Decimal
8214
225828654
2008-07-15T16:17:49Z
Epbr123
1395162
Reverted edits by [[Special:Contributions/OrgasGirl|OrgasGirl]] to last version by BOTarate (using [[WP:HG|Huggle]])
{{otheruses}}
{{Table Numeral Systems}}
The '''decimal''' ('''base ten''' or occasionally '''denary''') [[numeral system]] has [[10 (number)|ten]] as its [[Base (mathematics)|base]]. It is the most widely used numeral system, perhaps because humans have ten digits over both hands.
== Decimal notation ==
Decimal notation is the writing of [[number]]s in the base-ten [[numeral system]], which uses various symbols (called [[numerical digit|digits]]) for no more than ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large. These digits are often used with a [[decimal separator]] which indicates the start of a fractional part, and with one of the sign symbols + (positive) or − (negative) in front of the numerals to indicate sign.
There are only two truly positional decimal systems in ancient civilization, the [[Counting rods|Chinese counting rods]] system and Hindu-Arabic numeric system, both required no more
than ten symbols. Other numeric systems require more or fewer symbols.
The [[algorism|decimal system]] is a [[positional notation|positional numeral system]]; it has positions for units, tens, hundreds, ''etc.'' The position of each digit conveys the multiplier (a power of ten) to be used with that digit—each position has a value ten times that of the position to its right.
[[10 (number)|Ten]] is the number which is the count of fingers and thumbs on both hands (or toes on the feet). In many languages the word [[digit]] or its translation is also the anatomical term referring to fingers and toes. In English, decimal (decimus < [[Latin|Lat.]]) means ''tenth'', decimate means ''reduce by a tenth'', and denary (denarius < Lat.) means ''the [[Unit of measurement|unit]] of ten''.
The symbols for the digits in common use around the [[globe]] today are called [[Hindu-Arabic numerals|Arabic numerals]] by Europeans and [[Indian numerals]] by Arabs, the two groups' terms both referring to the culture from which they learned the system. However, the symbols used in different areas are not identical; for instance, Western Arabic numerals (from which the European numerals are derived) differ from the forms used by other Arab cultures.
=== Alternative notations ===
Some cultures do, or used to, use other numeral systems, including [[pre-Columbian]] [[Mesoamerica]]n cultures such as the [[Maya numerals|Maya]], who use a [[vigesimal]] system (using all twenty fingers and [[toe]]s), some [[Nigeria]]ns who use several [[duodecimal]] (base 12) systems, the [[Babylonia]]ns, who used [[sexagesimal]] (base 60), and the [[Yuki tribe|Yuki]], who reportedly used [[octal]] (base 8).
[[Computer]] hardware and software systems commonly use a [[Binary numeral system|binary representation]], internally. 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 the equivalent decimal values for presentation to and manipulation by humans.
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 [[binary-coded decimal]], but there are other decimal representations in use (see [[IEEE 754r]]), especially in database implementations. Decimal arithmetic is used in computers so that decimal fractional results can be computed exactly, which is not possible using a binary fractional representation.
This is often important for financial and other calculations<ref>http://www2.hursley.ibm.com/decimal/decifaq.html</ref>.
=== Decimal fractions ===
A '''decimal fraction''' is a [[Fraction (mathematics)|fraction]] where the [[denominator]] is a [[exponentiation|power]] of ten.
Decimal fractions are commonly expressed without a denominator, the [[decimal separator]] being inserted into the numerator (with [[leading zero]]s added if needed), at the position from the right corresponding to the power of ten of the denominator. e.g., 8/10, 83/100, 83/1000, and 8/10000 are expressed as: 0'''.'''8, 0'''.'''83, 0'''.'''083, and 0'''.'''0008. In English-speaking and many Asian countries, a period ('''.''') is used as the decimal separator; in many other languages, a comma is used.
The '''integer part''' or '''integral part''' of a decimal number is the part to the left of the decimal separator (see also [[floor function]]). The part from the decimal separator to the right is the fractional part; if considered as a separate number, a zero is often written in front. Especially for negative numbers, we have to distinguish between the fractional part of the notation and the fractional part of the number itself, because the latter gets its own minus sign. It is usual for a decimal number whose [[absolute value]] is less than one to have a leading zero.
