Real Numbers
📌 Introduction
Real Numbers are the foundation of Mathematics. In this chapter, students learn about Euclid’s Division Lemma, Euclid’s Division Algorithm, Fundamental Theorem of Arithmetic, LCM and HCF, and irrational numbers. This chapter is frequently tested in CBSE board examinations.
🔹 Key Concepts Covered
- Euclid’s Division Lemma
- Euclid’s Division Algorithm
- Fundamental Theorem of Arithmetic
- Prime Factorisation
- LCM and HCF
- Irrational Numbers
- Decimal Expansion of Rational Numbers
🔹 Euclid’s Division Lemma
Statement:
For any two positive integers a and b, there exist unique integers q and r such that: a=bq+r,  0≤r<ba = bq + r,\; 0 \le r < ba=bq+r,0≤r<b
Where:
- a = dividend
- b = divisor
- q = quotient
- r = remainder
🔹 Euclid’s Division Algorithm
It is a systematic method to find the HCF of two positive integers.
Steps:
- Apply Euclid’s division lemma.
- Replace the divisor with the remainder.
- Repeat until remainder becomes zero.
- The last divisor is the HCF.
🔹 Fundamental Theorem of Arithmetic
Statement:
Every composite number can be expressed as a product of primes, and this factorisation is unique (apart from the order).
Example: 180=22×32×5180 = 2^2 \times 3^2 \times 5180=22×32×5
🔹 LCM and HCF
LCM×HCF=Product of two numbers\text{LCM} \times \text{HCF} = \text{Product of two numbers}LCM×HCF=Product of two numbers
This relation is important for board exams.
🔹 Irrational Numbers
Numbers which cannot be expressed in the form pq\frac{p}{q}qp​ where p and q are integers and q ≠0.
Examples: 2,3,π\sqrt{2}, \sqrt{3}, \pi2​,3​,π
🔹 Decimal Expansion of Rational Numbers
- Terminating decimals: When denominator has only prime factors 2 and/or 5.
- Non-terminating recurring decimals: Otherwise.
🔹 NCERT Solutions – Exercise Wise
Exercise 1.1
- Questions based on Euclid’s Division Algorithm
- Finding HCF of given numbers
Exercise 1.2
- Problems based on Fundamental Theorem of Arithmetic
- Finding LCM and HCF using prime factorisation
Exercise 1.3
- Irrational numbers
- Decimal expansion questions
(You can create separate pages for each exercise later)
🔹 Important Questions (Board Focused)
1. Find the HCF of 135 and 225 using Euclid’s Division Algorithm.
Answer: 45
2. Prove that √5 is irrational.
(Very important proof question for CBSE)
3. Find the LCM and HCF of 306 and 657 using prime factorisation.
🔹 Solved Example
Question:
Use Euclid’s Division Algorithm to find the HCF of 867 and 255.
Solution:
867 = 255 × 3 + 102
255 = 102 × 2 + 51
102 = 51 × 2 + 0
∴ HCF = 51
🔹 Practice Questions
- Find the HCF of 65 and 117.
- Express 140 as a product of its prime factors.
- State whether 0.375 is terminating or non-terminating.