Points to Remember: Real Numbers Notes | Study Mathematics (Maths) Class 10 - Class 10

 Page 1


Real 
Numbers 
Page 2


Real 
Numbers 
Real Numbers ( 2, -3,    ,    2, 0 )
1
3
Learning Numbers is the ?rst step to enter the world of Mathematics. Below are the di?erent form of numbers 
Terminating
( -3, 7,      )
1
3
(Repeating)
(..., -3, -2, -1, 0, 1, 2, 3, ....)
(2.5, 1.275, 0.8912)
1.252525...,
0.777777...,
4.32753275...
(-1, -2, -3, -4, -5.....)
Irrational numbers
(0, 1, 2, 3, 4, 5....)
Zero (0)
(1, 2, 3, 4,5,.....)
(Exactly 2 divisors) (Neither prime nor composite) (Minimum 3 divisors)
1
2
-2
4
0.5 , ,
2  , 3  ,
18
( )
Decimal form
3.14159...,
0.247912864...,
Non-terminating
Non-repeating
[
]
[ ]
( )
Rational Numbers
Integers Fractions / Decimals
Whole Numbers
Non Terminating
Negative Integers
Positive Integers
Prime Numbers One Composite Numbers
REAL NUMBERS 2
Page 3


Real 
Numbers 
Real Numbers ( 2, -3,    ,    2, 0 )
1
3
Learning Numbers is the ?rst step to enter the world of Mathematics. Below are the di?erent form of numbers 
Terminating
( -3, 7,      )
1
3
(Repeating)
(..., -3, -2, -1, 0, 1, 2, 3, ....)
(2.5, 1.275, 0.8912)
1.252525...,
0.777777...,
4.32753275...
(-1, -2, -3, -4, -5.....)
Irrational numbers
(0, 1, 2, 3, 4, 5....)
Zero (0)
(1, 2, 3, 4,5,.....)
(Exactly 2 divisors) (Neither prime nor composite) (Minimum 3 divisors)
1
2
-2
4
0.5 , ,
2  , 3  ,
18
( )
Decimal form
3.14159...,
0.247912864...,
Non-terminating
Non-repeating
[
]
[ ]
( )
Rational Numbers
Integers Fractions / Decimals
Whole Numbers
Non Terminating
Negative Integers
Positive Integers
Prime Numbers One Composite Numbers
REAL NUMBERS 2
Methods to nd HCF
Euclid’s Division Lemma
The HCF of (c, d) where c > d,
Apply Euclid’s Division Lemma to c and d.
c = dq + r, 0 = r < d (where q & r are whole numbers)
If r = 0, d is the HCF of (c, d),
if r ? 0, then apply the division lemma to d & r, d > r.
Continue this process till the remainder is zero.
The divisor at this stage will be HCF of (c, d).
Every composite number expressed as a product or factors of
prime numbers and this factorization, is unique.
We nd LCM and HCF of two positive integers using the
fundamental theorem of arithmetic, which is also called Prime
Factorization method.
For any two positive integers c & d.
HCF (c, d) x LCM (c, d) = c x d 
Fundamental Theorem of Arithmetic
Step 1 :
Step 3 :
Step 2 :
c x d
LCM (c, d)
so, HCF (c, d) =
Gain an understanding of an important concept in number theory - The Euclid’s Division Lemma and learn its application in nding out the
Highest Common Factor (HCF).
2
2
2
5
2
2
2
5
Product of the common & uncommon prime factors involved
in the numbers.
It is a smallest common dividend for two or more divisors.
Example: LCM of (8, 20) = 2 x 2 x 2 x 5
= 40
Example: HCF of (8, 20) = 2 x 2
= 4
LCM
?
?
Product of the common prime factors involved in the
numbers.
It is a greatest common divisor for two or more dividends.
HCF
?
?
LCM and HCF are the two important concepts in Algebra. This chart will help you understand how to nd LCM and HCF of two or more than two
positive integers.
REAL NUMBERS 3
Page 4


