Number System Converter

Number Systems Overview

Number systems are methods of representing numbers using different bases. Each system uses a specific set of digits and follows positional notation where each digit's value depends on its position.

Supported Number Systems

Binary (Base 2)

Uses only digits 0 and 1. Fundamental to digital systems and computer architecture.

• Digits: 0, 1
• Example: 1010₂ = 10₁₀
• Used in: Computer memory, digital circuits

Decimal (Base 10)

Standard number system used in everyday life with digits 0-9.

• Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Example: 42₁₀
• Used in: Mathematics, daily calculations

Octal (Base 8)

Uses digits 0-7. Historically used in computing as a shorthand for binary.

• Digits: 0, 1, 2, 3, 4, 5, 6, 7
• Example: 52₈ = 42₁₀
• Used in: Unix permissions, legacy systems

Hexadecimal (Base 16)

Uses digits 0-9 and letters A-F. Widely used in programming and digital systems.

• Digits: 0-9, A, B, C, D, E, F
• Example: 2A₁₆ = 42₁₀
• Used in: Memory addresses, color codes, programming

Conversion Methods

To Decimal Conversion

Multiply each digit by the base raised to its position power:

• Binary 1010₂: 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 8 + 0 + 2 + 0 = 10₁₀
• Hex 2A₁₆: 2×16¹ + 10×16⁰ = 32 + 10 = 42₁₀

From Decimal Conversion

Repeatedly divide by the target base and collect remainders:

• 42₁₀ to Binary: 42÷2=21 R0, 21÷2=10 R1, 10÷2=5 R0, 5÷2=2 R1, 2÷2=1 R0, 1÷2=0 R1 → 101010₂

Applications

Computer Science

• Memory addressing
• Data representation
• Assembly programming
• Debugging and analysis

Digital Electronics

• Logic circuit design
• Microcontroller programming
• Data communication
• Error detection codes

Examples & Practice

Example 1: Convert 255₁₀

• Binary: 11111111₂
• Octal: 377₈
• Hexadecimal: FF₁₆

Example 2: Convert 1100₂

• Decimal: 12₁₀
• Octal: 14₈
• Hexadecimal: C₁₆

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