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Class 10 Mathematics Revision Notes for Real Numbers of Chapter 1
Mathematics is considered a scoring subject amongst all the other subjects in Class 10 board examinations. All the topics in the subject are designed according to the official CBSE Syllabus given by NCERT. The subject contains different numerical equations and Formulas that students should practise regularly to score well in the examination.
In order to make this part easy, Extramarks has provided Class 10 Mathematics Chapter 1 Notes on its official website. It contains CBSE sample papers, important questions, etc., to prepare for the board examination. Candidates studying in Class 10 should refer to the NCERT books to study subjects such as Mathematics, Science, etc.
Class 10 Mathematics Revision Notes for Real Numbers of Chapter 1
Access Class 10 Mathematics Chapter 1 – Real Numbers Notes
Real Numbers
The combination of rational and irrational numbers in the number system is known as real numbers. All the numerical operations such as addition, subtraction, etc., can be performed on them. In addition, these numbers can also be represented on the number line. Because complex numbers cannot be expressed on a number line, they are referred to as imaginary numbers.
Examples that can help you understand the real numbers are 23, 13, ⅚, etc.
There are four main properties of real numbers. They are as follows:
 Identity property
 Distributive property
 Associative property
 Commutative property
Euclid’s Division Lemma
Theorem: For given positive integers a and b, there exist unique integers q and r satisfying a = bq + r where 0 ≤ r < b.
 Algorithm
It is defined as the series of steps that provide a procedure to solve a particular type of problem.
 Lemma
It is a proven statement which is used to prove another statement.
Euclid’s Division Algorithm
One can obtain the Highest Common Factor (HCF) of two particular positive integers, let’s say c and d, with c > d by following the steps given below:
 Step 1: Firstly, apply Euclid’s Division Lemma to c and d. Then, we will find the whole numbers q and r such that c = dq + r, 0 ≤ r < d.
 Step 2: Now, if r = 0, the d is the HCF of c and d. If r is not equal to 0, then one has to apply the division lemma to d and r.
 Step 3: After that, continue the process until the remainder is zero. The division obtained at this stage will be the required HCF.
Important points:
 Euclid’s Division Lemma and algorithm are closely related to each other. Sometimes people call the former the division algorithm as well.
 Euclid’s Division Algorithm is mostly used for positive integers, but it can also be extended to all integers except zero.
The Fundamental Theorem of Arithmetic
Theorem
Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors are present.
Finding Least Common Multiple (LCM)
One can find the LCM by studying the following example properly. It is as follows:
To find the LCM of 36 and 56,
 Step 1: 36 = 2x2x3x3
56 = 2x2x2x7
 Step 2: The common factors are 2×2
 Step 3: The uncommon prime factors are 3×3 for 36 and 2×7 for 56
 Step 4: So the LCM of 36 and 56 is 2x2x3x3x2x7 = 504
Finding HCF
One can find HCF (Highest Common Factor) by two methods. They are Prime factorisation and Euclid’s Division Algorithm. We will discuss Prime factorisation in detail here.
 Prime factorisation
For given two numbers, we can express them as products of their respective prime factors. Now select the prime factors that are common to both the numbers.
Example:
To find the HCF of 20 and 24
20 = 2x2x5 and 24 = 2x2x2x3
The common factor for both is 2×2, which is equal to 4. So the HCF of 20 and 24 is 4.
Rational and Irrational Numbers
 Rational numbers
The group of numbers that can be expressed as positive integers, negative integers, fractions, and zero are known as rational numbers.
They can be written as p/q, where q is not equal to 0.
Example: 3/2 is a rational number.
 Irrational numbers
The group of numbers that cannot be expressed as the fractions or ratios of two particular numbers are known as irrational numbers.
Example: √8 = 2.828…
Properties of rational and irrational numbers
Various types of properties are associated with rational and irrational numbers. They are as follows:
 The product of two irrational numbers is not always irrational.
