Real Numbers

Learning Objectives

• Understand and apply Euclid's Division Algorithm to find the highest common factor (HCF) of two integers by expressing one number as a multiple of another plus a remainder.
• Utilize the Fundamental Theorem of Arithmetic to express every composite number uniquely as a product of prime numbers, apart from the order of the factors.
• Calculate the HCF and least common multiple (LCM) of integers using the Prime Factorization Method, leveraging their prime factorization.
• Prove the irrationality of numbers such as $\sqrt{2}$ and $\sqrt{3}$ using the Fundamental Theorem of Arithmetic and the method of contradiction.
• Explore the Properties of Rational and Irrational Numbers, particularly how the sum, difference, product, and quotient of these numbers behave.

Chapter Notes

Euclid's Division Algorithm

• Definition: A method for finding the highest common factor (HCF) of two integers.
• Process:
  • For any two positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a = bq + r$, where $0 \leq r < b$.
  • This is essentially the long division process.
• Application: Used to compute the HCF of two positive integers.

Fundamental Theorem of Arithmetic

• Statement: Every composite number can be expressed as a product of prime numbers, and this factorization is unique, apart from the order of the factors.
• Example:
  • $32760 = 2^3 \times 3^2 \times 5 \times 7 \times 13$
• Importance: This theorem is crucial for proving properties of numbers, such as irrationality.

Prime Factorization Method for HCF and LCM

• HCF: The product of the smallest powers of all common prime factors.
• LCM: The product of the greatest powers of all prime factors involved.
• Example:
  • For numbers 6 and 20:
    • $6 = 2^1 \times 3^1$
    • $20 = 2^2 \times 5^1$
    • HCF(6, 20) = $2^1 = 2$
    • LCM(6, 20) = $2^2 \times 3^1 \times 5^1 = 60$

Proof of Irrationality Using Fundamental Theorem

• Theorem: If $p$ is a prime and $p$ divides $a^2$, then $p$ divides $a$.
• Proof by Contradiction:
  • Assume $\sqrt{2}$ is rational, i.e., $\sqrt{2} = \frac{a}{b}$ where $a$ and $b$ are coprime.
  • $2b^2 = a^2$ implies $a$ is divisible by 2.
  • Substitute $a = 2c$, then $b^2 = 2c^2$, implying $b$ is also divisible by 2.
  • This contradicts the assumption that $a$ and $b$ are coprime.
  • Hence, $\sqrt{2}$ is irrational.

Properties of Rational and Irrational Numbers

• Sum/Difference:
  • The sum or difference of a rational and an irrational number is irrational.
• Product/Quotient:
  • The product and quotient of a non-zero rational and an irrational number is irrational.
• Example:
  • Prove $5\sqrt{3}$ is irrational:
    • Assume $5\sqrt{3}$ is rational, i.e., $5\sqrt{3} = \frac{a}{b}$.
    • Rearrange to show $\sqrt{3}$ is rational, contradicting its known irrationality.
    • Hence, $5\sqrt{3}$ is irrational.