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Mathematics Class 10 - Real Numbers Notes

Comprehensive study notes for Class 10 - Real Numbers olympiad preparation

Real Numbers

Welcome to the chapter on Real Numbers for Class 10. In this chapter, you will learn about real numbers, their properties, and how to solve problems using them. By the end of this chapter, you will understand the importance of real numbers in mathematics and daily life.

Introduction

Real numbers include all the numbers you have learned so far: natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Real numbers can be represented on the number line.

Key Concepts

  • Natural Numbers: Counting numbers (1, 2, 3, ...).
  • Whole Numbers: Natural numbers and 0 (0, 1, 2, 3, ...).
  • Integers: Positive and negative whole numbers, including 0 (..., -2, -1, 0, 1, 2, ...).
  • Rational Numbers: Numbers that can be written as p/q, where p and q are integers and q ≠ 0.
  • Irrational Numbers: Numbers that cannot be written as p/q (e.g., √2, π).
  • Real Numbers: All rational and irrational numbers together.

Properties of Real Numbers

  • Real numbers can be added, subtracted, multiplied, and divided (except by zero).
  • The set of real numbers is denoted by R.
  • Every real number has a unique position on the number line.

Euclid’s Division Lemma

For any two positive integers a and b, there exist unique integers q and r such that:

a = bq + r, where 0 ≤ r < b

This lemma is useful for finding the Highest Common Factor (HCF) of two numbers.

Fundamental Theorem of Arithmetic

Every composite number can be written as a product of prime numbers in a unique way (apart from the order of the primes).

  • Example: 60 = 2 × 2 × 3 × 5

Irrational Numbers

  • Numbers like √2, √3, and π are irrational.
  • Their decimal expansion is non-terminating and non-repeating.

Decimal Expansion of Real Numbers

  • Terminating decimals: The decimal expansion ends (e.g., 1/4 = 0.25).
  • Non-terminating, repeating decimals: The decimal goes on forever but repeats (e.g., 1/3 = 0.333...).
  • Non-terminating, non-repeating decimals: The decimal goes on forever without repeating (e.g., √2 = 1.414213...).

Practice Questions

  1. Classify the following numbers as rational or irrational: 3/5, √5, 0.333..., π.
  2. Express 36 as a product of prime numbers.
  3. Find the HCF of 56 and 72 using Euclid’s division lemma.
  4. Write the decimal expansion of 7/8. Is it terminating or non-terminating?
  5. Is the sum of a rational and an irrational number always irrational? Give an example.

Challenge Yourself

  • Prove that √3 is an irrational number.
  • Find the LCM and HCF of 24 and 36 using prime factorization.

Did You Know?

  • The decimal expansion of π never ends and never repeats!
  • Zero is considered a rational number because it can be written as 0/1.

Glossary

  • Rational Number: A number that can be written as p/q, where q ≠ 0.
  • Irrational Number: A number that cannot be written as p/q.
  • Prime Number: A number greater than 1 with only two factors: 1 and itself.
  • Composite Number: A number with more than two factors.

Answers to Practice Questions

  1. 3/5: Rational, √5: Irrational, 0.333...: Rational, π: Irrational
  2. 36 = 2 × 2 × 3 × 3
  3. HCF of 56 and 72 is 8
  4. 7/8 = 0.875 (Terminating decimal)
  5. Yes, e.g., 1 (rational) + √2 (irrational) = 1 + √2 (irrational)

Real numbers are everywhere! Practice using them to solve problems in mathematics and daily life.

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