Decimal
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Numeral systems 

Hindu–Arabic numeral system 
East Asian 
Alphabetic 
Former 
Positional systems by base 
Nonstandard positional numeral systems 
List of numeral systems 
The decimal numeral system (also called baseten positional numeral system, and occasionally called denary) is the standard system for denoting integer and noninteger numbers. It is the extension to noninteger numbers of the Hindu–Arabic numeral system.^{[1]} The way of denoting numbers in the decimal system is often referred to as Decimal notation.^{[2]}
A decimal numeral, or just decimal, or, improperly decimal number, refers generally to the notation of a number in the decimal system, which contains a decimal separator (for example 10.00 or 3.14159). Sometimes these terms are used for any numeral in the decimal system. A decimal may also refer to any digit after the decimal separator, such as in "3.14 is the approximation of π to two decimals".
The numbers that may be represented in the decimal system are the decimal fractions, that is the fractions of the form a/10^{n}, where a is an integer, and n is a nonnegative integer.
The decimal system has been extended to infinite decimals, for representing any real number, by using an infinite sequence of digits after the decimal separator (see Decimal representation). In this context, the usual decimals are sometimes called terminating decimals. A repeating decimal, is an infinite decimal, that, after some place repeats indefinitely the same sequence of digits (for example 5.123144144144144... = 5.123144).^{[3]} An infinite decimal represents a rational number if and only if it is a repeating decimal or is a finite decimal.
Contents
Origin
Many numeral systems of ancient civilisations, use ten and its powers for representing numbers, probably because there are ten fingers on two hands and people started counting by using their fingers. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. 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 the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some noninteger numbers, called decimal fractions or decimal numbers for forming the decimal numeral system.
Decimal notation
For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign "–". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;^{[4]} the decimal separator is the dot "." in many countries (including English speaking ones), but may be a comma "," in other countries (mainly in Europe).
For representing a nonnegative number, a decimal consists of
 either a (finite) sequence of digits such as 2017, or in full generality,

 (in this case, the (entire) decimal represents an integer)
 or two sequence of digits separated by a decimal mark such as 3.14159, 15.00, or in full generality
It is generally assumed that, if m > 0, the first digit a_{m} is not zero, but, 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 b_{n} =0, it may be removed, and conversely, trailing zeros may be added without changing the represented number. For example, 15 = 15.0 = 15.00 and 5.2 = 5.20 = 5.200. sometimes, the unnecessary zeros are used for indicating the accuracy of a measure. For example, 15.00m may indicate that the measurement error is less than one centimeter, while 15m may mean that the length is roughly fifteen meters, and that the error may exceed 10 cm.
For representing a negative number, a minus sign is placed before a_{m}.
The numeral represents the number
Therefore, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal system is a positional numeral system
Decimal fractions
The numbers that are represented by decimal numeral are the decimal fractions (sometimes called decimal numbers), that is, the rational numbers that may be expressed as a fraction, the denominator of which is a power of ten.^{[5]} For example, the numerals represent the fractions 8/10, 1489/100, 24/100000. More generally, a decimal with n digits after the separator represents the fraction with denominator 10^{n}, whose numerator is the integer obtained by removing the separator.
Expressed as a fully reduced fraction, the decimal numbers are those, whose denominator is a product of a power of 2 by a power of 5. Thus the smallest denominators of decimal numbers are
The integer part, or integral part of a decimal is the integer written to the left of the decimal separator (see also truncation). For a nonnegative decimal, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the fractional 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 in computing, that the integer part is not written (for example .1234, instead of 0.1234). In normal writing, this is generally avoided because of the risk of confusion between the decimal mark and other punctuation.
Real number approximation
Decimal numerals do not allow an exact representation for all real numbers, so e.g. for the real number π. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates the real π, being less than 10^{5} off; and so decimals are widely used in science, engineering and everyday life.
More precisely, for every real number x, and every positive integer n, there are two decimals l and u, with at most n digits after the decimal mark, such that l ≤ x ≤ u and (u – l) = 10^{n}.
Numbers are very often obtained as the result of a measurement. As measurements are generally afflicted with some measurement error with a known upper bound, the result of a measurement is well represented by a decimal with n digits after the decimal mark, as soon as the absolute measurement error is bounded from above by 10^{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 real number, the numeral 0.080 is to suggest 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 also significant figures).
