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number-systems

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🗂️ Maths, Logic & Crypto · View mindmap

Number Systems

A number system is a method of representing quantities using a consistent set of symbols and rules. The fundamental characteristic of any number system is its base, which determines how many unique digits are available and at what point positional values increment. In a base-n system, each position represents a power of n, and digits range from 0 to n−1. The decimal system (base 10), familiar from everyday use, employs digits 0–9. However, other bases are equally valid and useful for different applications.

Common Bases

The binary system (base 2) uses only digits 0 and 1 and forms the foundation of digital computing and information storage. The hexadecimal system (base 16) uses digits 0–9 and letters A–F, offering a compact way to represent binary data and is widely used in programming and cryptography. The octal system (base 8) occasionally appears in computing contexts. Each system can express the same quantity; for example, the number 255 in decimal equals 11111111 in binary and FF in hexadecimal.

Conversion and Positional Notation

Converting between number systems relies on understanding positional notation. In any base-n system, a number is calculated as the sum of each digit multiplied by its corresponding power of n. For instance, the decimal number 345 represents (3 × 10²) + (4 × 10¹) + (5 × 10⁰). This same principle applies across all bases, allowing systematic conversion between systems through division and remainder operations.

Number systems are fundamental to mathematics, computer science, and cryptography. The choice of base affects how efficiently certain operations can be performed and how naturally a system suits particular applications.

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