[[Trailing zero]]s after the decimal point are not necessary, although in science, engineering and [[statistics]] they can be retained to indicate a required precision or to show a level of confidence in the accuracy of the number: Whereas 0'''.'''080 and 0'''.'''08 are numerically equal, in engineering 0'''.'''080 suggests a measurement with an error of up to 1 part in two thousand (±0'''.'''0005), while 0'''.'''08 suggests a measurement with an error of up to 1 in two hundred (see ''[[Significant figures]]'').
=== Other rational numbers ===
Any [[rational number]] which cannot be expressed as a decimal fraction has a unique infinite decimal expansion ending with [[recurring decimal]]s.
Ten is the product of the first and third [[prime number]]s, is one greater than the square of the second prime number, and is one less than the fifth prime number. This leads to plenty of simple decimal fractions:
:1/2 = 0.5
:1/3 = 0.333333… (with 3 repeating)
:1/4 = 0.25
:1/5 = 0.2
:1/6 = 0.166666… (with 6 repeating)
:1/8 = 0.125
:1/9 = 0.111111… (with 1 repeating)
:1/10 = 0.1
:1/11 = 0.090909… (with 09 repeating)
:1/12 = 0.083333… (with 3 repeating)
:1/81 = 0.012345679012… (with 012345679 repeating)
Other prime factors in the denominator will give longer recurring [[sequence]]s, see for instance [[7 (number)|7]], [[13 (number)|13]].
That a rational number must have a [[finite set|finite]] or recurring decimal expansion can be seen to be a consequence of the [[long division]] [[algorithm]], in that there are only q-1 possible nonzero [[remainder]]s on division by q, so that the recurring pattern will have a period less than q. For instance to find 3/7 by long division:
<u> .4 2 8 5 7 1 4 ..</u>.
7 ) 3.0 0 0 0 0 0 0 0
<u> 2 8 </u> 30/7 = 4 r 2
2 0
<u> 1 4 </u> 20/7 = 2 r 6
6 0
<u> 5 6 </u> 60/7 = 8 r 4
4 0
<u> 3 5 </u> 40/7 = 5 r 5
5 0
<u> 4 9 </u> 50/7 = 7 r 1
1 0
<u> 7 </u> 10/7 = 1 r 3
3 0
<u> 2 8 </u> 30/7 = 4 r 2 (again)
2 0
etc
The converse to this observation is that every [[recurring decimal]] represents a rational number ''p''/''q''. This is a consequence of the fact the recurring part of a decimal representation is, in fact, an infinite [[geometric series]] which will sum to a rational number. For instance,
:<math>0.0123123123\cdots = \frac{123}{10000} \sum_{k=0}^\infty 0.001^k = \frac{123}{10000}\ \frac{1}{1-0.001} = \frac{123}{9990} = \frac{41}{3330}</math>
=== Real numbers ===
{{further|[[Decimal representation]]}}
Every [[real number]] has a (possibly infinite) decimal representation, i.e., it can be written as
:<math> x = \mathop{\rm sign}(x) \sum_{i\in\mathbb Z} a_i\,10^i</math>
where
* sign() is the [[sign function]],
* ''a<sub>i</sub>'' ∈ { 0,1,…,9 } for all ''i'' ∈ '''Z''', are its '''decimal digits''', equal to zero for all ''i'' greater than some number (that number being the [[common logarithm]] of |x|).
Such a sum converges as ''i'' decreases, even if there are infinitely many nonzero ''a<sub>i</sub>''.
[[Rational number]]s (e.g. p/q) with [[prime factor]]s in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique [[recurring decimal]] representation.
Consider those rational numbers which have only the factors 2 and 5 in the denominator, i.e. which can be written as p/(2<sup>a</sup>5<sup>b</sup>). In this case there is a terminating decimal representation. For instance 1/1=1, 1/2=0.5, 3/5=0.6, 3/25=0.12 and 1306/1250=1.0448. Such numbers are the only real numbers which don't have a unique decimal representation, as they can also be written as a representation that has a recurring 9, for instance 1=0.99999…, 1/2=0.499999…, etc.
This leaves the [[irrational number]]s. They also have unique infinite decimal representation, and can be characterised as the numbers whose decimal representations neither terminate nor recur.
So in general the decimal representation is unique, if one excludes representations that end in a recurring 9.
Naturally, the same [[trichotomy]] holds for other base-n [[Positional notation|positional numeral system]]s:
* Terminating representation: rational where the denominator divides some n<sup>k</sup>
* Recurring representation: other rational
* Non-terminating, non-recurring representation: irrational
and a version of this even holds for irrational-base numeration systems, such as [[golden mean base]] representation.
