The World of Numbers Notes

Q: How to prove √2 is irrational?

Assume √2 = p/q in lowest terms, square both sides to get 2q² = p² and conclude p is even, so p = 2k. Substituting shows q is also even, contradicting lowest terms; hence √2 is irrational.

Q: What is Euclid's division lemma and how to use it to find HCF?

Euclid's lemma states a = bq + r with 0 ≤ r < b. To find HCF, divide larger by smaller, replace (a,b) with (b,r) and repeat; the last nonzero remainder is the HCF.

Q: Difference between rational and irrational numbers?

Rational numbers can be written as p/q (q ≠ 0) and have terminating or repeating decimal expansions; irrational numbers cannot be written as p/q and have non‑terminating, non‑repeating decimals.

Q: How to represent √5 on the number line?

Construct a right triangle with legs 1 and 2 so the hypotenuse equals √5, use a compass to transfer that length from the origin along the ray, and mark the point on the number line.

Q: Where to find class 9 mathematics The World of Numbers notes, important questions and MCQ?

Search Merit Spring ROH for Class 9 Mathematics The World of Numbers NCERT notes; use the site search to locate important questions and MCQ practice, and try the chapter tests and mock tests available on the platform.

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About these notes

Class 9 Mathematics The World of Numbers NCERT — concise notes that explain number systems, Euclid's division lemma, HCF–LCM methods and standard irrationality proofs. Main sub-topics: natural/whole/integers/rational/irrational; prime (अभाज्य संख्या) and composite (भाज्य संख्या); Euclid's lemma a = bq + r and HCF by repeated division; Fundamental Theorem of Arithmetic and prime factorisation (example: 2³ × 3² × 5); decimal expansions, rationalisation of surds and placing real numbers on the number line. For Class 9 CBSE/Bihar Board students preparing for school tests, chapter tests, quick revision, important questions and MCQ practice on Merit Spring ROH.

Key points you will learn

Class 9 Mathematics The World of Numbers NCERT notes — concise revision points and definitions
Euclid's Division Lemma: a = bq + r with 0 ≤ r < b; use repeated division to get HCF
HCF by repeated division: last nonzero remainder is HCF (महत्तम समापवर्तक)
HCF × LCM = product of two numbers; LCM = product/HCF (लघुत्तम समापवर्त्य)
Fundamental Theorem of Arithmetic and prime factorisation, e.g. 2³ × 3² × 5
Rational numbers → terminating or repeating decimals; irrationals → non‑terminating, non‑repeating
Rationalisation of denominators (remove radicals) and locating √n on number line via right‑triangle constructions

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