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Class 9 → Mathematics → Chapter 1: Number System
I. Chapter Summary:
This chapter introduces different types of numbers such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It explains how these numbers together form the real number system. Students learn about decimal expansions of rational numbers, representation of irrational numbers on the number line, and operations involving real numbers. The laws of exponents are also revised and extended. This chapter builds a strong foundation for higher mathematics.
II. Key Concepts Covered:
- Number System:
Collection of all types of numbers used in mathematics. - Natural Numbers (ℕ):
Counting numbers → 1, 2, 3, … - Whole Numbers (𝕎):
Includes 0 → 0, 1, 2, 3, … - Integers (ℤ):
Positive and negative numbers → …, -2, -1, 0, 1, 2, … - Rational Numbers (ℚ):
Numbers of the form p/q where q ≠ 0. - Irrational Numbers:
Numbers that cannot be expressed as p/q (e.g., √2, π). - Real Numbers (ℝ):
Combination of rational and irrational numbers. - Decimal Expansion of Rational Numbers:
Either terminating or non-terminating recurring. - Representation on Number Line:
Real numbers can be represented geometrically. - Laws of Exponents:
Rules for simplifying powers.
III. Important Questions:
(A) MCQs (1 Mark)
- Which of the following is irrational?
(a) 1/2
(b) 0.25
(c) √2 ✅
(d) 5 - The decimal expansion of 1/3 is:
(a) Terminating
(b) Non-terminating recurring ✅
(c) Irrational
(d) Whole number - Which of the following is a rational number?
(a) √5
(b) π
(c) 7/8 ✅
(d) √3 - The set of real numbers includes:
(a) Rational only
(b) Irrational only
(c) Both rational and irrational ✅
(d) Integers only
(B) Short Answer Questions (2/3 Marks)
- Define rational and irrational numbers with examples.
- Write decimal expansion of 1/8 and classify it.
- Represent √2 on number line. (CBSE PYQ 2020)
- State laws of exponents.
(C) Long Answer Questions (5 Marks)
- Prove that √2 is irrational. (CBSE PYQ 2019)
- Explain decimal expansion of rational numbers.
- Represent real numbers on number line with examples.
- Simplify expressions using laws of exponents.
(D) HOTS Questions
- Why is π considered an irrational number?
- Can the sum of a rational and irrational number be rational? Explain.
IV. Key Formulas/Concepts:
Laws of Exponents:
$a^m times a^n = a^{m+n}$
$frac{a^m}{a^n} = a^{m-n}$
$(a^m)^n = a^{mn}$
$a^0 = 1$
$a^{-m} = frac{1}{a^m}$
Example:
$2^3 times 2^2 = 2^{5} = 32$
V. Deleted Portions (CBSE 2025–2026):
- Some advanced proofs related to irrational numbers
- Higher-level problems on exponents
(Refer to NCERT official website for latest updates)
VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026):
| Unit/Chapter | Estimated Marks | Type of Questions Typically Asked |
|---|---|---|
| Number System | 6–8 Marks | MCQs, Short Answer, Long Answer |
Note: This is an estimate; actual marks may vary.
VII. Previous Year Questions (PYQs):
1 Mark
- Identify a rational number. (2018)
2/3 Marks
- Represent √3 on number line. (2020)
- Define irrational numbers. (2019)
5 Marks
- Prove √2 is irrational. (2019)
- Explain decimal expansion of rational numbers. (2020)
VIII. Real-World Application Examples:
- Banking & Finance: Rational numbers used in calculations
- Measurement: Irrational numbers like π used in circles
- Engineering: Real numbers used in calculations
IX. Student Tips & Strategies for Success:
Time Management:
- Practice daily numerical questions
- Revise exponent laws regularly
Exam Preparation:
- Focus on concepts like irrational numbers
- Practice proofs carefully
Stress Management:
- Practice step-wise solving
- Avoid rote learning
X. Career Guidance & Exploration:
After Class 9:
Science Stream:
- Engineering, Data Science
Commerce Stream:
- Finance, Accounting
Arts Stream:
- Statistics, Economics
XI. Important Notes:
- Practice is key in Mathematics
- Focus on conceptual clarity
- Learn formulas and apply them
- Refer NCERT regularly
- Solve previous year questions