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Numeral systems |
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List of numeral systems |

A **numeral system** 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.

The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number *eleven* in the decimal or base-10 numeral system (today, the most common system globally), the number *three* in the binary or base-2 numeral system (used in modern computers), and the number *two* in the unary numeral system (used in tallying scores).

The number the numeral represents is called its value. Not all number systems can represent the same set of numbers; for example, Roman numerals cannot represent the number 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.

For example, the usual decimal representation gives every nonzero natural number a unique representation as a finite sequence of digits, beginning with a non-zero digit.

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.