==History==
There follows a chronological list of recorded decimal writers.
===Decimal writers===
* ''c.'' 3500 - 2500 BC [[Elamite Empire|Elamites]] of [[Iran]] possibly used early forms of decimal system.<ref>http://www.chn.ir/english/eshownews.asp?no=1622</ref><ref>http://www.mpiwg-berlin.mpg.de/Preprints/P183.PDF</ref>
* ''c.'' 2900 BC [[Egypt]]ian hieroglyphs show counting in powers of 10 (1 million + 400,000 goats, ''etc.'') – see Ifrah, below
* ''c.'' 2600 BC [[Indus Valley Civilization]], earliest known physical use of decimal [[fractions]] in ancient weight system: 1/20, 1/10, 1/5, 1/2. See [[Ancient Indus Valley weights and measures]]
* ''c.'' 1400 BC [[History of China|Chinese]] writers show familiarity with the concept: for example, 547 is written 'Five hundred plus four decades plus seven of days' in some manuscripts
* ''c.'' 1200 BC In [[History of India|ancient India]], the [[Vedas|Vedic]] text ''[[Yajur-Veda]]'' states the [[Exponentiation|powers]] of 10, up to 10<sup>55</sup>
* ''c.'' 400 BC [[Pingala]] – develops the binary number system for Sanskrit prosody, with a clear mapping to the base-10 decimal system
* ''c.'' 250 BC [[Archimedes]] writes the ''[[Sand Reckoner]]'', which takes decimal calculation up to 10<sup>80,000,000,000,000,000</sup>
* ''c.'' 100–200 The ''[[Indian mathematics#Satkhandagama|Satkhandagama]]'' written in [[India]] – earliest use of decimal logarithms
* ''c.'' 476–550 [[Aryabhata]] – uses an alphabetic cipher system for numbers that used zero
* ''c.'' 598–670 [[Brahmagupta]] – explains the [[Hindu-Arabic numerals]] (modern number system) which uses decimal [[integer]]s, [[negative]] integers, and [[0 (number)|zero]]
* ''c.'' 780–850 [[Muḥammad ibn Mūsā al-Ḵwārizmī]] – first to expound on [[algorism]] outside [[India]]
* ''c.'' 920–980 [[Al-Uqlidisi|Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi]] – earliest known direct mathematical treatment of decimal fractions.
* ''c.'' 1300–1500 The [[Kerala School]] in [[South India]] – decimal [[floating point]] numbers
* 1548/49–1620 [[Simon Stevin#Decimal fractions|Simon Stevin]] – author of ''De Thiende'' ('the tenth')
*1561–1613 [[Bartholemaeus Pitiscus]] – (possibly) decimal point notation.
* 1550–1617 [[John Napier]] – use of decimal logarithms as a computational tool
* 1765 [[Johann Heinrich Lambert]] – discusses (with few if any proofs) patterns in decimal expansions of rational numbers and notes a connection with Fermat's little theorem in the case of prime denominators
*1800 [[Karl Friedrich Gauss]] – uses number theory to systematically explain patterns in recurring decimal expansions of rational numbers (e.g., the relation between period length of the recurring part and the denominator, which fractions with the same denominator have recurring decimal parts which are shifts of each other, like 1/7 and 2/7) and also poses questions which remain open to this day (e.g., a special case of [[Artin's conjecture on primitive roots]]: is 10 a generator modulo ''p'' for infinitely many primes ''p''?).
*1925 [[Louis Charles Karpinski]] – ''The History of Arithmetic'' <ref>''The History of Arithmetic'', [[Louis Charles Karpinski]], 200pp, Rand McNally & Company, 1925. </ref>
*1959 [[Werner Buchholz]] – ''Fingers or Fists? (The Choice of Decimal or Binary representation)''<ref>''Fingers or Fists? (The Choice of Decimal or Binary representation)'', [[Werner Buchholz]], Communications of the ACM, Vol. 2 #12, pp3–11, ACM Press, December 1959.</ref>
*1974 [[Hermann Schmid]] – ''Decimal Computation''<ref>''Decimal Computation'', [[Hermann Schmid]], John Wiley & Sons 1974 (ISBN 047176180X); reprinted in 1983 by Robert E. Krieger Publishing Company (ISBN 0898743184)</ref>
*2000 [[Georges Ifrah]] – ''The Universal History of Numbers: From Prehistory to the Invention of the Computer''<ref>''Histoire universelle des chiffres'', [[Georges Ifrah]], Robert Laffont, 1994 (The Universal History of Numbers: From prehistory to the invention of the computer, [[Georges Ifrah]], John Wiley and Sons Inc., 2000. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk)</ref>
*2003 [[Mike Cowlishaw]] – ''Decimal Floating-Point: Algorism for Computers''<ref>''Decimal Floating-Point: Algorism for Computers'', [[Mike Cowlishaw|Cowlishaw, M. F.]], Proceedings 16th IEEE Symposium on Computer Arithmetic, ISBN 0-7695-1894-X, pp104-111, IEEE Comp. Soc., June 2003</ref>.
