Editing Numeral system (section)

{{short description|Notation for expressing numbers}}
{{About|expressing numbers with symbols|different kinds of numbers|Number system|expressing numbers with words|Numeral (linguistics)}}
{{more footnotes|date=January 2011}}
{{Numeral systems}}
[[File:Numeral Systems of the World.svg|264px|thumb|right|Numbers written in different numeral systems]]

A '''numeral system''' is a writing system for expressing [[Number|numbers]]; that is, a [[mathematical notation]] for representing numbers of a given set, using [[Numerical digit|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|decimal or base-10]] numeral system (today, the most common system globally), the number ''three'' in the [[binary number|binary or base-2]] numeral system (used in modern computers), and the number ''two'' in the [[unary numeral system]] (used in [[Tally marks|tallying]] scores).

The number the numeral represents is called its ''value''. Additionally, not all number systems can represent the same set of numbers; for example, [[Roman numerals|Roman]], [[Greek numerals|Greek]], and [[Egyptian numerals]] don't have a representation of the number [[zero]].

Ideally, a numeral system will:
*Represent a useful set of numbers (e.g. all [[integer]]s, or [[rational number]]s)
*Give every number represented a unique representation (or at least a standard representation)
*Reflect the [[algebra|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 system]]s'', but that name is ambiguous, as it could refer to different systems of numbers, such as the system of [[real number]]s, the system of [[complex number]]s, various [[hypercomplex number]] systems, the system of [[p-adic number|''p''-adic numbers]], etc. Such systems are, however, not the topic of this article.