numeralsystem

A numeral system, or number system, is a writing system for expressing numbers. It provides a set of symbols and rules for combining them to represent quantities. In most modern systems, numbers are written using place-value notation, where the value of a digit depends on its position within a base-b representation.

A base, or radix, defines how many distinct digits the system uses. In a base-b system, numbers are written as sequences of digits d_k ... d_0 with 0 ≤ d_i < b, and the value equals sum d_i · b^i. This framework supports compact representation and arithmetic through consistent digit weights.

Common bases include decimal (base-10), binary (base-2), octal (base-8), and hexadecimal (base-16). Decimal uses digits 0–9; binary uses 0 and 1; hexadecimal uses 0–9 and A–F. Some systems are non-positional, such as Roman numerals, where symbols denote quantities through combinations rather than place value.

Historically, numeral systems evolved from tally marks and non-positional symbols to positional systems. The Hindu-Arabic numeral system, developed in South Asia and transmitted to the Islamic world and Europe, introduced zero and place-value notation. This advance greatly simplified calculation and laid the foundation for modern mathematics and computing.

Numeral systems underpin computing, digital encoding, scientific notation, and everyday arithmetic. Converting numbers between bases or to and from decimal is a common task in computer science and engineering. Some systems tailor to specific applications, such as hexadecimal for byte-oriented data or binary for logic operations.