Class 10 Mathematics Chapter 1 Real Numbers

  • Home
  • Class 10 Mathematics Chapter 1 Real Numbers
ptitle-particle2
ptitle-particle1

[ays_quiz id=”59″]

Class 10 → Mathematics → Chapter 1: Real Numbers

I. Chapter Summary:

Chapter 1: Real Numbers introduces students to the different types of numbers, focusing on natural numbers, whole numbers, integers, rational numbers, and irrational numbers. The chapter also delves into the properties of real numbers, such as closure, commutative, associative, distributive properties, and how they apply to operations like addition, subtraction, multiplication, and division. It also explains the Euclidean algorithm for finding the greatest common divisor (GCD) of two numbers and introduces the decimal expansion of real numbers.

II. Key Concepts Covered:

  1. Real Numbers:

  • Definition of real numbers as the set of rational and irrational numbers.
  • Representation of real numbers on the number line.

  1. Properties of Real Numbers:

  • Closure, commutative, associative, and distributive properties for addition and multiplication.

  1. Euclidean Algorithm:

  • A method for finding the greatest common divisor (GCD) of two numbers using division.

  1. Rational and Irrational Numbers:

  • Rational numbers: Numbers that can be expressed as a fraction of two integers (a/b).
  • Irrational numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π).

  1. Decimal Expansion:

  • Terminating decimals and non-terminating recurring decimals.
  • Non-terminating non-recurring decimals as representations of irrational numbers.

  1. Laws of Exponents:

  • Laws such as $a^m times a^n = a^{m+n}, quad (a^m)^n = a^{m times n}, quad a^0 = 1$

III. Important Questions:

(A) Multiple Choice Questions (1 Mark):

  1. Which of the following is an irrational number?

  • (a) 1/3
  • (b) √2
  • (c) 2/7
  • (d) 3.5
  • Answer: (b) √2

PYQ (2019): Which of the following is an example of a non-terminating decimal expansion? (√2)

  1. The GCD of 56 and 98 is:

  • (a) 28
  • (b) 14
  • (c) 7
  • (d) 49
  • Answer: (b) 14

PYQ (2020): Find the GCD of 56 and 98 using the Euclidean algorithm.

  1. Which of the following is a rational number?

  • (a) √3
  • (b) π
  • (c) 1/4
  • (d) √5
  • Answer: (c) 1/4

PYQ (2018): Identify the rational number from the list.

  1. Which of the following is an example of a terminating decimal?

  • (a) 1/3
  • (b) 22/7
  • (c) 0.75
  • (d) √3
  • Answer: (c) 0.75

PYQ (2021): Identify the terminating decimal.

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

  1. Find the HCF of 56 and 98 using the Euclidean algorithm.

  • PYQ (2018): Show how the Euclidean algorithm works for finding the GCD of two numbers.

  1. Explain why √2 is an irrational number.

  • PYQ (2020): Why is √2 irrational? Provide a proof by contradiction.

  1. Write the decimal expansion of 2/5. Is it terminating or non-terminating?

  • PYQ (2019): Convert 2/5 to its decimal form and state whether it is terminating.

  1. State and prove the laws of exponents with examples.

  • PYQ (2019): Prove that $a^m times a^n = a^{m+n}$

 (C) Long Answer Questions (5 Marks):

  1. Using the Euclidean algorithm, find the HCF of 120 and 84.

  • PYQ (2021): Find the HCF of two numbers using the Euclidean algorithm and verify your result.

  1. Explain the properties of real numbers with examples.

  • PYQ (2020): State and explain the commutative and associative properties with respect to real numbers.

  1. Prove that the sum of a rational and an irrational number is always irrational.

  • PYQ (2018): Prove that √3 + 1 is irrational.

  1. Explain the concept of terminating and non-terminating decimals with examples.

  • PYQ (2021): Provide examples of terminating and non-terminating decimals and explain their properties.

(D) HOTS (Higher Order Thinking Skills):

  1. If a number has a terminating decimal expansion, is it always a rational number? Justify your answer.

  • Answer: Yes, because any number with a terminating decimal can be expressed as a fraction, making it rational.

  1. How would you find the least common multiple (LCM) using the Euclidean algorithm?

  • Answer: The LCM of two numbers can be found using the formula LCM $(a, b) = frac{a times b}{text{GCD}(a, b)}$

IV. Key Formulas/Concepts:

  • GCD of two numbers (a, b):

  • Use the Euclidean algorithm: GCD(a, b) = GCD(b, a % b)

  • Exponent Laws:

  • $a^m times a^n = a^{m+n}$
  • $(a^m)^n = a^{m times n}$
  • $a^0 = 1$

  • Rational Numbers: Numbers that can be expressed as a/b, where a and b are integers and b ≠ 0.

  • Irrational Numbers: Numbers that cannot be written as a fraction (e.g., π, √2).

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 Type of Questions Typically Asked
Real Numbers 6-8 Marks 1 Mark: MCQs, 2/3 Marks: Short Answer, 5 Marks: Long Answer

VII. Previous Year Questions (PYQs):

2018:

  • 1 Mark: Identify the irrational number from the list.

  • 2 Marks: Find the HCF of 56 and 98 using the Euclidean algorithm.

  • 5 Marks: Prove that the sum of a rational and an irrational number is irrational.

2019:

  • 1 Mark: Convert 2/5 to its decimal form.
  • 3 Marks: Prove the laws of exponents.
  • 5 Marks: Explain the properties of real numbers.

2020:

  • 2 Marks: Why is √2 irrational?
  • 3 Marks: Prove that √3 + 1 is irrational.
  • 5 Marks: Find the GCD of 120 and 84 using the Euclidean algorithm.

VIII. Real-World Application Examples to Connect with Topics:

  1. Rational Numbers in Finance: Rational numbers are used to represent fractions of money, such as tax calculations, discounts, and interest rates.

  2. Irrational Numbers in Engineering: The value of π, used in calculations related to circles and spheres, is an irrational number.

  3. Exponent Laws in Science: The laws of exponents are crucial in physics, especially in calculations involving large and small numbers, such as in the study of atomic structure.

IX. Student Tips & Strategies for Success (Class-Specific):

  • Time Management: Allocate specific time slots for each chapter and take regular breaks. Prioritize weak areas.
  • Exam Preparation: Practice solving MCQs, short-answer questions, and long-answer questions from previous years.
  • Stress Management: Keep calm, take deep breaths, and stay organized. Regular revision will build confidence.

X. Career Guidance & Exploration (Class-Specific):

For Classes 9–10:

  • Academic Streams: After Class 10, students can opt for Science (Physics, Chemistry, Biology/Maths), Commerce (Business Studies, Accountancy, Economics), or Arts (Literature, History, Political Science).
  • Foundational Entrance Exams: Begin preparing for NTSE, Olympiads, and other talent search exams.

For Classes 11–12:

  • Specific Careers: For Science students, fields like Engineering, Medicine, Research, and Data Science are options. For Commerce, Accounting, Economics, and Business Management. Arts students can pursue Journalism, Fine Arts, or Design.
  • Top Entrance Exams: Prepare for NEET, JEE, CUET, and other competitive exams for top universities.

XI. Important Notes:

  • Always refer to the official CBSE website for the latest updates.
  • Consistent revision and practice are key to success.
  • Focus on understanding the concepts deeply rather than rote memorization.

Create your account

[ct-user-form form_type="register"]