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Real numbers Class Xth, Cheat Sheet of Mathematics

Mathematics

Master Real Numbers in Minutes — Not Hours! Struggling with concepts like Euclid’s Division Lemma, HCF & LCM, or irrational numbers? This Class 10 Real Numbers Cheat Sheet is designed to make everything crystal clear at a glance. Key formulas simplified Step-by-step methods you won’t forget Smart shortcuts to solve questions faster Clean, easy-to-revise layout (perfect before exams!) No unnecessary theory — just what you actually need to score better and understand deeply. Whether you're revising last minute or building strong basics, this cheat sheet turns confusion into confidence. Study smart. Revise faster. Score higher.

Typology: Cheat Sheet

2025/2026

Uploaded on 03/30/2026

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Chapter 1: Real Numbers (Class 10) 1. Definition of Real Numbers (Real Numbers are the union of all Rational & Irrational numbers) * Rational Rational (Q): Can be written as & q (9#0) | e.g., 2,75, +, 0.4 * Trrational: Cannot be written as 2 (e.g. V2,V3,0) 7 2. Euclid’s Division Lemma _ For any two positive integers a & b, there _ exist unique integers q & r satisfying: a=bqtr where O 2b*= 0a’. Step 3: This implies a? is divisible by 2. (Therefore, a is also divisible by 2). Step 4: Let a = 2k (for some integer b. Substituting in 2b? =a’, we get 2b? = (2k)*= 4k? > b*= 2k’. Step 5: This implies b? is divisible by 2. (Therefore, b is also divisible by 2). Step 6: So, both a and b have 2 as a common factor. Step 7: This contradicts the assumption that a, b are co-primes. Therefore, our initial assumption is wrong. ¥2 is Irrational. Decimal Expansion of Rational Numbers Let x =4 (Rational). Factorize q into primes. Case 1: Terminating Decimals (if q=2"5™) Case 2: Non-terminating Repeating (if q#2°5™)