Real 
Numbers 
Real Numbers ( 2, -3,    ,    2, 0 )
1
3
Learning Numbers is the ?rst step to enter the world of Mathematics. Below are the di?erent form of numbers 
Terminating
( -3, 7,      )
1
3
(Repeating)
(..., -3, -2, -1, 0, 1, 2, 3, ....)
(2.5, 1.275, 0.8912)
1.252525...,
0.777777...,
4.32753275...
(-1, -2, -3, -4, -5.....)
Irrational numbers
(0, 1, 2, 3, 4, 5....)
Zero (0)
(1, 2, 3, 4,5,.....)
(Exactly 2 divisors) (Neither prime nor composite) (Minimum 3 divisors)
1
2
-2
4
0.5 , ,
2  , 3  ,
18
( )
Decimal form
3.14159...,
0.247912864...,
Non-terminating
Non-repeating
[
]
[ ]
( )
Rational Numbers
Integers Fractions / Decimals
Whole Numbers
Non Terminating
Negative Integers
Positive Integers
Prime Numbers One Composite Numbers
REAL NUMBERS 2
Methods to nd HCF
Euclid’s Division Lemma
The HCF of (c, d) where c > d,
Apply Euclid’s Division Lemma to c and d.
c = dq + r, 0 = r < d (where q & r are whole numbers)
If r = 0, d is the HCF of (c, d),
if r ? 0, then apply the division lemma to d & r, d > r.
Continue this process till the remainder is zero.
The divisor at this stage will be HCF of (c, d).
Every composite number expressed as a product or factors of
prime numbers and this factorization, is unique.
We nd LCM and HCF of two positive integers using the
fundamental theorem of arithmetic, which is also called Prime
Factorization method.
For any two positive integers c & d.
HCF (c, d) x LCM (c, d) = c x d 
Fundamental Theorem of Arithmetic
Step 1 :
Step 3 :
Step 2 :
c x d
LCM (c, d)
so, HCF (c, d) =
Gain an understanding of an important concept in number theory - The Euclid’s Division Lemma and learn its application in nding out the
Highest Common Factor (HCF).
2
2
2
5
2
2
2
5
Product of the common & uncommon prime factors involved
in the numbers.
It is a smallest common dividend for two or more divisors.
Example: LCM of (8, 20) = 2 x 2 x 2 x 5
= 40
Example: HCF of (8, 20) = 2 x 2
= 4
LCM
?
?
Product of the common prime factors involved in the
numbers.
It is a greatest common divisor for two or more dividends.
HCF
?
?
LCM and HCF are the two important concepts in Algebra. This chart will help you understand how to nd LCM and HCF of two or more than two
positive integers.
REAL NUMBERS 3
 
Introduction to 
Real numbers
Finding HCF using Euclid's
Division Lemma
Scan the QR Codes to watch our free videos
PLEASE KEEP IN MIND
 In a layman’s language, LCM of two or more than two
numbers means, the smallest number which is divisible by these two
numbers or more than two numbers.
 
 HCF of two or more than two numbers means, the highest
number which divides these two or more than two numbers.
 If the denominator of a rational number is in the form of 2
n
 or
5
m
 or 2
n
5
m
, Then the decimal expansion of that rational number is
terminating, or else it is non-terminating.
 Ex:        – terminating and       – non-terminating.
?
?
?
?
?
 If a number “n” is not a perfect square, then vn is an
irrational number.
 Arithmetic operations on rational  and irrational numbers 
 Rational + Rational = Rational
 Rational x Rational = Rational
 Irrational + Irrational = Irrational / Rational
 Irrational x Irrational = Irrational / Rational
 Irrational + Rational = Irrational
 Irrational x Rational = Irrational
 
2
5
2
7
.
.
.
.
.
.
REAL NUMBERS 4