Wrong shortcode initialized

I. Chapter Summary

This chapter develops students’ understanding of the Number System by exploring the different types of numbers (natural, whole, integers, rational, irrational), their properties, and representations on the number line. It introduces real numbers, techniques for approximating irrational numbers by rational numbers, and the laws of exponents for real powers. Mastery of this chapter lays the foundation for algebra, geometry, and higher-level problem solving.

II. Key Concepts Covered

Concept	Explanation
Natural, Whole, Integer	ℕ = {1,2,3…}, W = {0,1,2…}, ℤ = {…–2,–1,0,1,2…}
Rational Numbers (ℚ)	Numbers of form p/q, p∈ℤ, q≠0; decimal either terminating or repeating
Irrational Numbers	Cannot be expressed p/q; decimal non-terminating, non-repeating (√2, π)
Real Numbers (ℝ)	ℚ ∪ (Irrational); every point on number line
Representation on Number Line	Every real number corresponds to exactly one point
Decimal Approximations	Techniques: √2 ≈ 1.414, by successive interval bisection
Laws of Exponents (for $a > 0, , m, n in mathbb{R}$ )

• $a^m cdot a^n = a^{m+n}$
• $frac{a^m}{a^n} = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $(ab)^m = a^m b^m$
• $a^0 = 1, a^(–m)=1/a^m$

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

1. Which of these is an irrational number?
   • (a) $frac{22}{7}$
   • (b) 0.1010010001… ✔️
   • (c) 0.333…
   • (d) $-frac{3}{5}$
2. The decimal expansion of 5/8 is:
   • (a) 0.625 ✔️
   • (b) 0.6250…
   • (c) 0.0625
   • (d) 0.6(25)
     (PYQ 2019)
3. $a^{frac{1}{2}} cdot a^{frac{1}{3}}$ equals:
   • (a) $a^{frac{5}{6}}$✔️
   • (b) $a^{-frac{1}{6}}$
   • (c) $a^{frac{1}{5}}$
   • (d) $a^1$
4. Which set is uncountable?
   • (a) Natural numbers
   • (b) Rational numbers
   • (c) Real numbers ✔️
   • (d) Integers

(B) Short Answer Questions (2/3 Marks)

1. Prove that √3 is irrational. (PYQ 2018)
2. Express 0.272727… as a fraction in simplest form.
3. Using laws of exponents, simplify: $frac{2^3 cdot 2^{-1}}{2^{frac{1}{2}}}$.
4. Find the point on the number line representing $-frac{7}{4}$.

(C) Long Answer Questions (5 Marks)

1. State and prove the laws of exponents for real numbers m and n. (PYQ 2020)
2. Explain, with a construction, how to locate √5 on the number line.
3. Distinguish between rational and irrational numbers with three examples each.
4. Show that between any two distinct real numbers there are infinitely many rational numbers.

(D) HOTS (Higher Order Thinking Skills)

1. Design an algorithm (in steps) to approximate π to three decimal places using only bisecting intervals on the number line.
2. If $a^m = b^n quad text{for positive } a neq b text{ and } m, n neq 0$
   integers, what can you say about a and b? Analyze and justify.

IV. Key Formulas/Concepts

• Decimal to Fraction (repeating):
  If $x = 0.overline{abc}, quad text{then} quad x = frac{abc}{999}$
  .
• Interval Bisection for √k:
  Find a,b such that $a^2 < k < b^2$; midpoint $m = frac{a + b}{2}, quad text{test} quad m^2 < k$, iterate.
• Exponent Rules (see section II).

V. Deleted Portions (CBSE 2025–2026)

“No portions have been deleted from this chapter as per the rationalized NCERT textbooks.”

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)

Unit/Chapter	Estimated Marks	Question Types
Number System	6–7 Marks	1 MCQ, 1 Short Answer, 1 Long Answer, 1 HOTS/Data Analysis

VII. Previous Year Questions (PYQs)

Marks	Question	Year
1	Which set of numbers is uncountable?	PYQ 2019
2	Express 0.6363… as a fraction.	PYQ 2018
3	Prove that √7 is irrational.	PYQ 2020
5	Show that between any two real numbers there exist infinitely many rational numbers.	PYQ 2019

VIII. Real-World Application Examples

• Computer Graphics: Real numbers approximate pixel coordinates; irrational slopes appear in diagonal lines.
• Engineering: Measurements (√2 in constructing right angles) use irrational approximations.
• Finance: Exponential growth/decay (compound interest) uses laws of exponents.

IX. Student Tips & Strategies for Success

• Time Management:
  • Spend one day on theory (definitions, proofs).
  • One day on constructions (√k on number line).
  • One day on exponent exercises and mixed problems.
• Exam Preparation:
  • Memorize and practice laws of exponents until automatic.
  • Solve interval bisection examples for square-root constructions.
  • Practice converting repeating decimals to fractions.
• Stress Management:
  • Break proofs into bullet points.
  • Use number-line diagrams as visual anchors.

X. Career Guidance & Exploration

• For Classes 9–10:
  • Streams: Science (engineering), Commerce (finance), Arts (data analytics).
  • Foundational Exams: NTSE, RMO (RMO).
• For Classes 11–12:
  • Careers: Engineering (IIT-JEE), Data Science, Cryptography, Pure Mathematics (C.U.E.T).
  • Top Institutions: IITs, IISc, Amity, Delhi University (Mathematics).

XI. Important Notes

• Always refer to the official CBSE website for any last-minute updates.
• Focus on conceptual clarity—understand why proofs and constructions work.
• Regular revision and practice of varied problems are key to success.