How to Convert Between Number Systems?

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To convert between different number systems, first convert the original number to decimal (base-10) using the positional notation method, then convert from decimal to the target number system. Decimal serves as the intermediate base since it's the most familiar number system.

The conversion process involves understanding the place values and using the appropriate conversion algorithms for each number system.

Number System Conversion Principles

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1. Positional Notation

Each digit in a number has a value based on its position. The rightmost digit represents the base^0 position, the next digit represents base^1, and so on. For example, in binary 1011 = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11 in decimal.

2. Decimal as Intermediate Base

• Source → Decimal → Target
• This two-step process ensures accuracy
• Decimal is the most familiar base for calculations

3. Conversion Algorithms

To convert from decimal to another base, repeatedly divide by the target base and collect remainders in reverse order. To convert to decimal, multiply each digit by its place value and sum the results.

4. Common Number Systems

Binary (base-2) uses digits 0-1, Octal (base-8) uses 0-7, Decimal (base-10) uses 0-9, and Hexadecimal (base-16) uses 0-9 and A-F.

5. Precision and Validation

• Validate input for the source number system
• Handle negative numbers and fractions
• Consider precision for floating-point conversions

6. Applications

• Computer programming and digital electronics
• Data encoding and cryptography
• Mathematical research and education
• Network addressing and data compression

Common Number Systems

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Binary (Base-2):

Uses digits 0 and 1. Fundamental to computer science and digital electronics. Each position represents a power of 2.

Decimal (Base-10):

Uses digits 0-9. The standard number system used in everyday life. Each position represents a power of 10.

Hexadecimal (Base-16):

Uses digits 0-9 and letters A-F. Commonly used in computer programming and digital electronics for compact representation.

Octal (Base-8):

Uses digits 0-7. Historically used in computing and still used in some Unix file permissions.

Other Bases:

Base-3 through Base-15, Base-32, Base-36, and Base-64 are used in specialized applications like data encoding, cryptography, and mathematical research.