Real Numbers

REAL NUMBERS

1.1 Introduction

• In Class IX, students were introduced to real numbers and irrational numbers.

• The chapter continues this discussion, focusing on two main properties of positive integers in Section 1.2:

  • Euclid’s Division Algorithm:

    • States that any positive integer a can be divided by another positive integer b, yielding a remainder r that is smaller than b.

    • This concept is likened to the long division method.

    • It has various applications, particularly in determining the Highest Common Factor (HCF) of two integers.

  • Fundamental Theorem of Arithmetic:

    • Asserts that every composite number can be expressed uniquely as a product of primes.

    • This theorem is instrumental in various mathematical applications.

    • Its applications include proving the irrationality of certain numbers (e.g., $\sqrt{2}, \sqrt{3}, \sqrt{5}$) and understanding the nature of decimal expansions of rational numbers.

    • The prime factorization of the denominator of a rational number reveals the type of decimal expansion.

1.2 The Fundamental Theorem of Arithmetic

• Definition: The theorem states that every composite number can be expressed (factorized) uniquely as a product of prime numbers, disregarding the order of the factors.

• Any natural number can be written as a product of its prime factors. For example:

  • $2 = 2$

  • $4 = 2 \times 2$

  • $253 = 11 \times 23$

• Discusses combinations of primes:

  • Combination of primes (e.g., 2, 3, 7, 11, 23) leads to various integers through multiple products.

• Example calculations demonstrate product combinations, e.g.,

  • $7 \times 11 \times 23 = 1771$

  • $3 \times 7 \times 11 \times 23 = 5313$

  • $2 \times 3 \times 7 \times 11 \times 23 = 10626$

  • Also discusses the infinite nature of primes, leading to infinitely many integers through combinations.

• Historical Background:

  • The theorem can be traced back to the works of Euclid, and its first proof was given by mathematician Carl Friedrich Gauss.

  • Gauss is recognized as one of the greatest mathematicians in history.

• Practical Factorization Example:

  • Factorizing $32760$ as a product of primes:

  • $32760 = 2^3 \times 3^2 \times 5 \times 7 \times 13$

  • Another example is factorizing $123456789$.

• Statement of the Theorem:

  • Theorem 1.1 (Fundamental Theorem of Arithmetic): Every composite number can be expressed uniquely as a product of primes, except for the order of the prime factors.

  • Example of uniqueness: $2 \times 3 \times 5 \times 7$ is the same as $3 \times 5 \times 7 \times 2$.

• Prime Factorisation Notation:

  • For any composite number $x$, factorization follows:

  • $x = p_1^{k_1} p_2^{k_2} … p_n^{k_n}$, where $p_i$ are prime numbers in ascending order.

1.3 Applications of the Fundamental Theorem

• Example 1: Show that for any natural number $n$, $4n$ cannot end in 0.

  • Explanation: For $4n$ to end with a 0, it must be divisible by 5, which contradicts the factorization $4n = (2)^{2n}$ that contains only the prime number 2.

• HCF and LCM Calculation using Prime Factorization:

  • Example 2: Find HCF and LCM of 6 and 20.

  • $6 = 2^1 \times 3^1$

  • $20 = 2^2 \times 5^1$

  • HCF: $2^{1}$ (smallest power of common factor)

  • LCM: $2^{2} \times 3^{1} \times 5^{1} = 60$

• Example 3: Finding HCF and LCM of 96 and 404:

  • $96 = 2^5 \times 3$

  • $404 = 2^2 \times 101$

• Example 4: Find HCF and LCM of 6, 72, and 120:

  • Factorizations are: $6 = 2^1 \times 3^1$, $72 = 2^3 \times 3^2$, $120 = 2^3 \times 3^1 \times 5^1$

  • HCF: $2^1 \times 3^1 = 6$

  • LCM: $2^{3} \times 3^{2} \times 5^1 = 360$

• Product of HCF and LCM:

  • $HCF(a, b) \times LCM(a, b) = a \times b$ holds true for two numbers, but not for three numbers.

1.4 Revisiting Irrational Numbers

• Definition: An irrational number cannot be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers, and $q \neq 0$.

• Examples: Numbers such as $\sqrt{2}, \sqrt{3}, \sqrt{5}$.

• Need proof for the irrationality of these numbers:

• Theorem 1.2: If a prime number $p$ divides $a^2$, then $p$ divides $a$.

• Proof Method: Proof by contradiction.

• Example proof for $\sqrt{2}$ as irrational:

  • Assume $\sqrt{2} = \frac{r}{s}$. Then square to find a contradiction.

  • Similar proof applies for $\sqrt{3}$.

1.5 Conclusion

• Summary of key points:

  • The Fundamental Theorem of Arithmetic: Every composite number can be factored uniquely into primes.

  • An important theorem regarding divisibility of primes in terms of square numbers.

  • Proved that $\sqrt{2}$ and $\sqrt{3}$ are irrational.