Real numbers include all rational and irrational numbers. Euclid's Division Lemma states: for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. This lemma is used to find the HCF (Highest Common Factor) of two numbers by repeatedly applying division. For example, HCF(135, 225): 225 = 1×135 + 90 → 135 = 1×90 + 45 → 90 = 2×45 + 0 → HCF = 45. The algorithm always terminates because the remainder strictly decreases at each step.

To find HCF of two numbers: divide the larger by the smaller, then divide the divisor by the remainder, and repeat until remainder is 0. The last divisor is the HCF. Example: HCF(135, 225) = 45.

The Fundamental Theorem of Arithmetic states that every composite number greater than 1 can be expressed as a product of prime numbers in a unique way (ignoring the order of factors). This is also called the Unique Prime Factorisation Theorem. For example: 360 = 2³ × 3² × 5. Applications: (1) Finding HCF — take the product of smallest powers of common primes; (2) Finding LCM — take the product of greatest powers of all primes. HCF × LCM = Product of the two numbers (for two numbers only).

HCF = product of the smallest powers of common prime factors. LCM = product of the greatest powers of all prime factors. Relation: HCF × LCM = Product of the two numbers (for two numbers only).

An irrational number cannot be expressed as p/q where p, q are integers and q ≠ 0. Proof that √2 is irrational: Assume √2 = p/q in lowest terms. Then 2q² = p², so p² is even → p is even. Let p = 2m. Then 2q² = 4m² → q² = 2m² → q is even. But this contradicts p/q being in lowest terms. So √2 is irrational. Similarly, √3, √5, √7 are irrational. Sum or product of a non-zero rational and an irrational is always irrational. π and e are also irrational.

To find HCF and LCM using prime factorisation: first express each number as a product of prime factors. HCF = product of the smallest power of each common prime factor. LCM = product of the greatest power of each prime factor (common or not). Example: 12 = 2²×3 and 18 = 2×3². HCF = 2¹×3¹ = 6; LCM = 2²×3² = 36. Verification: HCF × LCM = 6 × 36 = 216 = 12 × 18. ✓ This method is more efficient than the division method for larger numbers.

A rational number p/q (in lowest terms) has a terminating decimal if q has only 2 and 5 as prime factors. Otherwise it is non-terminating repeating. Irrational numbers have non-terminating non-repeating decimal expansions.

The real number line contains both rational numbers (integers, fractions like 3/4, 22/7) and irrational numbers (√2, √3, π). Between any two rational numbers, there are infinitely many rational and irrational numbers — this property is called density. Rational + Rational = Rational. Rational × Irrational = Irrational (if rational ≠ 0). Irrational + Irrational can be rational (e.g., √2 + (−√2) = 0). So care must be taken when combining irrational numbers.