Real Number System and Place Value

Real numbers (ℝ) are divided into several subsets: • Natural Numbers (ℕ): Counting numbers starting from 1: {1, 2, 3, 4, 5, ...}. • Whole Numbers (𝕎): Natural numbers together with zero: {0, 1, 2, 3, 4, ...}. • Integers (ℤ): Positive whole numbers, zero, and negative whole numbers: {..., -3, -2, -1, 0, 1, 2, 3, ...}. • Rational Numbers (ℚ): Any number that can be expressed as a fraction a/b where a and b are integers and b ≠ 0. Includes terminating and recurring decimals. • Irrational Numbers: Numbers that cannot be expressed as simple fractions (e.g., √2, √3, π). Their decimals are non-terminating and non-recurring.

There is a critical distinction between "Place Value" and "Value of a Digit": • Place Value refers to the position a digit occupies (e.g. Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten Millions, Hundred Millions, Billions). • Value refers to the actual quantity represented by the digit, calculated as: Digit × Place Value. Example: In the number 7,842,519: • The place value of 8 is "Hundred Thousands". • The value of 8 is 8 × 100,000 = 800,000.

• Prime Number: A whole number greater than 1 that has exactly two factors: 1 and itself (e.g. 2, 3, 5, 7, 11, 13, 17, 19, 23). Note: 2 is the ONLY even prime number! • Composite Number: A number having more than two factors (e.g. 4, 6, 8, 9, 10, 12). • Prime Factorization: Expressing a composite number as a product of prime numbers. Example: Express 72 as a product of prime factors: 72 = 2 × 36 = 2 × 2 × 18 = 2 × 2 × 2 × 9 = 2 × 2 × 2 × 3 × 3. In Index Notation: 72 = 2³ × 3².

To find the HCF and LCM of numbers (e.g., 24 and 36): First, express each number in prime factor index form: 24 = 2³ × 3¹ 36 = 2² × 3² • Highest Common Factor (HCF): Take the COMMON prime factors with the LOWEST powers. HCF = 2² × 3¹ = 4 × 3 = 12. • Lowest Common Multiple (LCM): Take ALL prime factors with the HIGHEST powers. LCM = 2³ × 3² = 8 × 9 = 72.