वास्तविक संख्याएँ (Real Numbers) Notes…

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About these notes

Class 10 Math वास्तविक संख्याएँ (Real Numbers) notes give concise definitions, Euclid’s division lemma, Fundamental Theorem of Arithmetic, HCF (महत्तम समापवर्तक) and LCM (लघुत्तम समापवर्त्य) methods, decimal-expansion rules and worked examples for quick revision. Main sub-topics: Euclid’s division algorithm, unique prime factorisation (example 2³ × 3² × 5), HCF/LCM by prime powers, proofs of irrationality (√2, √3), terminating vs non‑terminating recurring decimals and real number line representation. For Class 10 CBSE and Bihar Board students preparing for school tests, board exams and practising with chapter test and mock tests on Merit Spring ROH.

Key points you will learn

Class 10 Math वास्तविक संख्याएँ (Real Numbers) notes
Euclid’s division lemma and use to compute HCF by repeated remainder method
Fundamental Theorem of Arithmetic: unique prime factorisation (e.g. 2³ × 3² × 5)
Find HCF and LCM via highest/lowest prime powers; for two numbers a × b = HCF × LCM
Proofs of irrationality (method used for √2, √3)
Decimal expansion rule: rational → terminating or recurring; denominator prime factors 2 and/or 5 give termination
Represent real numbers on number line and compare magnitudes; density of rationals and irrationals
Exam tip: convert to lowest terms before applying rules, and check factors when deciding decimal type

What is inside this chapter

Notes for this chapter will be added soon.

Questions students ask

Where to find Class 10 Real Numbers notes, important questions and MCQ?

Merit Spring ROH chapter page hosts concise notes, important questions, MCQ and a chapter test; use the site’s mock tests section to practise board-pattern questions and check preparation.

How to prove √2 is irrational?

Start by assuming √2 = p/q in lowest terms, square both sides to get 2q² = p², deduce p is even so q is even too, contradicting lowest-terms — hence √2 is irrational.

Difference between HCF and LCM?

HCF (महत्तम समापवर्तक) is the largest common divisor of integers, LCM (लघुत्तम समापवर्त्य) is the smallest common multiple; for two numbers a and b, a × b = HCF × LCM.

How to tell if a decimal will terminate or repeat?

A rational number p/q in lowest terms has a terminating decimal expansion exactly when q’s prime factors are only 2 and/or 5; otherwise the decimal is non‑terminating recurring.

How to use Euclid’s division lemma to find HCF?

Apply a = bq + r repeatedly: replace (a,b) with (b,r) each step until remainder 0; the last non‑zero remainder is the HCF.

More Math chapters

Real Numbers Polynomials Pair of Linear Equations in Two Variables Quadratic Equations Arithmetic Progressions Triangles All chapters →

Class 10 — other subjects

Science Social Science Computer ENGLISH HINDI SANSKRIT

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