UNIT – I : PURE ARITHMETIC
CHAPTER-1
RATIONAL AND IRRATIONAL NUMBERS
Topic-1 Properties of Rational and Irrational Numbers
Revision Notes
p
¾¾Rational Number : A number that can be expressed in the form , where p and q both are integers and q ≠ 0, is
q
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
In other words, is a rational number. If
q
(1) q ≠ 0
(2) p and q have no common factor other than 1(one) i.e. p and q are co-primes.
It is called in the lowest terms or simplest form or irreducible form.
(3) q is usually positive, whereas p may be positive, negative or zero.
In general, the set of rational number is denoted by the letter Q.
p
\ Q = : p , q ∈ z and q ≠ 0
q
Every integer (positive, negative or zero) and every decimal number (non-terminating non-recurring) is a
rational number.
p −p
Corresponding to every rational number , its negative rational number is .
q q
Properties of rational numbers
(1) It a, b are any two rational numbers, then a + b is also a rational numbers.
(2) If a, b are any two rational numbers, then a – b and b – a is also a rational number.
(3) If a, b are any two rational numbers, then a × b is also a rational number.
a
(4) If a, b are any two rational numbers and b ≠ 0, then is also a rational number.
b
(5) The collection of rational number is ordered i.e. If a, b are any two rational numbers, then either a < b or a > b
or a = b.
a c
(6) Two rational numbers and are equal, if and only if : a × d = b × c
b d
a c
Also, > ⇒a×d>b×c
b d
a c
and < ⇒a×d<b×c
b d
,2 Oswaal ICSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-IX
a+b
(7) If a, b are any two different rational numbers, then is also a rational number and it lies between a and b.
2
a+b
i.e. if a>b⇒a> >b
2
a+b
and if a<b⇒a< <b
2
p
Irrational Number : A number that cannot be expressed in the form , where p and q are both integers and q ≠ 0,
q
p and q have no common factors (except), is called an irrational number.
The square roots of prime natural numbers are irrational numbers, m is irrational, if m is the prime number.
A non-terminating and non-recurring decimal is an irrational number.
Ex - (i) 0·26561987 ...... (ii) 238·56575859 ......
The number p is also an irrational number.
Properties of irrational numbers
(1) For any two positive rational number x and y if x and y are irrationals then :
x > y ⇒x>y
y ⇒x<y
and x <
(2) a + b x = c + d x ⇒ a = c and b = d
(3) The negative of an irrational number is always irrational.
(4) If x is rational number and y is irrational number, then x + y, x – y and y – x are irrational numbers.
x y
(5) If x is a non-zero rational number and y is an irrational number, then xy, and are irrational numbers.
y x
(6) The sum of two irrational numbers may or may not be irrational.
Ex- (i) ( 6 + 2 5 ) + ( 9 − 2 5 ) = 15, which is not an irrational number.
(ii) ( 2 3 + 5) + (7 2 − 5) = 2 3 + 7 2 , which is an irrational number.
(7) The difference of two irrational number may or may not be irrational.
Ex- (i) ( 9 − 5 ) − ( 3 − 5 ) = 6, which is not an irrational number.
(ii) ( 8 + 5 ) − ( 5 + 2 ) = 3 + 5 − 2 , which is an irrational number.
(8) The product of two irrational numbers, may or may not be irrational.
Ex- (i) ( 2 + 7 ) × ( 2 − 7 ) = 4 – 7 = – 3, which is not an irrational number.
(ii) ( 2 + 7 ) × ( 3 − 6 ) = 6 − 2 6 + 3 7 − 42 , which is an irrational number.
(9) The quotient of two irrational number may or may not be irrational number.
4 25 4 20
Ex - (i) 4 75 ÷ 7 3 = = ×5 = ,
7 7 7
which is not an irrational number.
4 25
(ii) 4 25 ÷ 2 5 = = 2 5 , which is an irrational number.
2 5
Real Number : The collection of all rational numbers together with all irrational numbers forms the collection of
real numbers. This collection is denoted by R.
i.e. R = Q ∪ Q
where Q is the set of rational number and Q is the set of irrational numbers.
Rational number (Q) is the set of all terminating or recurring decimals :
Irrational number (Q ) is the set of all non-terminating and non-recurring decimals.
, Oswaal ICSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-IX 3
Real Numbers (R)
↓ ↓
Rational Numbers (Q) Irrational Numbers
−7 3 ..... etc.
