Chapter 1 · Class IX Mathematics · NCERT Exercises

Number Systems — Exercises

Irrationals, Real Numbers & Laws of Exponents — All 5 Exercises Fully Solved

📂 5 Exercises 📝 24 Questions 🎓 Foundation

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Exercise Index

Natural, whole, integer, rational & irrational numbers; classifying numbers on number line

Irrational numbers; locating √2, √3, √5 on the number line using compass construction

4 Q Easy-Moderate

Real numbers and their decimal expansions; terminating vs non-terminating; p/q form

Operations on real numbers; surds; representing real numbers on number line (successive magnification)

Laws of exponents for real numbers; simplification; rationalisation of denominators

5 Q Moderate-Hard

5 exercise files · 24 total questions

Chapter at a Glance

CBSE BoardsNTSEOlympiad

9 Concepts

10 Formulas

Foundation Difficulty

8–10% Weightage

Before You Begin

Prerequisites

✓Basic number system (Class VIII)
✓Fractions and decimals
✓Square roots and cube roots

Have Ready

🔧Compass and ruler for number line construction
🔧Notebook for decimal proofs

Exercise Topic Map

Exercise 1.1 Classify each number; recall ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ; find rational between two rationals using mean

Exercise 1.2 Locate √n geometrically: draw right triangle with legs 1 & √(n−1); arc gives √n

Exercise 1.3 Long divide to get decimal; identify repeating block; p/q: denominator only 2s & 5s → terminating

Exercise 1.4 a+b, a−b, a×b, a÷b for irrationals; successive magnification to plot on number line

Exercise 1.5 aᵐ·aⁿ = aᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ; rationalise: multiply by conjugate (a−√b)/(a−√b)

Key Formulae — Recall Before Solving

\(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R};\quad \mathbb{Q} \cup \mathbb{Q}' = \mathbb{R}\)

\(a^m \cdot a^n = a^{m+n};\quad (a^m)^n = a^{mn};\quad a^m \cdot b^m = (ab)^m\)

\(a^{1/n} = \sqrt[n]{a};\quad a^{m/n} = (\sqrt[n]{a})^m\)

\(\dfrac{1}{a+\sqrt{b}} = \dfrac{a-\sqrt{b}}{a^2-b} \quad \text{(rationalisation)}\)

NCERT Solving Method

Step 1 — Classification: always place the number on the hierarchy ℕ→ℤ→ℚ→ℝ; check if it terminates/repeats for rational. Step 2 — Number line construction: for √n, use the Pythagorean approach; draw carefully with compass. Step 3 — Decimal expansion: perform long division; observe the remainder pattern; the decimal repeats when a remainder repeats. Step 4 — Rationalisation: identify the conjugate; multiply numerator and denominator; use (a²−b²) identity. Step 5 — Exponent laws: convert all roots to fractional exponents first; then apply laws systematically.

Continue Your Preparation

📖NotesTheory & concepts 🧠MCQsTopic-wise practice ⚖️True–FalseConcept check ➡️Next ChapterPolynomials

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NCERT Class 9 Maths Chapter 1 Number Systems Notes, Formulas & Solutions

NCERT Class 9 Maths Chapter 1 Number Systems Notes, Formulas & Solutions — Complete Notes & Solutions · academia-aeternum.com

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