Grade IX Number Systems

This chapter builds the foundation for all higher mathematics. It covers definitions of rational and irrational numbers, real numbers, their properties, decimal expansions, representation on the number line, operations (including rationalization), and laws of exponents. The chapter has 5 main sections (as per the latest NCERT textbook) and 5 exercises.

Objectives by the end of the plan:

  • Classify numbers as rational/irrational/real.
  • Prove simple irrationality and work with decimal expansions.
  • Represent numbers on the number line geometrically.
  • Perform operations and rationalize denominators.
  • Apply exponent laws to real numbers confidently.
  • Score 90%+ in any test on this chapter.

Resources Required

  • NCERT Class 9 Maths Textbook (Chapter 1).
  • Notebook for notes, proofs, and diagrams.
  • Graph paper/ruler/compass for number line constructions.
  • NCERT Solutions (for self-checking).
  • Optional: Khan Academy or BYJU’s videos for visuals (10-15 min per day).

Total Duration: 8 days (1.5–2 hours/day). Revise daily for 15 minutes. Practice without a calculator.

Day 1: Introduction, Rational & Irrational Numbers (Sections 1.1 & 1.2)

Key Concepts & Notes

  • Natural (N), Whole (W), Integers (Z), Rational (Q = p/q, p,q integers, q ≠ 0, coprime), Irrational (cannot be p/q), Real (R = Q ∪ Irrationals).
  • Between any two rationals, there are infinitely many rationals.
  • √p is irrational if p is not a perfect square (proof by contradiction: assume p/q in lowest terms → p² = k q² leads to contradiction on common factors).
  • Every real number corresponds to a unique point on the number line.

Sample Problems (Solve in notebook)

  1. Is zero a rational number? Can you write it in the form p/q? Solution: Yes, 0 = 0/1 (or any 0/q).
  2. Find five rational numbers between 1 and 2. Solution: 7/6, 8/6, 9/6, 10/6, 11/6 (or 1.1, 1.2, ..., 1.9 as decimals, but prefer fractions).
  3. Prove that √2 is irrational (standard board question). Solution: Assume √2 = p/q (p,q coprime, q ≠ 0). Then p² = 2q² ⇒ p even (p=2k) ⇒ 4k²=2q² ⇒ q even. Contradiction. Hence irrational.
  4. Give an example of two irrationals whose sum is rational. Solution: √2 and –√2 (sum = 0).

Practice: NCERT Exercise 1.1 (all questions) + Exercise 1.2 (Q1–Q3).

Day 2: Real Numbers and their Decimal Expansions (Section 1.3)

Key Concepts & Notes

  • Rational decimals: Terminating (denominator factors only 2 & 5) or non-terminating recurring.
  • Irrational decimals: Non-terminating and non-recurring.
  • To convert recurring decimal to p/q: Let x = decimal, multiply by 10^k (k=period length), subtract, solve.
  • Irrationals between two rationals: Use non-recurring decimals.

Sample Problems

  1. Express 1/7 in decimal form and say whether it is terminating or recurring. Solution: 0.142857142857… (bar over 142857) → recurring, hence rational.
  2. Convert 0.6̅ (0.666…) to p/q form. Solution: x = 0.6̅ → 10x = 6.6̅ → 9x = 6 → x = 2/3.
  3. Find an irrational number between 1/7 and 2/7. Solution: 0.15015001500015… (non-recurring, non-terminating).
  4. Without actual division, predict the decimal nature of 13/3125. Solution: 3125 = 5^5 → terminating decimal.

Practice: NCERT Exercise 1.3 (all).

Day 3: Representing Real Numbers on the Number Line

Key Concepts & Notes

  • Rationals: Divide the segment equally (e.g., 5/3 = 1 + 2/3).
  • Irrationals: Geometric construction using Pythagoras theorem (square root spiral or right triangle method).

Sample Problems (Draw diagrams)

  1. Locate √2 on the number line (describe steps). Solution: Draw OA = 1 unit. At A, draw perpendicular AB = 1 unit. Join OB = √2. Use compass to mark arc from O intersecting number line at P. OP = √2.
  2. Locate 3.√5 approximately on the number line (up to 1 decimal). Solution: √5 ≈ 2.236 → 3 × 2.236 = 6.708 → mark between 6 and 7, closer to 6.7.
  3. Visualize 4.26 on the number line up to 3 decimal places. Solution: Between 4 and 5; first mark 4.2, then divide next segment into 10, mark 4.26.

Practice: NCERT Exercise 1.2 (Q4–Q5) + construct square root spiral for √2, √3, √5, √6.

