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Real Numbers Notes

Home / Class Notes / Class 10 / Math / Real Numbers
✓ Free notes Class 10 Math 1 study file

About these notes

Class 10 Math Real Numbers CBSE notes give concise explanations, worked steps and practice tips for Euclid’s division lemma, prime factorisation, Euclid’s algorithm, HCF (महत्तम समापवर्तक)–LCM (लघुत्तम समापवर्त्य) relations, proofs of irrationality and decimal classification. Main sub-topics: Euclid’s division lemma a = bq + r, prime-factor trees, HCF by Euclid’s algorithm, HCF×LCM relation, proving √2/√5 irrational, terminating vs recurring decimals and remainder-to-HCF problems. For Class 10 students preparing for CBSE term exams, school tests and chapter tests; use site notes plus separate important questions and MCQ practice on Merit Spring ROH.

Key points you will learn

  • Class 10 Math Real Numbers — Euclid’s division lemma: a = bq + r with 0 ≤ r < b and typical applications
  • Prime factorisation: write numbers as product of primes (e.g. 2³ × 3² × 5) and use common primes to find HCF
  • Euclid’s algorithm: repeated division replacing (a,b) by (b, remainder) until remainder 0; last non-zero remainder is HCF
  • HCF × LCM = product of two numbers (useful to get LCM once HCF known)
  • Proving irrationality by contradiction (assume √p = p/q in lowest terms → derive common prime factor) — method for √2, √5
  • Classify decimals: rational ↔ terminating or repeating; irrational ↔ non-terminating, non-repeating
  • Remainder problems: convert 'number leaves remainders r1,r2' to (number − r1), (number − r2) and take their HCF
  • Quick tip: check prime factors to decide last digit patterns — e.g. 15ⁿ always ends with 5 (no factor 2 ⇒ no factor 10)

What is inside this chapter

📄

Real Numbers

📥 Open notes

Questions students ask

Can 15ⁿ (n a natural number) end with the digit 0?

No. 15ⁿ = (3·5)ⁿ has no factor 2, so it cannot have factor 10; every power of 15 therefore ends with 5. Check prime factors: absence of 2 prevents trailing 0.

How do I find HCF using Euclid’s algorithm?

Use repeated division: divide larger by smaller, replace the pair by (smaller, remainder), and repeat until remainder is 0; the last non-zero remainder is the HCF. Write each division step clearly.

How to prove √5 is irrational?

Assume √5 = p/q in lowest terms. Then p² = 5q² so p divisible by 5; write p = 5k, substitute to get q² = 5k² so q divisible by 5, contradicting lowest terms.

Where can I get Class 10 Math Real Numbers notes and MCQ practice?

Find the chapter notes on the Merit Spring ROH chapter page; for extra practice use the site-wide chapter tests and mock tests for MCQ practice and timed revision to prepare for school and term exams.

What is the difference between rational and irrational numbers (decimal behaviour)?

Rational numbers have decimals that terminate or repeat (recurring); irrational numbers have non-terminating, non-repeating decimals. Use prime-factor test of denominator (after simplification) to decide if decimal terminates.

More Math chapters

Polynomials Pair of Linear Equations in Two Variables Quadratic Equations Arithmetic Progressions Triangles Coordinate Geometry All chapters →

Class 10 — other subjects

Science Social Science Computer ENGLISH HINDI SANSKRIT