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# 07 - Let's Recap - Real numbers - Class 10 - Maths Class 10 Notes | EduRev

## Class 10 : 07 - Let's Recap - Real numbers - Class 10 - Maths Class 10 Notes | EduRev

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Real Numbers

Real Numbers are combination of five different types of numbers as described in detail below.
 Natural Numbers, commonly referred to as counting numbers i.e. 1, 2, 3, 4, 5, 6, 7,
....etc.
 Whole Numbers are combination of natural numbers and zero (0) i.e. 0, 1, 2, 3, 4, 5, 6,
7, ....etc.
 Integers are combination of Natural and Whole Numbers and their negatives.
For e.g. …-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7... Continue infinitely in both
directions.
 Rational Numbers are made of ratios of integers and referred to as fractions. Rational
numbers take on the general form of a/b where a and b can be any integer except b ? 0.
Thus, the Rational numbers include all Integers, Whole numbers and Natural numbers.
 Irrational Numbers are made of special, unique numbers that cannot be represented as a
ratio of Integers. Examples of Irrational numbers include (3.14159265358979...) and the
square root of 2 (1.4142135623730950...).
Algorithm -An algorithm is a series of well defined steps which gives a procedure for solving a
type of problem.
Lemma - A lemma is a proven statement used for proving another statement.
Euclid’s Division lemma:
For any two given positive integers a and b there exist unique integers q and r such that
a=bq + r, where 0= r<b.
Here a, b, q and r are respectively called as dividend, divisor, quotient and remainder. Euclid’s
Division Lemma can be used to find H.C.F of two positive integers.
Euclid’s division Algorithm: It is a technique to compute Highest Common Factor(H.C.F) of
two given positive integers, consider c and d are two positive integers, with c > d. We use
following the steps to find H.C.F of c and d:
Step I: Apply Euclid’s division lemma, to c and d, so we find whole numbers, q and r such
that c =dq +r, 0=r<d
Step II: If r=0, d is the H.C.F of c and d. If r?0 apply division lemma to d and r

Page 2

Real Numbers

Real Numbers are combination of five different types of numbers as described in detail below.
 Natural Numbers, commonly referred to as counting numbers i.e. 1, 2, 3, 4, 5, 6, 7,
....etc.
 Whole Numbers are combination of natural numbers and zero (0) i.e. 0, 1, 2, 3, 4, 5, 6,
7, ....etc.
 Integers are combination of Natural and Whole Numbers and their negatives.
For e.g. …-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7... Continue infinitely in both
directions.
 Rational Numbers are made of ratios of integers and referred to as fractions. Rational
numbers take on the general form of a/b where a and b can be any integer except b ? 0.
Thus, the Rational numbers include all Integers, Whole numbers and Natural numbers.
 Irrational Numbers are made of special, unique numbers that cannot be represented as a
ratio of Integers. Examples of Irrational numbers include (3.14159265358979...) and the
square root of 2 (1.4142135623730950...).
Algorithm -An algorithm is a series of well defined steps which gives a procedure for solving a
type of problem.
Lemma - A lemma is a proven statement used for proving another statement.
Euclid’s Division lemma:
For any two given positive integers a and b there exist unique integers q and r such that
a=bq + r, where 0= r<b.
Here a, b, q and r are respectively called as dividend, divisor, quotient and remainder. Euclid’s
Division Lemma can be used to find H.C.F of two positive integers.
Euclid’s division Algorithm: It is a technique to compute Highest Common Factor(H.C.F) of
two given positive integers, consider c and d are two positive integers, with c > d. We use
following the steps to find H.C.F of c and d:
Step I: Apply Euclid’s division lemma, to c and d, so we find whole numbers, q and r such
that c =dq +r, 0=r<d
Step II: If r=0, d is the H.C.F of c and d. If r?0 apply division lemma to d and r

Real Numbers

Step III: Continue the process till the remainder is zero. The divisor at this stage will be the
required H.C.F
The Fundamental theorem of Arithmetic:
Every composite number can be expressed (factorised) as a product of primes, and this
factorization is unique, apart from the order in which the prime factors occur.
Ex. 28 = 2 x 2 x 7 ; 27 = 3 x 3 x 3
Theorem : Sum or difference of a rational and irrational number is irrational.
Theorem : The product and quotient of a non-zero rational and irrational number is irrational.
Theorem : If p is a prime and p divides a
2
, then p divides “a” where a is a positive integer.
Theorem : If p is a prime number then v is an irrational number.
Theorem:  Let x be a rational number whose decimal expansion terminates. Then x can be
expressed in the form of


where p and q are co-prime and the prime factorisation of q is the
form of 2
n
.5
m
where n, m are non negative integers.
Example: 	0.7=


