CBSE Notes Class 10 Maths Chapter 1 Real Numbers

The CBSE Notes prepared for the CBSE Class 10 Maths, which are in line with the NCERT curriculum, ensure that students acquire the fundamental concepts that are then expanded upon to become more detailed explanations. Chapter 1 deals entirely with "Real Numbers," which is a vital topic for Class 10 that introduces basic ideas on numbers, their properties, and applications. This chapter provides crucial information on the ideas of divisibility, the Euclidean division method, and rational and irrational numbers.

1.0Download CBSE Notes for Class 10 Maths Chapter 1 Real Numbers - Free PDF!!

Students can download free PDF of CBSE Notes for Class 10 the first chapter of Maths, Real Numbers. This resource is designed for easy comprehension of the concepts and can be accessed from any location.

2.0CBSE Class 10 Math Chapter 1 Real Numbers - Revision Notes

Students' understanding of numbers and their types is expanded in Class 10 Math Chapter 1 as the real number system is further developed for them. Here is a broad summary of the main concepts covered in this chapter:

1. Important Concepts in Real Numbers

• Natural Numbers and Whole Numbers: Understanding the sequence of natural numbers (1, 2, 3…) & the inclusion of zero to form whole numbers.
• Integers and Rational Numbers: Entering into understanding positive and negative integers and that rational numbers can also be represented as fractions.
• Irrational Numbers: A number that cannot be written as a simple fraction; an example is the square root of 2 and pi (π). The decimal representations of such numbers continue infinitely in a non-repeating pattern.
• Prime Factorisation: Using the fundamental theorem of arithmetic, every composite number can be expressed as a unique product of primes, except for the order of factors.
• Divisibility and Euclidean Division: Introduction to divisibility rules and Euclidean division lemma for solving division problems effectively.

2. Definitions

• Real Numbers: All the numbers on a number line, both rational and irrational.
• Rational Numbers: Numbers that can be written in the form p/q, where p and q are integers, and q ≠ 0.
• Irrational Numbers: These numbers can't be written as a simple fraction and contain either non-recurring or non-terminating decimal expansions.
• Euclidean Algorithm: This is the method to find out the greatest common divisor, or GCD, of two numbers on the basis of division.

3. Formulas

Some key formulas in Maths regarding the set of real numbers are as follows:

• Prime Factorisation: Every integer greater than one is either a prime number or may be uniquely expressed as a product of prime numbers.
• Euclidean Division Algorithm: If a and b are positive integers, there exist unique integers q & r such that

$\mathbf{a}=\mathbf{bq}+\mathbf{r}\text{ where }0\le \mathbf{r}<\mathbf{b}$

• LCM and HCF Calculation:  Prime factorisation may be used to find out the Least Common Multiple (LCM) & the Highest Common Factor (HCF) of two numbers. $2^{\mathrm{m}}\times 5^{\mathrm{n}}$
• Expressing Rational Numbers as Decimals: Rational numbers with terminating decimals have denominators that can be expressed in form

4. Tips and Tricks to Remember Concepts

• Prime Factorisation Shortcut: Start with smaller numbers and practice breaking down numbers into prime factors.
• Rational vs. Irrational: Remember, if a decimal repeats or terminates, it's rational. If it goes on without a pattern, it's irrational.
• Euclidean Algorithm: Use division repeatedly until you reach zero to find the GCD.

3.0CBSE Class 10 Math Chapter 1 Real Numbers - Key Notes

Rational Numbers

• The numbers of the form p/q, where p and q are integers and q ≠ 0, are known as rational numbers. Examples:  4/7, 3/2, 5/8, 0/1, −2/3. The set of all rational numbers is denoted by Q.
• Every rational number can be expressed in decimal form in terminating decimals or non-terminating repeating decimals. Examples: 1/5 = 0.2, 1/3 = 0.333…, 22/7 = 3.1428714287

Irrational Numbers

Those numbers which when expressed in decimal form are neither terminating nor repeating are known as irrational numbers. Examples: √2, √3, √5, π

Euclid’s Division Lemma

Euclid’s division lemma states that: “For any two positive integers a and b, there exist unique integers q and r such that” a = bq + r, where 0 ≤ r < b

Euclid’s Division Algorithm

• The Euclid’s Algorithm is an efficient method for computing the greatest common divisor (GCD) or highest common factor (HCF).
• To obtain the HCF of two positive integers, say c and d, where c > d, follow the steps below:

Step 1: Apply Euclid’s division lemma to c and d.
Find whole numbers q and r such that c = dq + r, where 0 ≤ r < d.

