Chapter 1: Real Numbers

1.1 Introduction to Real Numbers

Real Numbers: Continuation of Class IX study, focusing on positive integers.
Key Properties: Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.
Euclid’s Division Algorithm: States that any positive integer 'a' divided by 'b' leaves a remainder 'r' smaller than 'b'. Used primarily for computing HCF.
Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of primes. Used to prove irrationality of numbers and analyze decimal expansions of rational numbers.

1.2 The Fundamental Theorem of Arithmetic

1.2.1 Prime Factorisation

Natural Numbers: Can be written as a product of their prime factors (e.g., 2=2, 4=2x2).
Composite Numbers: Can be obtained by multiplying prime numbers (e.g., 7x11x23 = 1771).
Factor Tree: A method used to find the prime factors of a large number (e.g., 32760 = 2³ × 3² × 5 × 7 × 13).

1.2.2 Theorem 1.1: Fundamental Theorem of Arithmetic

Statement: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
Equivalent Version: The prime factorisation of a natural number is unique, except for the order of its factors.
Ascending Order: For uniqueness, prime factors are often written in ascending order (e.g., x = p₁p₂...p_n where p₁ ≤ p₂ ≤ ... ≤ p_n).
Historical Context: First recorded as Proposition 14 of Book IX in Euclid’s Elements, with the first correct proof by Carl Friedrich Gauss.

1.2.3 Applications of the Fundamental Theorem of Arithmetic

1.2.3.1 Checking for Digit Zero

Rule: If a number (like 4ⁿ) ends with the digit zero, its prime factorisation must contain the prime 5.
Example (4ⁿ): 4ⁿ = (2²)ⁿ = 2²ⁿ. The only prime factor is 2, so it can never end with the digit zero.

1.2.3.2 Finding HCF and LCM by Prime Factorisation

HCF (Highest Common Factor): Product of the smallest power of each common prime factor involved in the numbers.
LCM (Least Common Multiple): Product of the greatest power of each prime factor, involved in the numbers.
Example (6, 20): 6 = 2¹ × 3¹, 20 = 2² × 5¹. HCF(6, 20) = 2¹ = 2. LCM(6, 20) = 2² × 3¹ × 5¹ = 60.
Relationship (Two Numbers): For any two positive integers 'a' and 'b', HCF(a, b) × LCM(a, b) = a × b.
Relationship (Three Numbers): HCF(p, q, r) × LCM(p, q, r) ≠ p × q × r.
Formulas for Three Numbers:
- LCM (p, q, r) = (p × q × r × HCF(p, q, r)) / (HCF(p, q) × HCF(q, r) × HCF(p, r))
- HCF (p, q, r) = (p × q × r × LCM(p, q, r)) / (LCM(p, q) × LCM(q, r) × LCM(p, r))

1.3 Revisiting Irrational Numbers

1.3.1 Definition and Properties

Irrational Number 's': Cannot be written in the form ^p⁄_q, where p and q are integers and q ≠ 0.
Examples: √2, √3, √15, π, 0.101101110...
Locating Irrationals: Can be located on the number line.

1.3.2 Theorem 1.2: Divisibility by a Prime

Statement: Let 'p' be a prime number. If 'p' divides a², then 'p' divides 'a', where 'a' is a positive integer.
Proof Basis: Relies on the uniqueness part of the Fundamental Theorem of Arithmetic.

1.3.3 Theorem 1.3: Proof that √2 is Irrational

Proof Technique: Proof by contradiction.
Assumption: Assume √2 is rational, implying √2 = ^a⁄_b where a and b are coprime integers (b ≠ 0).
Steps:
1. Square both sides: 2b² = a², implying 2 divides a².
2. By Theorem 1.2, 2 divides 'a'. So, a = 2c for some integer 'c'.
3. Substitute 'a': 2b² = (2c)² ⇒ 2b² = 4c² ⇒ b² = 2c².
4. This implies 2 divides b², and by Theorem 1.2, 2 divides 'b'.
5. Contradiction: Both 'a' and 'b' have 2 as a common factor, which contradicts the assumption that 'a' and 'b' are coprime.
Conclusion: The initial assumption was incorrect, therefore √2 is irrational.

1.3.4 Proof that √3 is Irrational (Example 5)

Proof Technique: Similar proof by contradiction as for √2.
Assumption: Assume √3 is rational, implying √3 = ^a⁄_b where a and b are coprime integers (b ≠ 0).
Steps:
1. Square both sides: 3b² = a², implying 3 divides a².
2. By Theorem 1.2, 3 divides 'a'. So, a = 3c for some integer 'c'.
3. Substitute 'a': 3b² = (3c)² ⇒ 3b² = 9c² ⇒ b² = 3c².
4. This implies 3 divides b², and by Theorem 1.2, 3 divides 'b'.
5. Contradiction: Both 'a' and 'b' have 3 as a common factor, contradicting the assumption that 'a' and 'b' are coprime.
Conclusion: √3 is irrational.

1.3.5 Operations with 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 irrational number is irrational.
Example (5 - √3): Assume 5 - √3 is rational. Then √3 = 5 - ^a⁄_b = ^(5b-a)⁄_b, which would be rational. This contradicts √3 being irrational. Thus, 5 - √3 is irrational.
Example (3√2): Assume 3√2 is rational. Then √2 = ^a⁄_(3b), which would be rational. This contradicts √2 being irrational. Thus, 3√2 is irrational.

1.4 Summary of Chapter 1

Fundamental Theorem of Arithmetic: Every composite number can be uniquely factorised as a product of primes (order disregarded).
Prime Divisibility Theorem: If a prime 'p' divides a², then 'p' divides 'a' (for positive integer 'a').
Irrationality Proofs: Demonstrated proofs for numbers like √2 and √3 being irrational using proof by contradiction.
HCF/LCM of Three Numbers: Note that HCF(p, q, r) × LCM(p, q, r) ≠ p × q × r. Specific formulas exist for their relationship.