Numeral system

:Occasionally the term "number system" is used for this concept, but that is also the name of an altogether different concept; see number system.

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A numeral is a symbol or group of symbols that represents a number. Numerals differ from numbers just as words differ from the things they refer to. The symbols "11", "eleven" and "XI" are different numerals, all representing the same number. This article treats the various systems of numerals. See also number names.

Related Topics:
Number - Number names

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A numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the context that allows the numeral "11" to be interpreted as the binary numeral for three, the decimal numeral for eleven, or other numbers in other bases.

Related Topics:
Binary - Decimal - Base

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Ideally, a numeral system will:

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• Represent a useful set of numbers (e.g. all whole numbers, integers, or real 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 whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic. However, when decimal representation is used for the rational or real numbers, the representation is no longer unique: many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that recurs, such as 2.30999999... . Numerals which terminate have no non-zero digits after a given position. For example, numerals like 2.31 and 2.310 are taken to be the same, except in the natural sciences where differing precision is denoted.

Related Topics:
Decimal - Finite - Sequence - Digit - Algorithm - Rational - Recurs

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Numeral systems are sometimes called number systems, but that name is misleading: different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc., are not the topic of this article.

Related Topics:
Number system - Real number - Complex number - ''p''-adic numbers

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Numeral system: Types of numeral systems ►