Adecimal numeral (also often justdecimal or, less correctly,decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by adecimal separator (usually "." or "," as in25.9703 or3,1415).[3]Decimal may also refer specifically to the digits after the decimal separator, such as in "3.14 is the approximation ofπ totwo decimals". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value.
The numbers that may be represented in the decimal system are thedecimal fractions. That is,fractions of the forma/10n, wherea is an integer, andn is anon-negative integer. Decimal fractions also result from the addition of an integer and afractional part; the resulting sum sometimes is called afractional number.
Decimals are commonly used toapproximate real numbers. By increasing the number of digits after the decimal separator, one can make theapproximation errors as small as one wants, when one has a method for computing the new digits.
Originally and in most uses, a decimal has only a finite number of digits after the decimal separator. However, the decimal system has been extended toinfinite decimals for representing anyreal number, by using aninfinite sequence of digits after the decimal separator (seedecimal representation). In this context, the usual decimals, with a finite number of non-zero digits after the decimal separator, are sometimes calledterminating decimals. Arepeating decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g.,5.123144144144144... = 5.123144).[4] An infinite decimal represents arational number, thequotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.
Ten digits on two hands, the possible origin of decimal counting
Manynumeral systems of ancient civilizations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly theEgyptian numerals, then theBrahmi numerals,Greek numerals,Hebrew numerals,Roman numerals, andChinese numerals.[5] Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of theHindu–Arabic numeral system for representingintegers. This system has been extended to represent some non-integer numbers, calleddecimal fractions ordecimal numbers, for forming thedecimal numeral system.[5]
either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer:
or a decimal mark separating two sequences of digits (such as "20.70828")
.
Ifm > 0, that is, if the first sequence contains at least two digits, it is generally assumed that the first digitam is not zero. In some circumstances it may be useful to have one or more 0's on the left; this does not change the value represented by the decimal: for example,3.14 = 03.14 = 003.14. Similarly, if the final digit on the right of the decimal mark is zero—that is, ifbn = 0—it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number;[note 1] for example,15 = 15.0 = 15.00 and5.2 = 5.20 = 5.200.
For representing anegative number, a minus sign is placed befoream.
The numeral represents the number
.
Theinteger part orintegral part of a decimal numeral is the integer written to the left of the decimal separator (see alsotruncation). For a non-negative decimal numeral, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is thefractional part, which equals the difference between the numeral and its integer part.
When the integral part of a numeral is zero, it may occur, typically incomputing, that the integer part is not written (for example,.1234, instead of0.1234). In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation.
In brief, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal system is apositional numeral system.
Decimal fractions (sometimes calleddecimal numbers, especially in contexts involving explicit fractions) are therational numbers that may be expressed as afraction whosedenominator is apower of ten.[8] For example, the decimal expressions represent the fractions8/10,1489/100,79/100000,+1618/1000 and+314159/100000, and therefore denote decimal fractions. An example of a fraction that cannot be represented by a decimal expression (with a finite number of digits) is1/3, 3 not being a power of 10.
More generally, a decimal withn digits after theseparator (a point or comma) represents the fraction with denominator10n, whose numerator is the integer obtained by removing the separator.
It follows that a number is a decimal fractionif and only if it has a finite decimal representation.
Expressed asfully reduced fractions, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are
Decimal numerals do not allow an exact representation for allreal numbers. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximatesπ, being less than 10−5 off; so decimals are widely used inscience,engineering and everyday life.
More precisely, for every real numberx and every positive integern, there are two decimalsL andu with at mostn digits after the decimal mark such thatL ≤x ≤u and(u −L) = 10−n.
Numbers are very often obtained as the result ofmeasurement. As measurements are subject tomeasurement uncertainty with a knownupper bound, the result of a measurement is well-represented by a decimal withn digits after the decimal mark, as soon as the absolute measurement error is bounded from above by10−n. In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the error bounds. For example, although 0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see alsosignificant figures).
For areal numberx and an integern ≥ 0, let[x]n denote the (finite) decimal expansion of the greatest number that is not greater thanx that has exactlyn digits after the decimal mark. Letdi denote the last digit of[x]i. It is straightforward to see that[x]n may be obtained by appendingdn to the right of[x]n−1. This way one has
[x]n = [x]0.d1d2...dn−1dn,
and the difference of[x]n−1 and[x]n amounts to
,
which is either 0, ifdn = 0, or gets arbitrarily small asn tends to infinity. According to the definition of alimit,x is the limit of[x]n whenn tends toinfinity. This is written asor
x = [x]0.d1d2...dn...,
which is called aninfinite decimal expansion ofx.
Conversely, for any integer[x]0 and any sequence of digits the (infinite) expression[x]0.d1d2...dn... is aninfinite decimal expansion of a real numberx. This expansion is unique if neither alldn are equal to 9 nor alldn are equal to 0 forn large enough (for alln greater than some natural numberN).
If alldn forn >N equal to 9 and[x]n = [x]0.d1d2...dn, the limit of the sequence is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.:dN, bydN + 1, and replacing all subsequent 9s by 0s (see0.999...).