=



×

=

		

Theorem: Let x =


be a rational number such that the prime factorisation of q is not of the form
of 2
n
.5
m
, where n, m are non negative integers. Then x has a decimal expansion which is non
terminating repeating (recurring).
Example:


=

	×
=1.1666……
Theorem: For any two positive integers p and q, HCF (p, q) x LCM (p, q) = p x q
Example: 4 & 6; HCF (4, 6) = 2, LCM (4, 6) = 12;  HCF x LCM = 2 x 12 =24
?	p x q = 24

Page 3

Real Numbers

Real Numbers are combination of five different types of numbers as described in detail below.
 Natural Numbers, commonly referred to as counting numbers i.e. 1, 2, 3, 4, 5, 6, 7,
....etc.
 Whole Numbers are combination of natural numbers and zero (0) i.e. 0, 1, 2, 3, 4, 5, 6,
7, ....etc.
 Integers are combination of Natural and Whole Numbers and their negatives.
For e.g. …-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7... Continue infinitely in both
directions.
 Rational Numbers are made of ratios of integers and referred to as fractions. Rational
numbers take on the general form of a/b where a and b can be any integer except b ? 0.
Thus, the Rational numbers include all Integers, Whole numbers and Natural numbers.
 Irrational Numbers are made of special, unique numbers that cannot be represented as a
ratio of Integers. Examples of Irrational numbers include (3.14159265358979...) and the
square root of 2 (1.4142135623730950...).
Algorithm -An algorithm is a series of well defined steps which gives a procedure for solving a
type of problem.
Lemma - A lemma is a proven statement used for proving another statement.
Euclid’s Division lemma:
For any two given positive integers a and b there exist unique integers q and r such that
a=bq + r, where 0= r<b.
Here a, b, q and r are respectively called as dividend, divisor, quotient and remainder. Euclid’s
Division Lemma can be used to find H.C.F of two positive integers.
Euclid’s division Algorithm: It is a technique to compute Highest Common Factor(H.C.F) of
two given positive integers, consider c and d are two positive integers, with c > d. We use
following the steps to find H.C.F of c and d:
Step I: Apply Euclid’s division lemma, to c and d, so we find whole numbers, q and r such
that c =dq +r, 0=r<d
Step II: If r=0, d is the H.C.F of c and d. If r?0 apply division lemma to d and r

Real Numbers

Step III: Continue the process till the remainder is zero. The divisor at this stage will be the
required H.C.F
The Fundamental theorem of Arithmetic:
Every composite number can be expressed (factorised) as a product of primes, and this
factorization is unique, apart from the order in which the prime factors occur.
Ex. 28 = 2 x 2 x 7 ; 27 = 3 x 3 x 3
Theorem : Sum or difference of a rational and irrational number is irrational.
Theorem : The product and quotient of a non-zero rational and irrational number is irrational.
Theorem : If p is a prime and p divides a
2
, then p divides “a” where a is a positive integer.
Theorem : If p is a prime number then v is an irrational number.
Theorem:  Let x be a rational number whose decimal expansion terminates. Then x can be
expressed in the form of


where p and q are co-prime and the prime factorisation of q is the
form of 2
n
.5
m
where n, m are non negative integers.
Example: 	0.7=


=



×

=

		

Theorem: Let x =


be a rational number such that the prime factorisation of q is not of the form
of 2
n
.5
m
, where n, m are non negative integers. Then x has a decimal expansion which is non
terminating repeating (recurring).
Example:


=

	×
=1.1666……
Theorem: For any two positive integers p and q, HCF (p, q) x LCM (p, q) = p x q
Example: 4 & 6; HCF (4, 6) = 2, LCM (4, 6) = 12;  HCF x LCM = 2 x 12 =24
?	p x q = 24

Real Numbers

Types of Decimals:
Decimals

Terminating                                    Non-terminating

Non-terminating repeating             Non-terminating non-repeating

Rational numbers                                               Irrational numbers

Real numbers

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## Mathematics (Maths) Class 10

51 videos|346 docs|103 tests

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