Infinite decimal expansion
For a real number x, let us denote [x]_{n} greatest decimal that is not greater than x, and has n digits after the decimal mark. If d_{n} is the last digit of [x]_{n}, it may be shown that [x]_{n} may be obtained simply by writing d_{n} on the right of [x]_{n–1}. In other words, [x]_{n–1} and [x]_{n} differ only by the last digit of [x]_{n}, and one has
 [x]_{n} = [x]_{0}.d_{1}d_{2}...d_{n}.
One may increase n indefinitely, and this is a direct consequence of the definition of a limit that x is the limit of [x]_{n} when n tends to the infinity. This is written as
 x = [x]_{0}.d_{1}d_{2}...d_{n}...,
and is called the infinite decimal expansion of x
Conversely, for any integer [x]_{0} and any infinite sequence of digits (d_{n}), the infinite expression [x]_{0}.d_{1}d_{2}...d_{n}... is the infinite decimal expansion of a unique real number x, except if all d_{n} equal 9 for n large enough.
If all d_{n} equal 9 for n large enough, the limit of the sequence of the [x]_{0}.d_{1}d_{2}...d_{n} is the decimal fraction obtained by replacing the last d_{n} that is not a 9, by d_{n} – 1, and replacing all subsequent 9 by 0 (see 0.999...).
In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Decimal fractions have exactly two infinite decimal expansions, one containing only 0 after some place, which is obtained by the above definition of [x]_{n}, and the other containing only 9 after some place, which is obtained by replacing "not greater" by "less" in the definition of [x]_{n}.
Rational numbers
The long division allows computing the infinite decimal expansion of a rational number. If the rational number is a decimal fraction, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many 0. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainder 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 a repeating decimal. For example,
 1/81 = 0.012345679012... (with 012345679 repeating).
Conversely, every eventually repeating sequence of digit is the infinite decimal expansion of a rational number. This is a consequence of the fact that the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number. For example,
Decimal computation
Modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such as the ENIAC or the IBM 650, used decimal representation internally).^{[6]} 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 binarycoded decimal,^{[7]}^{[8]} especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for FloatingPoint Arithmetic).^{[9]}
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).^{[10]}^{[11]} See Arbitraryprecision arithmetic for exact calculations.
History
Many ancient cultures calculated with numerals based on ten, sometimes argued due to human hands typically having ten digits.^{[12]} Egyptian hieroglyphs, in evidence since around 3000 BC, used a purely decimal system,^{[13]} just as the Cretan hieroglyphs (ca. 1625−1500 BC) of the Minoans whose numerals are closely based on the Egyptian model.^{[14]}^{[15]} The decimal system was handed down to the consecutive Bronze Age cultures of Greece, including Linear A (ca. 18th century BC−1450 BC) and Linear B (ca. 1375−1200 BC) — the number system of classical Greece also used powers of ten, including, like the Roman numerals did, an intermediate base of 5.^{[16]} Notably, the polymath Archimedes (ca. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 10^{8}^{[16]} and later led the German mathematician Carl Friedrich Gauss to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery.^{[17]} The Hittites hieroglyphs (since 15th century BC), just like the Egyptian and early numerals in Greece, was strictly decimal.^{[18]}
Some ancient texts like the Vedas dating back to 1900–1700 BCE mention decimals and mathematical decimal fractions.^{[citation needed]}
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 nonpositional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20, to 90, 100, 200, to 900, 1000, 2000, 3000, 4000, to 10,000.^{[19]} The world's earliest positional decimal system was the Chinese rod calculus.^{[20]}
History of decimal fractions
Decimal fractions were first developed and used by the Chinese in the end of 4th century BC,^{[21]} and then spread to the Middle East and from there to Europe.^{[20]}^{[22]} The written Chinese decimal fractions were nonpositional.^{[22]} However, counting rod fractions were positional.^{[20]}
Qin Jiushao in his book Mathematical Treatise in Nine Sections (1247) denoted 0.96644 by




 寸
 096644



^{[23]}
J. Lennart Berggren notes that positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'lHasan alUqlidisi written in the 10th century.^{[24]} The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them.^{[25]} The Persian mathematician Jamshīd alKāshī claimed to have discovered decimal fractions himself in the 15th century.^{[24]} Al Khwarizmi introduced fraction to Islamic countries in the early 9th century, his fraction presentation was an exact copy of traditional Chinese mathematical fraction from Sunzi Suanjing.^{[20]} This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century Abu'lHasan alUqlidisi and 15th century Jamshīd alKāshī's work "Arithmetic Key".^{[20]}^{[26]}
A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century.^{[27]}
Natural languages
The ingenious method of expressing every possible number using a set of ten symbols emerged in India. Several Indian languages show a straightforward decimal system. Many IndoAryan and Dravidian languages have numbers between 10 and 20 expressed in a regular pattern of addition to 10.^{[28]}
The Hungarian 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 as tenone and 23 as twotenthree, and 89,345 is expressed as 8 (ten thousands) 万 9 (thousand) 千 3 (hundred) 百 4 (tens) 十 5 is found in Chinese, and in Vietnamese with a few irregularities. Japanese, Korean, and 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. For example, in English 11 is "eleven" not "tenone" or "oneteen".
