Notes on Real Numbers

1.1 Introduction

In this chapter, the concept of real numbers is revisited, particularly focusing on properties of positive integers. Two significant properties discussed are Euclid’s division algorithm and the Fundamental Theorem of Arithmetic. These properties lay foundational understanding for further discussions on rational and irrational numbers.

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 factors. This uniqueness is crucial, meaning that for any composite number, the prime factorization is consistent aside from the order of factors.

Example 1: Factorization

Consider the number 32760, which can be factorized as:

• Prime Factorization: 32760 = 2^3 × 3^2 × 5 × 7 × 13.
  This example shows that every composite number must be represented through primes and emphasizes the uniqueness of this representation.

Application

The theorem is instrumental in various areas of mathematics, including:

• Proving the irrationality of certain numbers such as 2, 3, and 5.
• Determining whether the decimal representation of rational numbers is terminating or non-terminating by analyzing the prime factorization of the denominator.

Example 2: Rational Numbers

If a rational number has a denominator that can be expressed as a product of primes that includes only 2's and 5's, it will have a terminating decimal expansion; otherwise, it will be non-terminating repeating.

1.3 Revisiting Irrational Numbers

Irrational numbers are defined as numbers that cannot be represented as a fraction /q, where p and q are integers, and q ≠ 0. Key examples include , 3, and 5. The chapter goes through the proof of the irrationality of 2 and 3, using the Fundamental Theorem of Arithmetic.

Proof of 2 and 3 being irrational

Assuming both are rational leads to contradictions when applying the theorem, implying that they cannot be perfectly expressed as fractions of integers with no common factors (that is, they can't be reduced to simpler forms).

Example 3: Proof of 2 being irrational

1. Assume 2 can be expressed as a rational number r/s.
2. Rearranging leads to conclusions that imply both r and s share a prime factor of 2, contradicting the assumption of being coprime (no common prime factors).

Example 4: Proof of 3 being irrational

Similar to the proof for 2, it leads to the conclusion that a and b must share a prime factor of 3, contradicting their assumed coprimality.

1.4 Summary

Through this chapter, we have revisited the Fundamental Theorem of Arithmetic, showed proofs regarding the irrationality of certain numbers, and highlighted important properties of real numbers.

1. Fundamental Theorem of Arithmetic: Every composite number can be expressed uniquely as a product of primes.
2. Divisibility properties: Using Euclid's algorithm for integers helps find relationships between numbers.
3. Irrational numbers: Numbers like √2 and √3 cannot be expressed as fractions of integers.
4. Applications: The prime factorization method is essential for finding HCF and LCM of numbers.
5. Terminating vs. Non-terminating: The form of the denominator in fractions determines decimal expansion behavior.
6. Contradictions: Many proofs of irrationality are based on reductio ad absurdum (proof by contradiction).