 The sum of two particular irrational numbers is not always irrational.
 The product of two rational numbers is rational.
 The sum of two rational numbers will always be a rational number.
Number theory
Now let us discuss the number theory in detail. It is as follows:
 If p is a prime number which divides a2, then p divides a also.
Example: If 3 divides 62, then the result will be 36, showing that 3 divides 6.
 The sum or difference of a specific rational and irrational number is always irrational.
 The product and quotient of a nonzero rational and irrational number will be irrational.
 √p is irrational when p is a prime number.
Decimal Expansions of Rational Numbers
One can understand the concept of decimal expansions of rational numbers through various theorems. So let’s start!
Theorem 1:
Let’s assume x is a rational number whose decimal expansion terminates. Then x can be expressed as p/q, where p and q are coprime numbers. In addition, the prime factorisation of q is of form 2n 5m where m and n are nonnegative integers.
Theorem 2:
If x = p/q is a rational number in a way that the prime factorisation of q is in the form of 2n 5m where m and n are nonnegative integers, then x will have a decimal expansion that terminates.
Theorem 3:
Let x = p/q, where both are coprime numbers, be a rational number. Now the prime factorisation of q is not in the form of 2n 5m where m and n are nonnegative integers, then x will have a decimal expansion, which is nonterminating repeating [recurring].
From the theorem mentioned above, you can easily observe that the decimal expansion of every rational number will be either terminating or nonterminating repeating.
Introduction to Real Numbers – Repeated
Real Numbers – Repeated
Euclid’s Division Lemma – Repeated
Euclid’s Division Algorithm – Repeated
The Fundamental Theorem of Arithmetic – Repeated
Prime Factorisation
The process of finding the prime factors of a number is known as prime factorisation.
Example:
 Prime factorisation of 12 is 2x2x3 = 22 x 3
 Prime factorisation of 18 is 2x3x3 = 2 x 32
Methods for the Prime Factorisation
The two methods to find the prime factors are as follows:
 Division Method
 Factor Tree Method
Division Method
One can follow the steps given below to find the prime factors of a number using the division method:
 Step 1: You should divide the given number by the smallest prime number.
 Step 2: Then, divide the quotient by the smallest prime number.
 Step 3: Now repeat the process until the quotient becomes 1.
 Step 4: Once the process is complete, multiply all the prime factors.
Example
Let us take the example of the number 460.
 Step 1: Divide the number by the least prime number, which is 2. The answer will be 230.
 Step 2: Now, divide 230 by 2 again. The answer will be 115.
 Step 3: Divide 115 by the least prime number, which is 5.
 Step 4: After dividing, you will get 23 as the prime number. Now by dividing the number by itself, you will get 1.
The prime factors of 460 will be 22 x 5 x 23
Factor Tree Method
Now let us discuss the second method, known as the factor tree method.
 Step 1: Start by considering the given number as the root of the tree.
 Step 2: Now, write down the factors as branches of the tree.
 Step 3: Again, factorise the composite factors and write the factors as branches of the tree.
 Step 4: Now repeat the step until you find all composite factors.
Fundamental Theorem of Arithmetic – Repeated
Method of Finding LCM – Repeated
Method of Finding HCF – Repeated
Applications of HCF & LCM in RealWorld Problems
You can use HCF in the following areas:
 To arrange different objects into groups and rows.
 To count how many people we can invite.
 To evenly divide any assortment of items into their largest grouping.
 To split different things into smaller sections.
You can use LCM in the following ways:
 To analyse the recurrence of an event.
 To purchase several items in order to have enough.
 About an event that is or will keep happening repeatedly.
Revisiting Irrational Numbers
A number which cannot be expressed in p/q form, where p and q are integers and q is not equal to zero, is known as an irrational number.
Examples: √2, π, etc.
Theorem 1:
Let us consider p as a prime number. So if p divides a2, then p divides a, where a is a positive integer.
