Chapter 1: Real Numbers
Euclid’s Division Algorithm and Fundamental Theorem of Arithmetic
This chapter revisits the structure of real numbers, starting with Euclid’s division algorithm. It states that any positive integer a can be divided by another positive integer b to produce a quotient and a remainder smaller than b. It’s useful for finding the highest common factor (HCF) of two numbers. The Fundamental Theorem of Arithmetic asserts that every composite number can be uniquely expressed as a product of prime numbers, regardless of the order. This theorem helps in identifying patterns, determining HCF and LCM, and analyzing number properties, like whether a number’s decimal expansion terminates or repeats.
Applications of Prime Factorization and Decimal Expansions
Prime factorization plays a key role in solving problems related to HCF and LCM. If the prime factorization of two or more numbers is known, the HCF is the product of the smallest powers of common primes, and the LCM is the product of the greatest powers. This chapter also explains how to determine if a rational number’s decimal expansion will terminate or repeat. If the denominator of a fraction in its lowest form has only 2s or 5s (or both) as prime factors, the decimal terminates. Otherwise, it results in a non-terminating repeating decimal.
Irrational Numbers and Their Proofs
The chapter includes logical proofs that numbers like √2, √3, and √5 are irrational using the method of contradiction. It also explains that the sum or product of a rational and an irrational number (non-zero) is always irrational. These proofs use the uniqueness of prime factorization and Theorem 1.2, which states that if a prime divides a², it must divide a. This foundational knowledge reinforces the understanding of number classification and the structure of the real number system.