Number System

Understanding Number Systems

A number system is a way of representing numbers using digits or symbols. The most common number system in everyday life is the decimal (base-10) system, which uses digits 0-9. However, computers use other systems like binary (base-2) and hexadecimal (base-16).

Decimal System (Base-10)

The decimal system is the standard number system we use daily. Each digit position represents a power of 10:

• Units place: 10⁰ = 1
• Tens place: 10¹ = 10
• Hundreds place: 10² = 100
• Thousands place: 10³ = 1000

Example: The number 2,345 = (2 × 10³) + (3 × 10²) + (4 × 10¹) + (5 × 10⁰)

Binary System (Base-2)

Binary uses only two digits: 0 and 1. Each position represents a power of 2. Computers use binary because it's easy to represent using electrical states (on/off).

Example: The binary number 1011₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11₁₀

Hexadecimal System (Base-16)

Hexadecimal uses digits 0-9 and letters A-F (where A=10, B=11, C=12, D=13, E=14, F=15). It's commonly used in computing.

Example: The hexadecimal number 2F₁₆ = (2 × 16¹) + (15 × 16⁰) = 32 + 15 = 47₁₀

Conversion Between Systems

Converting from decimal to binary: Divide by 2 repeatedly and note remainders.

Example: Convert 13₁₀ to binary: 13÷2=6 r1, 6÷2=3 r0, 3÷2=1 r1, 1÷2=0 r1. Reading remainders bottom-up: 1101₂

Key Concepts

• Different bases use different digits (0-9 for base-10, 0-1 for base-2, 0-F for base-16)
• Place value is determined by powers of the base
• Conversion uses repeated division or multiplication
• Understanding number systems is crucial for computer science