Theorem 2:
Prove √2 is irrational.
Let’s assume √2 is a rational number. So, we can find the integers r and s where s is not equal to 0. It can also be expressed as √2 = r/s.
Now, if s and r have a common factor of 1, then we will divide the common factor to get √2 = a/b, where a and b are coprime numbers.
So, b√2 = a.
Squaring both sides and rearranging them will get 2b2 = a2. So 2 divides a2.
Now it follows that 2 divides a.
So, we can write a = 2c for some integer c.
Substituting for a, we get 2b2 = 4c2 , that is, b2 = 2c2 .
As a result, 2 divides b2, and hence 2 divides b.
After this, we get 2 as a common factor.
But this will contradict the fact that a and b have no common factors other than 1. The contradiction arises due to our incorrect assumption that √2 is rational.
Hence, √2 is irrational.
Irrational Numbers – Repeated
Proof by Contradiction
A contradiction arises when we get a statement p, such that it is true and it’s negative p is also true. We can understand the concept of contradiction through an example. So let us start.
Let’s consider we have two statements, p and q.
Now, statement p: x = a/b, where a and b are coprime numbers.
Statement q: 2 divides both a and b.
In this case, we assume that the statement p is true and also manage to show that q is also true. Therefore, we get a contradiction as statement q implies that the negation of statement p is true.
Revisiting Rational Numbers and Their Decimal Expansions – Repeated
Rational Numbers – Repeated
Terminating and Nonterminating Decimals
The decimals that end at a certain point are known as terminating decimals. Example 0.2, 2.56, etc.
Nonterminating decimals are those which do not have an end. Examples are 0.3333333………….., 0.123897078………….., etc.
Nonterminating decimals can be of two types:
Recurring: The numbers in which the same part of the decimals keeps on repeating. Example: 0.142857142857….
NonRecurring: The numbers in which no repetition of the decimal takes place are known as nonrecurring.
Example: 3.1415926535…
Benefits of Notes of Real Numbers Class 10
The benefits of the Class 10 Mathematics Chapter 1 Notes are as follows:
 The Class 10 Chapter 1 Mathematics Notes are created in a way that students can easily understand the concepts in a short time.
 Students can increase their problemsolving skills by reading Chapter 1 Mathematics Class 10 Notes.
 From Class 10 Mathematics Notes Chapter 1, students will be able to study the important questions and score topics, which will help them score good marks.
 All the notes are created by using the material given in NCERT books that are prescribed by CBSE. So students can rely on them to prepare for the board examinations.
 Students can easily learn to use the formulas to solve numerical problems by following the notes.
 Extramarks recommends that students read the notes before the examination, as it will help them properly revise the chapter.
General Tips
While revising the chapters in Mathematics, students should read every point carefully. They can also refer to Class 10 Mathematics Chapter 1 Notes to learn the concepts in depth. Candidates studying in Class 10 should frequently read and revise the real numbers because the concepts given in this chapter may be covered again in further classes. They should consider Extramarks as their study partner and refer to all the study material, revision notes, etc., provided.
Conclusion
The CBSE Chapter 1 Mathematics Notes are designed according to the CBSE syllabus that candidates can refer to for preparing for board examinations. Students should thoroughly practise all the formulas, important questions, etc., to score well in the examination. All the necessary materials for the preparation are available on the website of Extramarks which candidates can refer to study.
FAQs (Frequently Asked Questions)
1. How can I get good grades in Class 10 Mathematics exam?
Students should practise all the mathematical numerical and formulas to score well. They should also focus on CBSE important questions and important topics from the syllabus to score well in the examination.
2. Should I refer to the past years’ question papers?
Refer to the CBSE sample papers when preparing for the Class 10 board examinations. It will give you an idea of the important topics and questions asked in the examination. You can also study past years’ CBSE question papers, which contain a wealth of information about the structure and type of questions.