Step 2:
If r = 0, then d is the HCF of c and d.
If r ≠ 0, apply the division lemma to d and r.

Step 3: Continue the process until the remainder becomes zero. The divisor at that stage will be the required HCF.

Fundamental Theorem of Arithmetic

“Every composite number can be expressed as a product of primes, and their decomposition is unique apart from the order in which the prime factors occur.”

Example: 12600 = 2³ × 3² × 5² × 7

HCF and LCM of Numbers

• The HCF of two or more numbers is the greatest number that divides each of them exactly.
• The least number which is exactly divisible by each one of the given numbers is called LCM.
• HCF and LCM of given numbers by Prime Factorization Method

HCF = Product of least powers of common factors

LCM = Product of highest powers of all the factors

Important Relation

Product of two numbers = HCF × LCM

$\mathrm{LCM}(p,q,r)=\frac{p\cdot q\cdot r\cdot \mathrm{HCF}(p,q,r)}{\mathrm{HCF}(p,q)\cdot \mathrm{HCF}(q,r)\cdot \mathrm{HCF}(p,r)}\text{and}\mathrm{HCF}(p,q,r)=\frac{p\cdot q\cdot r\cdot \mathrm{LCM}(p,q,r)}{\mathrm{LCM}(p,q)\cdot \mathrm{LCM}(q,r)\cdot \mathrm{LCM}(p,r)}$

Revisiting Irrational Numbers

• An irrational number is a real number that cannot be expressed as the ratio of two integers. Examples:  √2, √3, √15, π, e
• The decimal representation of irrational numbers is non-terminating and non-repeating. Example: 0.101001000100001…
• If p is a prime number and p divides a², then p also divides a, where a is a positive integer.

Revisiting Rational Numbers and Their Decimal Expansion

• A rational number is any number that can be expressed as the quotient a/b of two integers, where b ≠ 0.
• The decimal representation of a rational number is either Terminating or Non-terminating but repeating. Examples: 3/7, 2, 0, −5, 2.6, 2.7777
• If x = p/q and the prime factorisation of q is of the form 2ᵐ × 5ⁿ, where m, n are non-negative integers, then x has a terminating decimal expansion.
• If x = p/q be a rational number, such that the the prime factorisation of q is not of the form 2ᵐ × 5ⁿ, where m and n are non-negative integers then x has a non-terminating repeating decimal expansion.

4.0Key Features of CBSE Maths Notes for Class 10 Chapter 1

1. Comprehensive Coverage of Concepts: The CBSE Maths notes explain full concepts related to Real Numbers so that each topic has been covered in depth by the students. These notes cover definitions, properties, and practical examples to solidify understanding.
2. Solved Problems and Practical Examples: The examples that have been worked out give students an opportunity to see how each concept applies in practice. These examples will help show how problems that can arise in the exam should be approached and solved.
3. Exam-Oriented Approach: Important ideas and formulae to memorise for the tests are the main focus of the notes. According to this perspective, the same is outlined to assist students in highlighting the most crucial elements for achieving high scores. A few key pointers are underlined to facilitate rapid modifications.
4. Tables for Quick Revision: Important properties, formulas, and definitions are provided in tables that you can turn to for quick reference. This proves very handy for last-minute revisions in preparation for the exams.
5. Easy-to-Understand Language: Concepts are broken down into simple language so that  students are able to understand and remember.
6. Visual Aids and Illustrations: Visual aids like number lines and diagrams are used as much as possible to assist students in understanding the concept, especially when it comes to subjects like integers and rational numbers.

These CBSE Notes for Class 10 Maths are perfect for assisting students with regular practice, last-minute review, and in-depth comprehension.