Editing Decimal (section)

== Decimal computation ==

[[File:Decimal multiplication table.JPG|thumb|right|300px|Diagram of the world's earliest known multiplica&shy;tion table ({{circa|305 BCE}}) from the [[Warring States period]]]]

Most modern [[computer]] hardware and software systems commonly use a [[Binary numeral system|binary representation]] internally (although many early computers, such as the [[ENIAC]] or the [[IBM 650]], used decimal representation internally).<ref>"Fingers or Fists? (The Choice of Decimal or Binary Representation)", [[Werner Buchholz]], ''Communications of the ACM'', Vol. 2 #12, pp. 3–11, ACM Press, December 1959.</ref>
For external use by computer specialists, this binary representation is sometimes presented in the related [[octal]] or [[hexadecimal]] systems.

For most purposes, however, binary values are converted to or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.)

Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using some variant of [[binary-coded decimal]],<ref name="Schmid_1983">{{cite book |title=Decimal Computation |first=Hermann |author=Schmid<!-- General Electric Company, Binghamton, New York, US --> |author-link=Hermann Schmid (computer scientist) |orig-year=1974 |date=1983 |edition=1 (reprint) |publisher=Robert E. Krieger Publishing Company |location=Malabar, Florida |isbn=0-89874-318-4}}</ref><ref name="Schmid_1974">{{cite book |title=Decimal Computation |first=Hermann |author=Schmid<!-- General Electric Company, Binghamton, New York, USA --> |author-link=Hermann Schmid (computer scientist) |date=1974 |edition=1st |publisher=[[John Wiley & Sons]] |location=Binghamton, New York|isbn=0-471-76180-X |url-access=registration |url=https://archive.org/details/decimalcomputati0000schm }}</ref> especially in database implementations, but there are other decimal representations in use (including [[decimal floating point]] such as in newer revisions of the [[IEEE 754|IEEE 754 Standard for Floating-Point Arithmetic]]).<ref>''Decimal Floating-Point: Algorism for Computers'', [[Mike Cowlishaw|Cowlishaw, Mike F.]], Proceedings [[16th IEEE Symposium on Computer Arithmetic]], {{isbn|0-7695-1894-X}}, pp. 104–11, IEEE Comp. Soc., 2003</ref>

Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This is not possible in binary, because the negative powers of <math>10</math> have no finite binary fractional representation; and is generally impossible for multiplication (or division).<ref>{{Cite web |url=http://speleotrove.com/decimal/decifaq.html |title=Decimal Arithmetic – FAQ<!-- Bot generated title --> |access-date=2008-08-15 |archive-date=2009-04-29 |archive-url=https://web.archive.org/web/20090429044324/http://speleotrove.com/decimal/decifaq.html |url-status=live }}</ref><ref>[http://www.dec.usc.es/arith16/papers/paper-107.pdf Decimal Floating-Point: Algorism for Computers] {{Webarchive|url=https://web.archive.org/web/20031116234555/http://www.dec.usc.es/arith16/papers/paper-107.pdf |date=2003-11-16 }}, [[Cowlishaw]], M. F., ''Proceedings [[16th IEEE Symposium on Computer Arithmetic]]'' ([http://www.dec.usc.es/arith16/ ARITH&nbsp;16] {{Webarchive|url=https://web.archive.org/web/20100819005155/http://www.dec.usc.es/arith16/ |date=2010-08-19 }}), {{isbn|0-7695-1894-X}}, pp.&nbsp;104–11, IEEE Comp. Soc., June 2003</ref> See [[Arbitrary-precision arithmetic]] for exact calculations.