Incan languages such as Quechua and Aymara have an almost straightforward decimal system, in which 11 is expressed as ten with one and 23 as twoten with three.
Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.^{[29]}
Other bases
Units of information 
Some cultures do, or did, use other bases of numbers.
 PreColumbian Mesoamerican cultures such as the Maya used a base20 system (perhaps based on using all twenty fingers and toes).
 The Yuki language in California and the Pamean languages^{[30]} in Mexico have octal (base8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.^{[31]}
 The existence of a nondecimal 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 tencount or tentywise), such would be expected if normal counting is not decimal, and unusual if it were.^{[32]}^{[33]} Where this counting system is known, it is based on the long hundred of 120 in number, and a long thousand of 1200 in number. The descriptions like 'long' only appear after the small hundred of 100 in number appeared with the Christians. Gordon's Introduction to Old Norse p 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' is translated to 200, and the cognate to 'two hundred' is translated at 240. Goodare details the use of the long hundred in Scotland in the Middle Ages, giving examples, 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 hundredlike 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'.^{[citation needed]}
 Many or all of the Chumashan languages originally used a base4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.^{[34]}
 Many languages^{[35]} use quinary (base5) number systems, including Gumatj, Nunggubuyu,^{[36]} Kuurn Kopan Noot^{[37]} and Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
 Some Nigerians use duodecimal systems.^{[38]} So did some small communities in India and Nepal, as indicated by their languages.^{[39]}
 The Huli language of Papua New Guinea is reported to have base15 numbers.^{[40]} Ngui means 15, ngui ki means 15 × 2 = 30, and ngui ngui means 15 × 15 = 225.
 UmbuUngu, also known as Kakoli, is reported to have base24 numbers.^{[41]} Tokapu means 24, tokapu talu means 24 × 2 = 48, and tokapu tokapu means 24 × 24 = 576.
 Ngiti is reported to have a base32 number system with base4 cycles.^{[35]}
 The Ndom language of Papua New Guinea is reported to have base6 numerals.^{[42]} Mer means 6, mer an thef means 6 × 2 = 12, nif means 36, and nif thef means 36×2 = 72.
See also
References
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 ^ Lam Lay Yong & Ang Tian Se (2004) Fleeting Footsteps. Tracing the Conception of Arithmetic and Algebra in Ancient China, Revised Edition, World Scientific, Singapore.
 ^ The viniculum (overline) in 5.123144 indicates that the '144' sequence repeats itself indefinitely, i.e. 144144144144.... 5.123
 ^ In some countries, such as Arab speaking ones, other glyphs are used for the digits
 ^ "Decimal Fraction". Encyclopedia of Mathematics. Retrieved 20130618.
 ^ 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.
 ^ Schmid, Hermann (1983) [1974]. Decimal Computation (1 (reprint) ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company. ISBN 0898743184.
 ^ Schmid, Hermann (1974). Decimal Computation (1st ed.). Binghamton, New York, USA: John Wiley & Sons. ISBN 047176180X.
