Number systems are fundamental to mathematics and computer science, serving as the basis for representing and manipulating numerical values. While we're most familiar with the decimal system, which uses 10 symbols (0-9), there are several other systems, each with its unique properties and applications.

1 Binary System

The binary system is the foundation of modern computing. It uses only two symbols, 0 and 1, known as bits. Each digit in a binary number represents a power of 2. For example, the binary number 1011 represents (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0) = 11 in decimal. Binary numbers are crucial in computer science for representing digital data and performing arithmetic operations at the electronic level.

2 Octal System

The octal system, also known as base-8, uses eight symbols (0-7). Octal numbers are often used in computing for representing groups of bits, as they provide a more compact representation than binary. Each digit in an octal number represents a power of 8. For instance, the octal number 54 represents (5 × 8^1) + (4 × 8^0) = 44 in decimal.

3 Decimal System

The decimal system, familiar to most of us, uses ten symbols (0-9). Each digit in a decimal number represents a power of 10. For example, the decimal number 456 represents (4 × 10^2) + (5 × 10^1) + (6 × 10^0) = 456. Decimal numbers are used in everyday life for counting, arithmetic calculations, and financial transactions.

4 Hexadecimal System

The hexadecimal system, often used in computing, employs sixteen symbols: 0-9 followed by A-F, where A stands for 10, B for 11, and so on, up to F for 15. Each digit in a hexadecimal number represents a power of 16. For example, the hexadecimal number 2A7 represents (2 × 16^2) + (A × 16^1) + (7 × 16^0) = 679 in decimal. Hexadecimal numbers are commonly used in programming, digital electronics, and color representations (such as HTML color codes).