A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.
|List of numeral systems|
The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number eleven in the decimal numeral system (used in common life), the number three in the binary numeral system (used in computers), and the number two in the unary numeral system (e.g. used in tallying scores).
The number the numeral represents is called its value. Not all number systems can represent all numbers that are considered in the modern days; for example, Roman numerals have no zero.
Ideally, a numeral system will:
- Represent a useful set of numbers (e.g. all integers, or rational numbers)
- Give every number represented a unique representation (or at least a standard representation)
- Reflect the algebraic and arithmetic structure of the numbers.
Numeral systems are sometimes called number systems, but that name is ambiguous, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc. Such systems are, however, not the topic of this article.