– 3, 0, 5, ,
2 5 2 , 3 , − 6 , ...... etc.
↓ ↓
Non-integer Rational Numbers
Integers (I or Z) 5 7 −3
.... – 3, – 2, – 1, 0, 1, 2, 3, ........ , , , ... etc.
14 2 5
↓ ↓ ↓
Positive integers Negative Integers Whole Numbers (W)
→
1, 2, 3, ....... ......, – 3, – 2, – 1 0, 1, 2, 3, ........
↓ ↓
Zero/Null integer Natural Number (N)
0 1, 2, 3, 4, .........
Properties of real numbers :
(1) If a, b are any two real numbers, then a + b is also a real number.
(2) If a, b are any two real numbers, then a – b is also a real number.
(3) If a, b are any two real numbers, then a × b is also a real number.
a
(4) If a, b are any two real numbers and b ≠ 0, then is also a real number.
b
(5) The set of real numbers is ordered i.e. if a, b are any two real numbers, then either a > b or a < b or a = b. This
is called trichotomy law.
a+b
(6) If a, b are any two real number, then is a real number and it lies between a and b.
2
a+b
i.e. if a > b ⇒ a > >b
2
a+b
and if a < b ⇒ a < <b
2
Every real number (rational or irrational) can be represented by a unique point on the number line. Conversely,
every point on the number line represents a unique real number.
When numerator of a rational number divide by its denominator and remainder becomes zero after some steps,
then such decimal expansion, are called terminating decimal.
1 7 9
Ex - = 0.5, = 0.875, = 0.1125
2 8 80
If the denominator of a rational number can be expressed as 2m or 5n or 2m × 5n, where m and n both are whole
numbers, then the rational number is a terminating decimal.
9
Ex -
80
Since, 80 = 2 × 2 × 2 × 2 × 5 = 24 × 51
i.e. 80 can be expressed as 2 × 5
m n
9
\ Rational number is a terminating decimal.
80
When numerator of a rational number divide by its denominator and remainder never becomes zero, then such
decimal expansions are called non-terminating decimal.
10
Ex - (i) = 3.333 ....... (ii) 2 = 1.41421356237300 .......
3
, 4 Oswaal ICSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-IX
(i) When remainder never becomes zero and they repeat after a certain stage which force the decimal expansion
to go forever, then such decimal expansion are called non-terminating recurring (repeating) decimal.
1 67
Ex - (i) = 0.142857142857 ...... (ii) = 5.153846153846 .......
7 13
(ii) When remainder never becomes zero and they do not repeat after a certain stage and force the decimal
expansion to go forever, then such decimal expansion are called non-terminating non-recurring decimal.
Ex - (i) p = 3·14159265358979323846 ........ (ii) 0·1010010001100 ...........
All integers (positive, zero or negative) are terminating decimals.
The decimal expansion of a rational number is either terminating or non-terminating recurring (repeating).
Conversely, a number whose decimal expansion is terminating or non-terminating recurring is rational number.
p
The decimal expansion of a rational number , where p and q are integers, q > 0, p and q have no common
q
factors except 1 is :
(i) terminating if prime factors of q are 2 or 5 or both.
(ii) non-terminating recurring if q has a prime factor other then 2 or 5.
The decimal expansion of a irrational number is non-terminating non-recurring. Conversely, a number whose
decimal expansion is non-terminating non-recurring is irrational number.
All decimal numbers (terminating, recurring or non-terminating and non-recurring) are real numbers.
Representation of rational numbers on the number line : We know that the rational numbers include natural
numbers, whole numbers, integers and fractional numbers. All these numbers can be represented on the number
line as shown below :
(1) Number line for natural numbers :
(2) Number line for whole numbers :
(3) Number line for integers :
–∞ ∞∞ +∞
(4) Number line for fractions :
3 9 −1 −7
(i) For fractions , , and
4 4 4 4
–∞ +∞
2 7 −5
(ii) For fractions , and
5 4 3
–∞ +∞
Representation of irrational number on the number line : To represent the irrational number2 on the number
line l, construct a right angled triangle OAC, right angled at A, such that OA = AC = 1, then by Pythagoras
theorem,
OC2 = OA2 + AC2
OC2 = (1)2 + (1)2 = 2
OC = 2
CHAPTER-1
RATIONAL AND IRRATIONAL NUMBERS
Topic-1 Properties of Rational and Irrational Numbers
Revision Notes
p
¾¾Rational Number : A number that can be expressed in the form , where p and q both are integers and q ≠ 0, is
q
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
In other words, is a rational number. If
q
(1) q ≠ 0
(2) p and q have no common factor other than 1(one) i.e. p and q are co-primes.