Any such decimal fraction, i.e.:dn = 0 forn >N, may be converted to its equivalent infinite decimal expansion by replacingdN bydN − 1 and replacing all subsequent 0s by 9s (see0.999...).
In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of[x]n, and the other containing only 9s after some place, which is obtained by defining[x]n as the greatest number that isless thanx, having exactlyn digits after the decimal mark.
Long division allows computing the infinite decimal expansion of arational number. If the rational number is adecimal fraction, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has arepeating decimal. For example,
1/81 = 0.012345679012... (with the group 012345679 indefinitely repeating).
The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational.
For example, ifx is
0.4156156156...
then 10,000x is
4156.156156156...
and 10x is
4.156156156...
so 10,000x − 10x, i.e. 9,990x, is
4152.000000000...
andx is
4152/9990
or, dividing both numerator and denominator by 6,692/1665.
Most moderncomputer hardware and software systems commonly use abinary representation internally (although many early computers, such as theENIAC or theIBM 650, used decimal representation internally).[9]For external use by computer specialists, this binary representation is sometimes presented in the relatedoctal orhexadecimal 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 ofbinary-coded decimal,[10][11] especially in database implementations, but there are other decimal representations in use (includingdecimal floating point such as in newer revisions of theIEEE 754 Standard for Floating-Point Arithmetic).[12]
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 have no finite binary fractional representation; and is generally impossible for multiplication (or division).[13][14] SeeArbitrary-precision arithmetic for exact calculations.
The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1,000, 2,000, 3,000, 4,000, to 10,000.[26]The world's earliest positional decimal system was the Chineserod calculus.[27]
The world's earliest positional decimal system Upper row vertical form Lower row horizontal form
Starting from the 2nd century BCE, some Chinese units for length were based on divisions into ten; by the 3rd century CE these metrological units were used to express decimal fractions of lengths, non-positionally.[28] Calculations with decimal fractions of lengths wereperformed using positional counting rods, as described in the 3rd–5th century CESunzi Suanjing. The 5th century CE mathematicianZu Chongzhi calculated a 7-digitapproximation ofπ.Qin Jiushao's bookMathematical Treatise in Nine Sections (1247) explicitly writes a decimal fraction representing a number rather than a measurement, using counting rods.[29] The number 0.96644 is denoted
寸
.
Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East.[27]
Al-Khwarizmi introduced fractions to Islamic countries in the early 9th century CE, written with a numerator above and denominator below, without a horizontal bar. This form of fraction remained in use for centuries.[27][30]
Positional decimal fractions appear for the first time in a book by the Arab mathematicianAbu'l-Hasan al-Uqlidisi written in the 10th century.[31] The Jewish mathematicianImmanuel Bonfils used decimal fractions around 1350 but did not develop any notation to represent them.[32] The Persian mathematicianJamshid al-Kashi used, and claimed to have discovered, decimal fractions in the 15th century.[31]
A forerunner of modern European decimal notation was introduced bySimon Stevin in the 16th century. Stevin's influential bookletDe Thiende ("the art of tenths") was first published in Dutch in 1585 and translated into French asLa Disme.[33]
John Napier introduced using the period (.) to separate the integer part of a decimal number from the fractional part in his book on constructing tables of logarithms, published posthumously in 1620.[34]: p. 8, archive p. 32)
A method of expressing every possiblenatural number using a set of ten symbols emerged in India.[35] Several Indian languages show a straightforward decimal system.Dravidian languages have numbers between 10 and 20 expressed in a regular pattern of addition to 10.[36]
TheHungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty").
A straightforward decimal rank system with a word for each order (10十, 100百, 1000千, 10,000万), and in which 11 is expressed asten-one and 23 astwo-ten-three, and 89,345 is expressed as 8 (ten thousands)万 9 (thousand)千 3 (hundred)百 4 (tens)十 5 is found inChinese, and inVietnamese with a few irregularities.Japanese,Korean, andThai 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. For example, in English 11 is "eleven" not "ten-one" or "one-teen".
Incan languages such asQuechua andAymara have an almost straightforward decimal system, in which 11 is expressed asten with one and 23 astwo-ten with three.
Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.[37]
TheYuki language inCalifornia and the Pamean languages[38] inMexico haveoctal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.[39]
The existence of a non-decimal base in the earliest traces of the Germanic languages is attested by the presence of words and glosses meaning that the count is in decimal (cognates to "ten-count" or "tenty-wise"); such would be expected if normal counting is not decimal, and unusual if it were.[40][41] Where this counting system is known, it is based on the "long hundred" = 120, and a "long thousand" of 1200. The descriptions like "long" only appear after the "small hundred" of 100 appeared with the Christians. Gordon'sIntroduction to Old NorseArchived 2016-04-15 at theWayback Machine p. 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240.Goodare[permanent dead link] details the use of the long hundred in Scotland in the Middle Ages, giving examples such as calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.[42][43]
Many or all of theChumashan languages originally used abase-4 counting system, in which the names for numbers were structured according to multiples of 4 and16.[44]
Umbu-Ungu, also known as Kakoli, is reported to havebase-24 numbers.[51]Tokapu means 24,tokapu talu means 24 × 2 = 48, andtokapu tokapu means 24 × 24 = 576.