Day 4: Operations on Real Numbers (Section 1.4)

Key Concepts & Notes

  • Reals are closed under +, –, ×, ÷ (÷0 undefined).
  • Rational + Irrational = Irrational; Rational × Irrational (≠0) = Irrational.
  • Rationalize denominator: Multiply numerator & denominator by conjugate (a + b → a – b).
  • Useful identities: √(ab) = √a × √b, (√a + √b)(√a – √b) = a – b.

Sample Problems

  1. Rationalize the denominator: 1/(√5 + √2). Solution: [1 × (√5 – √2)] / [(√5 + √2)(√5 – √2)] = (√5 – √2)/(5 – 2) = (√5 – √2)/3.
  2. Simplify: (√3 + √2)². Solution: 3 + 2√6 + 2 = 5 + 2√6.
  3. Classify: √2 + √3 (irrational) and 2√3 × (1/√3) (rational = 2). Solution: First is irrational; second is rational.
  4. Rationalize: 1/(√7 – √3 + √2) (advanced, multi-step). Solution: First group as (√7) – (√3 + √2), multiply by conjugate step-by-step (refer NCERT Example 19).

Practice: NCERT Exercise 1.4 (all).

Day 5: Laws of Exponents for Real Numbers (Section 1.5)

Key Concepts & Notes

  • For a > 0, rational exponents: a^{m/n} = (a^m)^{1/n} = (a^{1/n})^m.
  • Laws (same as integers): a^p × a^q = a^{p+q}, (a^p)^q = a^{pq}, a^p / a^q = a^{p-q}, (ab)^p = a^p b^p, etc.

Sample Problems

  1. Simplify: 2^{1/3} × 2^{1/3} × 2^{1/3}. Solution: 2^{1/3 + 1/3 + 1/3} = 2^1 = 2.
  2. Find value: (81)^{1/4}. Solution: 81 = 3^4 → (3^4)^{1/4} = 3.
  3. Simplify: (5^{3/2} ÷ 5^{1/2}) × 5^{1/2}. Solution: 5^{3/2 – 1/2} × 5^{1/2} = 5^1 × 5^{1/2} = 5^{3/2}.
  4. Express 8^{2/3} in simplest form. Solution: (2^3)^{2/3} = 2^2 = 4.

Practice: NCERT Exercise 1.5 (all).

Day 6: Complete NCERT Exercise Practice + Mixed Problems

  • Solve every question from Exercises 1.1 to 1.5 again (timed: 45 minutes).
  • Check answers with official NCERT solutions.
  • Revise all proofs and constructions.

Day 7: Extra Important Questions & HOTS (Higher Order Thinking Skills)

Practice these (very likely in exams):

  1. Find three different irrational numbers between 5/7 and 9/11. Solution: Convert to decimals (≈0.714 and ≈0.818), pick 0.720720072..., 0.750750075..., 0.780780078... (non-recurring).
  2. Simplify: 1/(√9 + √8) + 1/(√9 – √8). Solution: Rationalize each → overall simplifies to nice rational (calculate step-by-step).
  3. If a = 2 + √3, find a + 1/a. Solution: 1/a = 2 – √3 → sum = 4.
  4. Express 0.4̅3̅2 as p/q. Solution: Let x = 0.432432… → 1000x = 432.432… → 999x = 432 → x = 432/999 = 16/37 (after simplifying).
  5. Locate √4.5 on number line geometrically.

More from previous year patterns: Rationalize 5/(√3 – √5), simplify √(5 + 2√5), prove 3 + 2√5 irrational.

Day 8: Full Revision + Mock Test

  • Make a one-page mind map: Rational vs Irrational table, decimal rules, exponent laws, rationalization steps.
  • Formulas to memorize: All exponent laws + identities for √a ± √b.
  • Take a 1-hour mock test (10 questions: 4 MCQ, 3 short, 3 long). Score yourself.
  • Weak areas → revise only those.

Quick Revision Checklist

  • Definitions + examples of each number type.
  • Decimal classification rule.
  • Proof of √2 irrational (memorize steps).
  • Rationalization technique (always conjugate).
  • Exponent laws application (base > 0).

Tips for Success

  • Draw neat diagrams for number line (compass mandatory).
  • Always show steps in proofs (assumption → contradiction).
  • Practice 2–3 questions daily even after the plan.
  • Common mistakes: Forgetting to rationalize fully, wrong period in decimals, negative bases with fractional exponents (avoid).

Follow this plan sincerely and you will master Number Systems completely. This chapter carries 8 marks in the annual exam and appears in almost every test. If you solve all NCERT + the samples above, you are exam-ready!