Page 1: Introduction
Real numbers consist of rational and irrational numbers.
This chapter revisits Euclid’s division algorithm, HCF/LCM, fundamental theorem of arithmetic, and proves irrationality of numbers like √2, √3, √5.
Important for board exams — many proof-based and numerical questions.
Page 2: Euclid’s Division Lemma
a = bq + r, where 0 ≤ r < b
Used to find HCF.
Page 3: Euclid’s Division Algorithm
To find HCF of two numbers:
- Apply lemma to a and b → a = bq + r
- Now apply to b and r
- Continue until r = 0
- Last non-zero remainder = HCF
Page 4: HCF and LCM Relation
Very useful in problems.
Page 5: Fundamental Theorem of Arithmetic
Prime factorisation is unique.
Page 6: Revisiting Rational and Irrational Numbers
Rational: p/q form (q ≠ 0)
Irrational: not p/q
Decimal expansion: terminating/repeating = rational; non-terminating non-repeating = irrational
Page 7: Proof that √2 is Irrational
2q² = p² → p even → p = 2k
2q² = 4k² → q² = 2k² → q even
Both even → contradiction.
Hence √2 irrational.
Page 8: Proof that √3 is Irrational
Page 9: Proof that √5 is Irrational
5q² = p² → p divisible by 5 → q also → contradiction.
Page 10: General Proof for Non-Square Primes
If p prime and not square, √p irrational.
Important pattern for exams.
Page 11: Decimal Expansion Rules
- Terminating: denominator factors 2 or 5 only
- Non-terminating repeating: other factors
Example: 1/8 = 0.125 (terminating), 1/6 = 0.1666... (repeating)
Page 12: Key Theorems Summary
- Euclid’s lemma and algorithm
- HCF × LCM = product
- Fundamental theorem of arithmetic
- Irrationality proofs
Page 13: Practice Questions - Easy (1-10)
- State Euclid’s lemma.
- HCF(96,404) using algorithm.
- LCM(12,15) using HCF.
- Prime factorisation of 96.
- Is 1/7 terminating?
- Why √2 irrational?
- Denominator for terminating decimal.
- HCF × LCM formula.
- Unique prime factorisation.
- r in Euclid’s lemma.
Page 14: Practice Questions - Medium (11-20)
- Find HCF and LCM of 26 and 91, verify product.
- Prove √3 irrational.
- Show 0.375 terminating.
- HCF of 135 and 225 using algorithm.
- Prove √5 irrational.
- LCM of three numbers using prime factorisation.
- Decimal of 1/20.
- Why unique factorisation?
- Find HCF(96,404) step by step.
- Prove 3 + √2 irrational.
Page 15: Practice Questions - Hard (21-30)
- Prove √p irrational for prime p.
- Show rational + irrational = irrational.
- Find LCM of 12,15,21.
- Prove 0.001001001... irrational.
- HCF of three numbers.
- Prove √2 + √3 irrational.
- Decimal expansion type for 1/11.
- Full proof √2 irrational with explanation.
- Application of Euclid’s algorithm.
- Prove HCF × LCM = product using factorisation.
Page 16: Important Proofs Recap
All irrationality proofs step by step.
Page 17: Common Mistakes
- Wrong assumption in proof
- Forgetting contradiction
- Wrong decimal type
- Missing steps in algorithm
- Confusing HCF and LCM
Page 18: Previous Year Questions
HCF/LCM, irrationality proof, decimal type.
Page 19: Exam Tips
- Show all steps in algorithm
- Write assumption clearly in proof
- Use contradiction method properly
- Verify HCF × LCM
- Memorise irrational proofs
Page 20: Quick Revision Sheet
All theorems, formulas, proofs outline.
Page 21: Final Motivation
Chapter 1 complete! Real Numbers is foundational for Class 10.
Master proofs — they are direct questions.
Class 10 Maths started strong 🦖
Page 22: Euclid’s Algorithm Steps
Detailed examples.
Page 23: Irrationality Proofs
All common ones.
Page 24: Decimal Expansion Rules
Examples and conditions.
Page 25: Prime Factorisation Table
Common numbers.
Page 26: Extra Numericals
HCF, LCM, proofs.
Page 27: Thank You & Copyright
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