Number base converter

In computing and mathematics, numbers can be represented in various systems, each with its unique base. The most common of these are decimal, hexadecimal, octal, and binary.

Decimal (Base-10): This is the standard system for denoting integer and non-integer numbers. It's also known as the base-10 system, primarily because it comprises ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this system, each digit in a number has a positional value, and the value of the digit is calculated as the digit times ten raised to the position's power. For example, in the decimal number 345, the digit 5 is in the 'ones' position, 4 is in the 'tens' position, and 3 is in the 'hundreds' position.

Hexadecimal (Base-16): The hexadecimal system, or base-16, uses sixteen distinct symbols. The first ten are the same as in the decimal system (0-9), and the next six are represented by the first six letters of the English alphabet (A-F). This system is particularly popular in computing as a human-friendly way of representing binary values. One hexadecimal digit represents four binary digits (bits), making it simpler to understand and communicate the values in large binary strings. For instance, the hexadecimal number 2F3 corresponds to the binary 001011110011.

Octal (Base-8): The octal number system, or base-8, uses eight symbols: 0 through 7. It's less common than hexadecimal but was historically used in computing because it's relatively easy to convert between octal and binary numbers. Each octal digit represents three binary digits. For example, the octal number 547 translates to the binary number 101100111.

Binary (Base-2): Binary is the most basic numbering system in computing and digital electronics. It uses only two symbols, 0 and 1, to represent all possible numbers. Each binary digit (or bit) represents a power of 2. The binary system forms the basis of all modern computing systems. In binary, the number 1010 represents the decimal number 10 (1×2^3 + 0×2^2 + 1×2^1 + 0×2^0).

Each of these systems has its specific use cases. Binary is fundamental to the operation of computers, hexadecimal provides a more human-readable format for binary data, octal was historically significant in some computing contexts, and decimal is the most commonly used system for everyday counting and arithmetic.