Ngiti is reported to have abase-32 number system with base-4 cycles.[45]
TheNdom language ofPapua New Guinea is reported to havebase-6 numerals.[52]Mer means 6,mer an thef means 6 × 2 = 12,nif means 36, andnif thef means 36×2 = 72.
^Sometimes, the extra zeros are used for indicating theaccuracy of a measurement. For example, "15.00 m" may indicate that the measurement error is less than one centimetre (0.01 m), while "15 m" may mean that the length is roughly fifteen metres and that the error may exceed 10 centimetres.
^"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.
^Schmid, Hermann (1983) [1974].Decimal Computation (1 (reprint) ed.). Malabar, Florida: Robert E. Krieger Publishing Company.ISBN0-89874-318-4.
^Bisht, R. S. (1982), "Excavations at Banawali: 1974–77", in Possehl, Gregory L. (ed.), HarappanCivilisation: A Contemporary Perspective, New Delhi: Oxford and IBH Publishing Co., pp. 113–24
^Georges Ifrah:From One to Zero. A Universal History of Numbers, Penguin Books, 1988,ISBN0-14-009919-0, pp. 200–13 (Egyptian Numerals)
^Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002,ISBN978-0-486-42165-0, p. 50
^Georges Ifrah:From One to Zero. A Universal History of Numbers, Penguin Books, 1988,ISBN0-14-009919-0, pp. 213–18 (Cretan numerals)
^Menninger, Karl:Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979,ISBN3-525-40725-4, pp. 150–53
^Georges Ifrah:From One to Zero. A Universal History of Numbers, Penguin Books, 1988,ISBN0-14-009919-0, pp. 218f. (The Hittite hieroglyphic system)
^Lam Lay Yong et al. The Fleeting Footsteps pp. 137–39
^abcLam Lay Yong, "The Development of Hindu–Arabic and Traditional Chinese Arithmetic",Chinese Science, 1996 p. 38, Kurt Vogel notation
^Joseph Needham (1959). "19.2 Decimals, Metrology, and the Handling of Large Numbers".Science and Civilisation in China. Vol. III, "Mathematics and the Sciences of the Heavens and the Earth". Cambridge University Press. pp. 82–90.
^Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997ISBN3-540-33782-2
^Lay Yong, Lam. "A Chinese Genesis, Rewriting the history of our numeral system".Archive for History of Exact Sciences.38:101–08.
^abBerggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Katz, Victor J. (ed.).The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 530.ISBN978-0-691-11485-9.
^Gandz, S.: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon (c. 1350), Isis 25 (1936), 16–45.
^B. L. van der Waerden (1985).A History of Algebra. From Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.
^Napier, John (1889) [1620].The Construction of the Wonderful Canon of Logarithms. Translated by Macdonald, William Rae. Edinburgh: Blackwood & Sons – via Internet Archive.In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many cyphers after it as there are figures after the period.
^Marcia Ascher. "Ethnomathematics: A Multicultural View of Mathematical Ideas". The College Mathematics Journal.JSTOR2686959.
^McClean, R. J. (July 1958), "Observations on the Germanic numerals",German Life and Letters,11 (4):293–99,doi:10.1111/j.1468-0483.1958.tb00018.x,Some of the Germanic languages appear to show traces of an ancient blending of the decimal with the vigesimal system.
^Voyles, Joseph (October 1987), "The cardinal numerals in pre-and proto-Germanic",The Journal of English and Germanic Philology,86 (4):487–95,JSTOR27709904.
^Stevenson, W.H. (1890). "The Long Hundred and its uses in England".Archaeological Review. December 1889:313–22.
^Poole, Reginald Lane (2006).The Exchequer in the twelfth century : the Ford lectures delivered in the University of Oxford in Michaelmas term, 1911. Clark, NJ: Lawbook Exchange.ISBN1-58477-658-7.OCLC76960942.
^There is a surviving list ofVentureño language number words up to 32 written down by a Spanish priest ca. 1819. "Chumashan Numerals" by Madison S. Beeler, inNative American Mathematics, edited by Michael P. Closs (1986),ISBN0-292-75531-7.
^abHammarström, Harald (17 May 2007). "Rarities in Numeral Systems". In Wohlgemuth, Jan; Cysouw, Michael (eds.).Rethinking Universals: How rarities affect linguistic theory(PDF). Empirical Approaches to Language Typology. Vol. 45. Berlin: Mouton de Gruyter (published 2010). Archived fromthe original(PDF) on 19 August 2007.
^Mazaudon, Martine (2002). "Les principes de construction du nombre dans les langues tibéto-birmanes". In François, Jacques (ed.).La Pluralité(PDF). Leuven: Peeters. pp. 91–119.ISBN90-429-1295-2. Archived fromthe original(PDF) on 2016-03-28. Retrieved2014-09-12.
