1. REAL NUMBERS

1. REAL NUMBERS

  • 1

    Real Numbers

    This section explores fundamental concepts related to real numbers, focusing on the Fundamental Theorem of Arithmetic and the properties of irrational numbers.

  • 1.1

    Introduction

    This section introduces the fundamental concepts of real numbers, focusing on the properties of positive integers, including Euclid's division algorithm and the Fundamental Theorem of Arithmetic.

  • 1.2

    The Fundamental Theorem Of Arithmetic

    The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of prime numbers.

  • 1.3

    Revisiting Irrational Numbers

    This section discusses the nature of irrational numbers, their properties, and proves the irrationality of specific numbers like 2 and 3 using the Fundamental Theorem of Arithmetic.

  • 1.4

    Summary

    This section summarizes the key concepts related to the Fundamental Theorem of Arithmetic and its implications in understanding real numbers.

    • Key Summary

    This chapter explores real numbers, focusing on the Fundamental Theorem of Arithmetic and its implications. It establishes that every composite number can be uniquely factored into prime numbers and discusses various applications of this theorem, including the proof of the irrationality of certain numbers. The chapter concludes with exercises and activities designed to reinforce understanding of these concepts.

    Key Takeaways

    • The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes uniquely, apart from the order of factors.
    • If a prime number divides the square of an integer, it also divides the integer itself.
    • The chapter provides proofs that certain numbers, such as 2 and 3, are irrational.

    Key Concepts

    • Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes, and this factorization is unique, except for the order of the factors.
    • Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers, such as the square roots of prime numbers.
    • Highest Common Factor (HCF): The largest positive integer that divides each of the given integers without leaving a remainder.
    • Lowest Common Multiple (LCM): The smallest positive integer that is divisible by each of the given integers.