==Natural languages==
A straightforward decimal system, in which 11 is expressed as ''ten-one'' and 23 as ''two-ten-three'', is found in [[Chinese language]]s except [[Wu (linguistics)|Wu]], and in [[Vietnamese language|Vietnamese]] with a few irregularities. [[Japanese language|Japanese]], [[Korean language|Korean]], and [[Thai language|Thai]] have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades.
Incan languages such as [[Quechua]] and [[Aymara language|Aymara]] have an almost straightforward decimal system, in which 11 is expressed as ''ten with one'' and 23 as ''two-ten with three''.
Some psychologists suggest irregularities of numerals in a language may hinder children's counting ability<ref>{{citation| last=Azar| first=Beth| year=1999| title=English words may hinder math skills development| url=http://www.apa.org/monitor/apr99/english.html |journal=American Psychology Association Monitor| volume=30| issue=4}}.</ref>.
==See also==
<div style="-moz-column-count:2; column-count:2;">
* [[Algorism]]
* [[Binary-coded decimal]]
* [[Decimal representation]]
* [[Decimal separator]]
* [[Dewey Decimal System]]
* [[Hindu-Arabic numeral system]]
* [[Numeral system]]
* [[Scientific notation]]
* [[SI prefix]]
* [[10 (number)]]
</div>
== References ==
{{reflist|2}}
== External links ==
* [http://www2.hursley.ibm.com/decimal/decifaq.html Decimal arithmetic FAQ]
* Tests: [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1352 Decimal Place Value] [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1353&CurriculumID=5 Sums] [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=739&CurriculumID=5 Fractions]
* [http://www.mathsisfun.com/worksheets/decimals.php Practice Decimal Arithmetic with Printable Worksheets]
* [http://www.mathsisfun.com/converting-decimals-fractions.html Converting Decimals to Fractions]
*[http://spot.colorado.edu/~gubermas/NCTM_pap.htm Cultural Aspects of Young Children's Mathematics Knowledge]
* [http://www2.hursley.ibm.com/decimal/decbibindex.html Decimal Bibliography]
[[Category:Elementary arithmetic]]
[[Category:Fractions]]
[[Category:Positional numeral systems| 10]]
[[ar:نظام عد عشري]]
[[be:Дзесятковая сістэма злічэння]]
[[be-x-old:Дзесятковая сыстэма зьлічэньня]]
[[ca:Nombre decimal]]
[[cs:Desítková soustava]]
[[da:Decimal]]
[[de:Dezimalsystem]]
[[el:Δεκαδικό σύστημα]]
[[es:Sistema decimal]]
[[eo:Dekuma sistemo]]
[[fr:Système décimal]]
[[ko:십진법]]
[[id:Sistem bilangan desimal]]
[[is:Tugakerfi]]
[[it:Sistema numerico decimale]]
[[he:השיטה העשרונית]]
[[ht:Sistèm desimal]]
[[lv:Decimālā skaitīšanas sistēma]]
[[hu:Tízes számrendszer]]
[[nl:Decimaal]]
[[ja:十進法]]
[[no:Titallsystemet]]
[[nn:Titalssystemet]]
[[pl:Dziesiętny system liczbowy]]
[[pt:Sistema de numeração decimal]]
[[qu:Chunkantin huchha llika]]
[[ru:Десятичная система счисления]]
[[simple:Decimal]]
[[sk:Desiatková číselná sústava]]
[[sl:Desetiški številski sistem]]
[[sr:Декадни систем]]
[[fi:Kymmenjärjestelmä]]
[[sv:Decimala talsystemet]]
[[th:เลขฐานสิบ]]
[[vi:Hệ thập phân]]
[[uk:Десяткова система числення]]
[[yi:דעצימאל]]
[[zh:十进制]]