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L-05, Ch-1st,

📘 Class 10 Maths Chapter 1 – Real Numbers | Full Chapter Summary In this chapter, “Real Numbers”, we study the classification of numbers, prime factorization, irrational numbers, and decimal expansions. This chapter strengthens the number system foundation and develops logical reasoning used in higher mathematics. 🔹 1. Revisiting Real Numbers Real numbers include all numbers on the number line: Natural numbers Whole numbers Integers Rational numbers Irrational numbers Together, rational and irrational numbers form the set of real numbers. 🔹 2. Fundamental Theorem of Arithmetic Statement: Every composite number can be expressed as a product of prime numbers, and this factorization is unique, apart from the order of factors. Example: This theorem is the basis for prime factorization and solving number problems. 🔹 3. Relationship Between HCF and LCM For any two positive integers a and b: This relation helps verify calculations and solve application problems. 🔹 4. Irrational Numbers and Proofs An irrational number cannot be written in the form Important result: 👉 √2 is irrational Proof idea (contradiction method): Assume √2 is rational → √2 = p/q (in simplest form) After squaring: 2q² = p² This implies p is even → p = 2k Substituting again leads to q also being even This contradicts the assumption that p/q is in lowest form. Therefore, √2 is irrational. Similar proofs apply to √3, √5, etc. 🔹 5. Decimal Expansion of Rational Numbers A rational number � (q ≠ 0) has: ✅ Terminating decimal expansion if q = � ✅ Non-terminating recurring decimal if q contains primes other than 2 or 5 Examples: 1/8 → terminating 1/3 → recurring This explains how decimal behavior depends on prime factors of the denominator. 🎯 Key Learning Outcomes Understanding rational and irrational numbers Proving numbers are irrational Applying the Fundamental Theorem of Arithmetic Using HCF–LCM relationship Identifying decimal expansion types Strengthening number system concepts 📥 Like, Share & Subscribe to @motion_maths for more NCERT Class 10 Maths summaries, solved examples, and concept explanations! 📌 Hashtags #Class10Maths #RealNumbers #IrrationalNumbers #NCERTMaths #CBSEClass10 #FundamentalTheoremOfArithmetic #HCFandLCM #MotionMaths #MathsWithSahil #MathsChapter1 #NCERTSolutions #MathsRevision #LearnMaths #BoardExamPreparation #Class10FullChapter #NumberSystem #StudyWithMe #EduYouTube