prime factorisation of the denominator of these rational numbers are not of the form 2
m
× 5
n
, where m,n are non-negative integers. Hence 
17
6
has non-terminating and repeating decimal expansion.
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4
( i i i )
17
8
 = 
3 0
17
2 5 ?
So, the denominator 8 of 
17
8
 is of the form 2
m
 × 5
n
, where m,n are non-negative integers.
Hence 
17
8
 has terminating decimal expansion.
( i v )
64
455
 = 
64
5 7 13 ? ?
Clearly, 455 is not of the form 2
m
 × 5
n
. So, the decimal expansion of 
64
455
 is non-terminating repeating.
? PROOF OF IRRATIONALITY OF 2 , 3 , 5 ,......
Ex.1 Prove that 
2
 is not a rational number or there is no rational whose square is 2.        (CBSE (outside Delhi ) 2008).
Sol. Let us find the square root of 2 by long division method as shown below.
2.000000000000
1.414215
  1
+1  1
24    100
  4      96
281    400
    1    281
2824
4
11900
11296
+
+
28282
2 +
+
60400
56564
282841
1
383600
282841
2828423
3
10075900
8485269
28284265 159063100
141421325
17641775
+5
28284270
?2  = 1.414215
Clearly, the decimal representation of 
2
 is neither terminating nor repeating.
We shall prove this by the method of contradiction.
If possible, let us assume that 
2
 is a rational number.
Then 
2
 = 
a
b
 where a, b are integers having no common factor other than 1.
?
? ?
2
2
a
2
b
? ?
?
? ?
? ?
 (squaring both sides)
2 = 
2
2
a
b
a
2
 = 2b
2
? 2 divides a
2
? 2 divides a
Therefore let a = 2c for some integer c.
? a
2
 = 4c
2
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? 2b
2
 = 4c
2
? b
2
 = 2c
2
? 2 divides b
2
? 2 divides b
Thus, 2 is a common factor of a and b.
But, it contradicts our assumption that a and b have no common factor other than 1.
So, our assumption that 2 is a rational, is wrong.
Hence, 2 is irrational.
Ex.2 Prove that 
3
3 is irrational.
Sol. Let 
3
3
 be rational = 
p
q
, where p and q ? Z and p, q have no common factor except 1 also q > 1.
?  
p
q
 = 
3
3
Cubing both sides
3
3
p
q
 = 3
Multiply both sides by q
2
3
p
q
 = 3q
2
, Clearly L.H.S is rational since p, q have no common factor.
? p
3
, q also have no common factor while R.H.S. is an integer.
? L.H.S ? R.H.S which contradicts our assumption that 
3
3
 is Irrational.
Ex.3 Prove that 2 + 3 is irrational.                                             [Sample paper (CBSE) 2008]
Sol. Let 2 + 3be a rational number equals to r
? 2 + 3 = r
3 = r – 2
Here L.H.S is an irrational number while R.H.S. r – 2 is rational. ? L.H.S ? R.H.S
Hence it contradicts our assumption that 2 + 3 is rational.
? 2 + 3 is irrational.
Ex.4 Prove that 2 + 3 is irrational.
Sol. Let 2 + 3 be rational number say 'x' ? x = 2 + 3
x
2
 = 2 + 3 + 2 3 · 2 = 5 + 2 6
? x
2
 = 5 + 2 6 ? 
6
 = 
2
x 5
2
?
As x, 5 and 2 are rationals ? 
2
x 5
2
?
 is a rational number.
?
6
 = 
2
x 5
2
?
 is a rational number
Which is contradication of the fact that 
6
 is a irrational number.
Hence our supposition is wrong ? 2 + 3 is an irrational number.