Real Numbers

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.

• Real numbers include both rational and irrational numbers.
• Euclid's division algorithm relates to divisibility and remainder in integer division.
• Every composite number can be uniquely factorized into primes (Fundamental Theorem of Arithmetic).
• The Fundamental Theorem of Arithmetic helps prove irrationality of certain numbers.
• Prime factorization of denominators reveals the nature of decimal expansions of rational numbers.
• The chapter builds on concepts introduced in Class IX.
• 📌 Real numbers: The set of all rational and irrational numbers.
• 📌 Rational numbers: Numbers expressible as p/q, where p and q are integers, q ≠ 0.
• 📌 Irrational numbers: Numbers that cannot be expressed as a ratio of two integers.

1.2 The Fundamental Theorem of Arithmetic

This section elaborates on the Fundamental Theorem of Arithmetic, which states that every composite number can be expressed as a product of prime numbers in a unique way, except for the order of the factors. The section begins by exploring the idea of constructing numbers by multiplying prime numbers, illustrating with examples such as 7 × 11 × 23 = 1771 and 2³ × 3 × 7³ = 8232. It then discusses the infinite nature of prime numbers and the question of whether every composite number can be expressed as a product of primes. Using factor trees, the section demonstrates the factorization of numbers like 32760 and 123456789 into their prime factors, confirming the conjecture that every composite number can indeed be factorized uniquely into primes. The theorem is formally stated and its historical context is given, highlighting Carl Friedrich Gauss's contribution. The uniqueness of prime factorization is emphasized, meaning that while the order of factors can vary, the prime factors themselves and their powers are unique. The section also revisits the method of finding HCF and LCM using prime factorization, illustrating with examples such as finding the HCF and LCM of 6 and 20, and 96 and 404. It concludes by noting that while the product of HCF and LCM equals the product of two numbers, this property does not extend straightforwardly to three or more numbers.

• Every composite number can be uniquely factorized into primes, except for order.
• Prime factorization can be done using factor trees.
• There are infinitely many prime numbers.
• Carl Friedrich Gauss provided the first correct proof of this theorem.
• HCF is the product of the smallest powers of common prime factors.
• LCM is the product of the greatest powers of all prime factors involved.
• 📌 Prime factorization: Expressing a number as a product of primes.
• 📌 HCF (Highest Common Factor): Greatest number dividing two or more numbers.
• 📌 LCM (Least Common Multiple): Smallest number divisible by two or more numbers.