 ^ Decimal FloatingPoint: Algorism for Computers, Cowlishaw, M. F., Proceedings 16th IEEE Symposium on Computer Arithmetic, ISBN 076951894X, pp104111, IEEE Comp. Soc., June 2003
 ^ Decimal Arithmetic – FAQ
 ^ Decimal FloatingPoint: Algorism for Computers, Cowlishaw, M. F., Proceedings 16th IEEE Symposium on Computer Arithmetic (ARITH 16), ISBN 076951894X, pp. 104–111, IEEE Comp. Soc., June 2003
 ^ Dantzig, Tobias (1954), Number / The Language of Science (4th ed.), The Free Press (Macmillan Publishing Co.), p. 12, ISBN 0029069904
 ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0140099190, pp. 200–213 (Egyptian Numerals)
 ^ Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, ISBN 9780486421650, p. 50
 ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0140099190, pp. 213–218 (Cretan numerals)
 ^ ^{a} ^{b} Greek numerals
 ^ Menninger, Karl: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, ISBN 3525407254, pp. 150–153
 ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0140099190, pp. 218f. (The Hittite hieroglyphic system)
 ^ Lam Lay Yong et al. The Fleeting Footsteps pp 137–139
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} Lam Lay Yong, "The Development of Hindu–Arabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p38, Kurt Vogel notation
 ^ "Ancient bamboo slips for calculation enter world records boo". The Institute of Archaeology, Chinese Academy of Social Sciences. Retrieved 10 May 2017.
 ^ ^{a} ^{b} Joseph Needham (1959). "Decimal System". Science and Civilisation in China, Volume III, Mathematics and the Sciences of the Heavens and the Earth. Cambridge University Press.
 ^ JeanClaude Martzloff, A History of Chinese Mathematics, Springer 1997 ISBN 3540337822
 ^ ^{a} ^{b} Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Katz, Victor J. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 530. ISBN 9780691114859.
 ^ 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.
 ^ Lam Lay Yong, "A Chinese Genesis, Rewriting the history of our numeral system", Archive for History of Exact Science 38: 101–108.
 ^ B. L. van der Waerden (1985). A History of Algebra. From Khwarizmi to Emmy Noether. Berlin: SpringerVerlag.
 ^ "Indian numerals". Ancient Indian mathematics. Retrieved 20150522.
 ^ Azar, Beth (1999). "English words may hinder math skills development". American Psychology Association Monitor. 30 (4). Archived from the original on 20071021.
 ^ Avelino, Heriberto (2006). "The typology of Pame number systems and the limits of Mesoamerica as a linguistic area" (PDF). Linguistic Typology. 10 (1): 41–60. doi:10.1515/LINGTY.2006.002.
 ^ Marcia Ascher. "Ethnomathematics: A Multicultural View of Mathematical Ideas". The College Mathematics Journal. Retrieved 20070413.

^ McClean, R. J. (July 1958), "Observations on the Germanic numerals", German Life and Letters, 11 (4): 293–299, doi:10.1111/j.14680483.1958.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 preand protoGermanic", The Journal of English and Germanic Philology, 86 (4): 487–495, JSTOR 27709904.
 ^ There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca. 1819. "Chumashan Numerals" by Madison S. Beeler, in Native American Mathematics, edited by Michael P. Closs (1986), ISBN 0292755317.
 ^ ^{a} ^{b} Hammarström, Harald (17 May 2007). "Rarities in Numeral Systems". In Wohlgemuth, Jan; Cysouw, Michael. Rethinking Universals: How rarities affect linguistic theory (PDF). Empirical Approaches to Language Typology. 45. Berlin: Mouton de Gruyter (published 2010). Archived from the original (PDF) on 19 August 2007.
 ^ Harris, John (1982). Hargrave, Susanne, ed. Facts and fallacies of aboriginal number systems (PDF). Work Papers of SILAAB Series B. 8. pp. 153–181. Archived from the original (PDF) on 20070831.
 ^ Dawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.
 ^ Matsushita, Shuji (1998). Decimal vs. Duodecimal: An interaction between two systems of numeration. 2nd Meeting of the AFLANG, October 1998, Tokyo. Archived from the original on 20081005. Retrieved 20110529.
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External links
 Decimal arithmetic FAQ
 Cultural Aspects of Young Children's Mathematics Knowledge