Real Numbers | Class 10 Mathematics Notes

1.1 Introduction

This chapter begins by revisiting the concept of real numbers, which you were introduced to in Class IX. Real numbers encompass both rational and irrational numbers. Rational numbers are those that can be expressed as a ratio p/q, where p and q are integers and q ≠ 0. Irrational numbers, on the other hand, cannot be expressed as such a ratio. The chapter focuses on two foundational concepts related to positive integers: Euclid's division algorithm and the Fundamental Theorem of Arithmetic. Euclid's division algorithm states that for any two positive integers a and b, a can be divided by b leaving a remainder r that is smaller than b. This is essentially the long division process and is fundamental to understanding divisibility properties of integers. The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of prime numbers, except for the order of the factors. This theorem is crucial as it allows us to prove the irrationality of certain numbers such as √2, √3, and √5, and to analyze the nature of decimal expansions of rational numbers by examining the prime factorization of their denominators. The chapter sets the stage for a deeper exploration of these concepts and their applications in number theory and beyond.

📊 Diagram: Figure 1062CH01: Chapter title page image.

🧪 Activity: No specific activity in this introductory section.

🔗 Connection: Leads to Section 1.2 where the Fundamental Theorem of Arithmetic is discussed in detail with examples and applications.