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| heading1 = Hindu–Arabic numeral system

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| heading2 = East Asian

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| heading3 = Alphabetic

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| heading5 = Positional systems by base

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| heading6 = Non-standard positional numeral systems

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| belowstyle = border-top:1px solid #aaa;border-bottom:1px solid #aaa; | below = List of numeral systems

}} 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. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.

The number the numeral represents is called its value.

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 of whole numbers gives every non zero whole number a unique representation as a finite sequence of digits, beginning by a non-zero digit. However, when decimal representation is used for the rational or real numbers, such numbers in general have an infinite number of representations, for example 2.31 can also be written as 2.310, 2.3100000, 2.309999999..., etc., all of which have the same meaning except for some scientific and other contexts where greater precision is implied by a larger number of figures shown.

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.


Numeral system sections
Intro  Main numeral systems  Generalized variable-length integers  See also  References  Sources  External links  

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