Chapter 1 Real Numbers – Class 10 NCERT Concepts

Introduction:
Real Numbers are the fundamental building blocks of mathematics. In Class 10, students embark on a comprehensive journey into the realm of real numbers. This chapter unravels the essence of real numbers, encompassing positive and negative integers, fractions, and irrational numbers. Essentially, real numbers are the numeric language of the real world, used to quantify everything around us, from natural numbers counting objects to irrational numbers computing intricate square roots.

Real Numbers:
Real numbers encompass a wide spectrum of numeric entities, including positive integers like 1, negative integers such as -1, fractions like ½, decimals like 1.75, and even irrational numbers like √2. In essence, any number we encounter in the tangible world falls under the umbrella of real numbers.

Properties of Real Numbers:
Real numbers are a harmonious blend of rational and irrational numbers. They find their place on the number line, facilitating easy visualization and comparison. Moreover, real numbers obey various mathematical properties, including closure, commutative, associative, and distributive properties.

Examples:

• • Closure Property: If you add, subtract, multiply, or divide two real numbers, the result is always another real number. For example, 2 + 3 = 5, 4 – 1 = 3, 2 * 5 = 10, and 6 / 2 = 3.

• • Commutative Property: The commutative property states that for addition and multiplication, changing the order of the numbers does not affect the result. For example, a + b = b + a, and a * b = b * a.

• • Associative Property: The associative property tells us that how we group numbers in addition and multiplication does not alter the result. For example, (a + b) + c = a + (b + c), and (a * b) * c = a * (b * c).

• • Distributive Property: The distributive property allows us to multiply a number by a sum or difference, as in a * (b + c) = a * b + a * c.

Euclid’s Division Lemma:
Euclid’s Division Lemma is a fundamental concept that states for any two integers a and b, there exists a unique pair of integers q and r, satisfying a = b × q + r, where 0 ≤ r < b. This lemma forms the basis of the Euclidean algorithm used to find the Highest Common Factor (HCF).

b = Divisor

q = Quotient

r = Remainder

Example:
Let’s use Euclid’s Division Lemma to find the quotient and remainder when 23 is divided by 5.

• • a = 23

• • b = 5

• • Using the lemma: 23 = 5 × 4 + 3

• • Here, q = 4 (quotient) and r = 3 (remainder).

Euclid’s Division Algorithm:
The Euclidean Division Algorithm is a systematic approach for determining the HCF of two numbers, primarily when one number is greater than the other. By repeatedly applying Euclid’s Division Lemma, the HCF can be efficiently computed.

Example:
Find the HCF of 48 and 18 using the Euclidean Division Algorithm.

• • We start by applying the division lemma: 48 = 18 × 2 + 12

• • Next, we apply the division lemma again: 18 = 12 × 1 + 6

• • Once more: 12 = 6 × 2 + 0

• • When we reach a remainder of 0, the divisor at this step (6) is the HCF. So, HCF(48, 18) = 6.

Fundamental Theorem of Arithmetic:
The Fundamental Theorem of Arithmetic unveils the uniqueness of prime factorization. It asserts that every natural number can be expressed as a product of prime numbers, and this factorization is unique, regardless of the order of prime factors.

Example:
Consider the number 60. Its prime factorization is 2 × 2 × 3 × 5. According to the Fundamental Theorem of Arithmetic, this factorization is unique for 60.

Method of Finding LCM:
The Lowest Common Multiple (LCM) is the smallest multiple shared by two or more numbers. It can be determined by identifying common prime factors and their respective powers among the numbers.

Example:
Find the LCM of 12 and 18.

• • Prime factorization of 12: 2 × 2 × 3

• • Prime factorization of 18: 2 × 3 × 3

• • LCM is the product of common and uncommon prime factors raised to their highest powers: LCM(12, 18) = 2 × 2 × 3 × 3 = 36.

Method of Finding HCF:
The Highest Common Factor (HCF) is the largest number that divides two or more numbers without leaving a remainder. It can be found using either prime factorization or the Euclidean algorithm.

Example:
Find the HCF of 24 and 36 using the prime factorization method.

• • Prime factorization of 24: 2 × 2 × 2 × 3

• • Prime factorization of 36: 2 × 2 × 3 × 3

• • HCF is the product of common prime factors: HCF(24, 36) = 2 × 2 × 3 = 12.

Relationship between LCM and HCF:
The product of two numbers equals the product of their HCF and LCM. This relationship provides a useful tool for solving mathematical problems.

Example:
For two numbers, 12 and 18, with HCF = 6 and LCM = 36, this relationship holds true: 12 × 18 = 6 × 36.

Applications of HCF & LCM:
HCF and LCM have practical applications in various real-world scenarios, including synchronizing events like bells ringing at different frequencies or determining the meeting point of two individuals traveling at different speeds.

Revisiting Irrational Numbers:
Irrational numbers are those that cannot be expressed as a fraction (p/q) and possess non-repeating, non-terminating decimal expansions. Prominent examples include √2, π, and e.

Number Theory and Interesting Results:
Number theory uncovers intriguing results such as the divisibility of prime numbers and the behavior of rational and irrational numbers when operated upon.

Proof by Contradiction:
Proof by contradiction is a method used to validate statements. It involves assuming the opposite of the statement and demonstrating a contradiction to prove the original statement’s validity.

Revisiting Rational Numbers and Their Decimal Expansions:
Rational numbers can be expressed as fractions (p/q), while their decimal expansions can be either terminating or recurring. Determining the nature of these decimals depends on the prime factors of the denominator.

Terminating and Non-Terminating Decimals:
Decimals can either terminate at a certain point or continue infinitely without repetition. Non-terminating decimals can further be categorized into recurring and non-recurring decimals.

Example:

• • Terminating Decimal: 0.25 (¼ in fractional form)

• • Recurring Decimal: 0.333… (

1/3 in fractional form)

• • Non-Recurring Decimal: π ≈ 3.1415926535…

Real Numbers for Class 10 Solved Examples:
Several solved examples illustrate the application of these concepts, aiding in a deeper understanding of real numbers.

Conclusion:
Class 10’s exploration of real numbers equips students with foundational knowledge that extends beyond mathematics, enriching their ability to understand and interact with the real world. These concepts serve as the building blocks for advanced mathematical studies and problem-solving in everyday life.