It is called in the lowest terms or simplest form or irreducible form.
(3) q is usually positive, whereas p may be positive, negative or zero.
In general, the set of rational number is denoted by the letter Q.
p
\ Q = : p , q ∈ z and q ≠ 0
q
Every integer (positive, negative or zero) and every decimal number (non-terminating non-recurring) is a
rational number.
p −p
Corresponding to every rational number , its negative rational number is .
q q
Properties of rational numbers
(1) It a, b are any two rational numbers, then a + b is also a rational numbers.
(2) If a, b are any two rational numbers, then a – b and b – a is also a rational number.
(3) If a, b are any two rational numbers, then a × b is also a rational number.
a
(4) If a, b are any two rational numbers and b ≠ 0, then is also a rational number.
b
(5) The collection of rational number is ordered i.e. If a, b are any two rational numbers, then either a < b or a > b
or a = b.
a c
(6) Two rational numbers and are equal, if and only if : a × d = b × c
b d
a c
Also, > ⇒a×d>b×c
b d
a c
and < ⇒a×d<b×c
b d
,2 Oswaal ICSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-IX
a+b
(7) If a, b are any two different rational numbers, then is also a rational number and it lies between a and b.
2
a+b
i.e. if a>b⇒a> >b
2
a+b
and if a<b⇒a< <b
2
p
Irrational Number : A number that cannot be expressed in the form , where p and q are both integers and q ≠ 0,
q
p and q have no common factors (except), is called an irrational number.
The square roots of prime natural numbers are irrational numbers, m is irrational, if m is the prime number.
A non-terminating and non-recurring decimal is an irrational number.
Ex - (i) 0·26561987 ...... (ii) 238·56575859 ......
The number p is also an irrational number.
Properties of irrational numbers
(1) For any two positive rational number x and y if x and y are irrationals then :
x > y ⇒x>y
y ⇒x<y
and x <
(2) a + b x = c + d x ⇒ a = c and b = d
(3) The negative of an irrational number is always irrational.
(4) If x is rational number and y is irrational number, then x + y, x – y and y – x are irrational numbers.
x y
(5) If x is a non-zero rational number and y is an irrational number, then xy, and are irrational numbers.
y x
(6) The sum of two irrational numbers may or may not be irrational.
Ex- (i) ( 6 + 2 5 ) + ( 9 − 2 5 ) = 15, which is not an irrational number.
(ii) ( 2 3 + 5) + (7 2 − 5) = 2 3 + 7 2 , which is an irrational number.
(7) The difference of two irrational number may or may not be irrational.
Ex- (i) ( 9 − 5 ) − ( 3 − 5 ) = 6, which is not an irrational number.
(ii) ( 8 + 5 ) − ( 5 + 2 ) = 3 + 5 − 2 , which is an irrational number.
(8) The product of two irrational numbers, may or may not be irrational.
Ex- (i) ( 2 + 7 ) × ( 2 − 7 ) = 4 – 7 = – 3, which is not an irrational number.
(ii) ( 2 + 7 ) × ( 3 − 6 ) = 6 − 2 6 + 3 7 − 42 , which is an irrational number.
(9) The quotient of two irrational number may or may not be irrational number.
4 25 4 20
Ex - (i) 4 75 ÷ 7 3 = = ×5 = ,
7 7 7
which is not an irrational number.
4 25
(ii) 4 25 ÷ 2 5 = = 2 5 , which is an irrational number.
2 5
Real Number : The collection of all rational numbers together with all irrational numbers forms the collection of
real numbers. This collection is denoted by R.
i.e. R = Q ∪ Q
where Q is the set of rational number and Q is the set of irrational numbers.
Rational number (Q) is the set of all terminating or recurring decimals :
Irrational number (Q ) is the set of all non-terminating and non-recurring decimals.
, Oswaal ICSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-IX 3
Real Numbers (R)
↓ ↓
Rational Numbers (Q) Irrational Numbers
−7 3 ..... etc.
