Chapter 1 Real Numbers

Notes

What are Real Numbers? Real numbers include all numbers that can be represented on the number line.

Types of Real Numbers: Rational Numbers (Q): Can be written in the form p/q (q ≠ 0) Examples: 1/2, -3/4, 5 Decimal expansion: terminating or repeating Irrational Numbers: Cannot be expressed as p/q Decimal expansion: non-terminating, non-repeating Examples: √2, √3, π 🔥 Euclid’s Division Lemma

For any two integers a and b (b ≠ 0):

👉 a = bq + r Where:

0 ≤ r < b q is the quotient r is the remainder 🔥 HCF (Highest Common Factor)

Methods to find HCF:

Prime Factorization Euclid’s Algorithm (most important) 🔥 Euclid’s Algorithm

Example: Find HCF of 135 and 225

225 = 135 × 1 + 90 135 = 90 × 1 + 45 90 = 45 × 2 + 0

👉 HCF = 45

🔥 Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of prime numbers uniquely.

Example: 60 = 2 × 2 × 3 × 5

🔥 Decimal Expansion of Rational Numbers

After simplification:

If denominator has only factors 2 or 5 → terminating decimal Otherwise → non-terminating repeating decimal

Examples: 1/8 = 0.125 (terminating) 1/3 = 0.333... (repeating)