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Class 10th Real Numbers Chp. 1 one shott 💥 Score 100/100 Markrs | By Indori Master


📘 CBSE Class 10 Maths Chapter 1 – Real Numbers (Detailed Description)
🧮 Chapter Summary:
The chapter “Real Numbers” revises and builds upon the number system introduced in earlier classes. It includes a deep dive into the Euclid’s Division Lemma, Fundamental Theorem of Arithmetic, and the properties of rational and irrational numbers, along with the behavior of decimal expansions.

   • Class 10th Real Numbers Chp. 1 one shott 💥...  

#realnumbers
#factorisation
#indore
#trending
#findthedifference
LCM & HCF
PRIME FACTORISATION


🧩 Topics Covered:
1. Euclid’s Division Lemma:
A method to find the HCF (Highest Common Factor) of two positive integers.

Lemma Statement:
For any two positive integers a and b, there exist unique integers q (quotient) and r
Used in: Efficient computation of HCF using the Euclidean algorithm.

2. The Fundamental Theorem of Arithmetic:
States that every composite number can be uniquely expressed as a product of prime numbers, apart from the order.



Finding LCM and HCF of numbers using prime factorization.

Solving problems related to number patterns.

3. Revisiting Irrational Numbers:

Proofs of irrationalit
The method involves contradiction using prime factorization.

Important: Sum or product of a rational and irrational number is usually irrational.

4. Rational Numbers and Their Decimal Expansions:
A rational number is a number that can be expressed as
𝑝
𝑞
q
p
​
.

Decimal expansion of rational numbers:

Terminating: Ends after a few digits (e.g., 0.75)

Non-Terminating Recurring: Never ends, but repeats in a pattern (e.g., 0.666…)

Condition:

If the denominator (after simplifying) is of the form
2
𝑛
×
5
𝑚
2
n
×5
m
, it’s terminating.

Otherwise, it’s non-terminating repeating.

5. HCF and LCM Using Prime Factorization:
To find HCF and LCM:

HCF: Take the lowest powers of common prime factors.

LCM: Take the highest powers of all prime factors.

Relation:

HCF
×
LCM
=
Product of the two numbers
HCF×LCM=Product of the two numbers
🧠 Important Concepts & Tricks:
Concept Key Point
Euclid’s Lemma Used for finding HCF efficiently
Fundamental Theorem Every number = unique product of primes
Irrational Numbers Cannot be written as
𝑝
𝑞
q
p
​

Decimal Expansion Rule Depends on factors of denominator (2 & 5)
HCF × LCM Rule Useful in problem-solving and competitive exams

🔎 Common Questions Asked in Exams:
Prove that
2
2
​
is irrational.

Use Euclid’s Lemma to find HCF of two numbers.

Express a number as a product of prime factors.

Determine whether the decimal expansion of a number is terminating or non-terminating.

🎯 Real-Life Applications:
Scheduling (using HCF & LCM)

Measurements and unit conversions

Repeating decimals in computer systems

Cryptography and coding (Prime Numbers)

📌 Conclusion:
Chapter 1 lays the foundation for various mathematical concepts that are not only useful in academics but also in real-world applications. Understanding the properties of numbers helps students to grasp more complex topics in algebra, arithmetic, and number theory.