– 3, 0, 5, ,
2 5 2 , 3 , − 6 , ...... etc.
↓ ↓
Non-integer Rational Numbers
Integers (I or Z) 5 7 −3
.... – 3, – 2, – 1, 0, 1, 2, 3, ........ , , , ... etc.
14 2 5
↓ ↓ ↓
Positive integers Negative Integers Whole Numbers (W)
→
1, 2, 3, ....... ......, – 3, – 2, – 1 0, 1, 2, 3, ........
↓ ↓
Zero/Null integer Natural Number (N)
0 1, 2, 3, 4, .........
Properties of real numbers :
(1) If a, b are any two real numbers, then a + b is also a real number.
(2) If a, b are any two real numbers, then a – b is also a real number.
(3) If a, b are any two real numbers, then a × b is also a real number.
a
(4) If a, b are any two real numbers and b ≠ 0, then is also a real number.
b
(5) The set of real numbers is ordered i.e. if a, b are any two real numbers, then either a > b or a < b or a = b. This
is called trichotomy law.
a+b
(6) If a, b are any two real number, then is a real number and it lies between a and b.
2
a+b
i.e. if a > b ⇒ a > >b
2
a+b
and if a < b ⇒ a < <b
2
Every real number (rational or irrational) can be represented by a unique point on the number line. Conversely,
every point on the number line represents a unique real number.
When numerator of a rational number divide by its denominator and remainder becomes zero after some steps,
then such decimal expansion, are called terminating decimal.
1 7 9
Ex - = 0.5, = 0.875, = 0.1125
2 8 80
If the denominator of a rational number can be expressed as 2m or 5n or 2m × 5n, where m and n both are whole
numbers, then the rational number is a terminating decimal.
9
Ex -
80
Since, 80 = 2 × 2 × 2 × 2 × 5 = 24 × 51
i.e. 80 can be expressed as 2 × 5
m n
9
\ Rational number is a terminating decimal.
80
When numerator of a rational number divide by its denominator and remainder never becomes zero, then such
decimal expansions are called non-terminating decimal.
10
Ex - (i) = 3.333 ....... (ii) 2 = 1.41421356237300 .......
3
, 4 Oswaal ICSE Revision Notes Chapterwise & Topicwise, MATHEMATICS, Class-IX
(i) When remainder never becomes zero and they repeat after a certain stage which force the decimal expansion
to go forever, then such decimal expansion are called non-terminating recurring (repeating) decimal.
1 67
Ex - (i) = 0.142857142857 ...... (ii) = 5.153846153846 .......
7 13
(ii) When remainder never becomes zero and they do not repeat after a certain stage and force the decimal
expansion to go forever, then such decimal expansion are called non-terminating non-recurring decimal.
Ex - (i) p = 3·14159265358979323846 ........ (ii) 0·1010010001100 ...........
All integers (positive, zero or negative) are terminating decimals.
The decimal expansion of a rational number is either terminating or non-terminating recurring (repeating).
Conversely, a number whose decimal expansion is terminating or non-terminating recurring is rational number.
p
The decimal expansion of a rational number , where p and q are integers, q > 0, p and q have no common
q
factors except 1 is :
(i) terminating if prime factors of q are 2 or 5 or both.
(ii) non-terminating recurring if q has a prime factor other then 2 or 5.
The decimal expansion of a irrational number is non-terminating non-recurring. Conversely, a number whose
decimal expansion is non-terminating non-recurring is irrational number.
All decimal numbers (terminating, recurring or non-terminating and non-recurring) are real numbers.
Representation of rational numbers on the number line : We know that the rational numbers include natural
numbers, whole numbers, integers and fractional numbers. All these numbers can be represented on the number
line as shown below :
(1) Number line for natural numbers :
(2) Number line for whole numbers :
(3) Number line for integers :
–∞ ∞∞ +∞
(4) Number line for fractions :
3 9 −1 −7
(i) For fractions , , and
4 4 4 4
–∞ +∞
2 7 −5
(ii) For fractions , and
5 4 3
–∞ +∞
Representation of irrational number on the number line : To represent the irrational number2 on the number
line l, construct a right angled triangle OAC, right angled at A, such that OA = AC = 1, then by Pythagoras
theorem,
OC2 = OA2 + AC2
OC2 = (1)2 + (1)2 = 2
OC = 2
