Decimal
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 This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation.
Numeral systems by culture 


Positional systems by base 
Decimal (10) 
Nonstandard positional numeral systems 
List of numeral systems 
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The decimal numeral system (also called base ten or occasionally denary) has ten as its base. It is the numerical base most widely used by modern civilizations.^{[1]}^{[2]}
Decimal notation often refers to a base10 positional notation such as the HinduArabic numeral system or rod calculus;^{[3]} however, it can also be used more generally to refer to nonpositional systems such as Roman or Chinese numerals which are also based on powers of ten.
Decimals also refer to decimal fractions, either separately or in contrast to vulgar fractions. In this context, a decimal is a tenth part, and decimals become a series of nested tenths. There was a notation in use like "tenthmetre", meaning the tenth decimal of the metre, currently an Angstrom. The contrast here is between decimals and vulgar fractions, and decimal divisions and other divisions of measures, like the inch. It is possible to follow a decimal expansion with a vulgar fraction; this is done with the recent divisions of the troy ounce, which has three places of decimals, followed by a trinary place.
Decimal notation
Decimal notation is the writing of numbers in a base10 numeral system. Examples are Greek numerals, Roman numerals, Brahmi numerals, and Chinese numerals, as well as the HinduArabic numerals used by speakers of many European languages. Roman numerals have symbols for the decimal powers (1, 10, 100, 1000) and secondary symbols for half these values (5, 50, 500). Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100 and another for 1000. Chinese numerals have symbols for 1–9, and additional symbols for powers of 10, which in modern usage reach 10^{72}.
However, when people who use HinduArabic numerals speak of decimal notation, they often mean not just decimal numeration, as above, but also decimal fractions, all conveyed as part of a positional system. Positional decimal systems include a zero and use symbols (called digits) for the ten values (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to represent any number, no matter how large or how small. These digits are often used with a decimal separator which indicates the start of a fractional part, and with a symbol such as the plus sign + (for positive) or minus sign − (for negative) adjacent to the numeral to indicate whether it is greater or less than zero, respectively.
Positional notation uses positions for each power of ten: units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier (power of ten) multiplied with that digit—each position has a value ten times that of the position to its right. There were at least two presumably independent sources of positional decimal systems in ancient civilization: the Chinese counting rod system and the HinduArabic numeral system (the latter descended from Brahmi numerals).
Ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). The English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal (decimus < Lat.) means tenth, decimate means reduce by a tenth, and denary (denarius < Lat.) means the unit of ten.
The symbols for the digits in common use around the globe today are called 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.
Decimal fractions
A decimal fraction is a fraction the denominator of which is a power of ten.^{[4]}
Decimal fractions are commonly expressed without a denominator, the decimal separator being inserted into the numerator (with leading zeros 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 Englishspeaking, some Latin American and many Asian countries, a period (.) or raised period (·) is used as the decimal separator; in many other countries, particularly in Europe, 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 truncation.) The part from the decimal separator to the right is the fractional part. It is usual for a decimal number that consists only of a fractional part (mathematically, a proper fraction) to have a leading zero in its notation (its numeral). This helps disambiguation between a decimal sign and other punctuation, and especially when the negative number sign is indicated, it helps visualize the sign of the numeral as a whole.
Trailing zeros 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: Although 0.080 and 0.08 are numerically equal, in engineering 0.080 suggests a measurement with an error of up to one part in two thousand (±0.0005), while 0.08 suggests a measurement with an error of up to one in two hundred (see significant figures).
Other rational numbers
Any rational number with a denominator whose only prime factors are 2 and/or 5 may be precisely expressed as a decimal fraction and has a finite decimal expansion.^{[5]}
 1/2 = 0.5
 1/20 = 0.05
 1/5 = 0.2
 1/50 = 0.02
 1/4 = 0.25
 1/40 = 0.025
 1/25 = 0.04
 1/8 = 0.125
 1/125 = 0.008
 1/10 = 0.1
If the rational number's denominator has any prime factors other than 2 or 5, it cannot be expressed as a finite decimal fraction,^{[5]} and has a unique eventually repeating infinite decimal expansion.
 1/3 = 0.333333… (with 3 repeating)
 1/9 = 0.111111… (with 1 repeating)
100 − 1 = 99 = 9 × 11:
 1/11 = 0.090909…
1000 − 1 = 9 × 111 = 27 × 37:
 1/27 = 0.037037037…
 1/37 = 0.027027027…
 1/111 = 0 .009009009…
also:
 1/81 = 0.012345679012… (with 012345679 repeating)
That a rational number must have a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are at most q1 possible nonzero remainders 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:
0.4 2 8 5 7 1 4 ... 7 ) 3.0 0 0 0 0 0 0 0 2 8 30/7 = 4 with a remainder of 2 2 0 1 4 20/7 = 2 with a remainder of 6 6 0 5 6 60/7 = 8 with a remainder of 4 4 0 3 5 40/7 = 5 with a remainder of 5 5 0 4 9 50/7 = 7 with a remainder of 1 1 0 7 10/7 = 1 with a remainder of 3 3 0 2 8 30/7 = 4 with a remainder of 2 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 that the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number. For instance,
Real numbers
Every real number has a (possibly infinite) decimal representation; i.e., it can be written as
where
 sign, which is related to the sign function,
 Z is the set of all integers (positive, negative, and zero), and
 a_{i} ∈ { 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 more and more negative values of i are included, even if there are infinitely many nonzero a_{i}.
Rational numbers (e.g., p/q) with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring decimal representation.
Nonuniqueness of decimal representation
Template:Refimprove section Consider those rational numbers which have only the factors 2 and 5 in the denominator, i.e., which can be written as p/(2^{a}5^{b}). 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 do not 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. The number 0 = 0/1 is special in that it has no representation with recurring 9.
This leaves the irrational numbers. They also have unique infinite decimal representations, 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.
The same trichotomy holds for other basen positional numeral systems:
 Terminating representation: rational where the denominator divides some n^{k}
 Recurring representation: other rational
 Nonterminating, nonrecurring representation: irrational
A version of this even holds for irrationalbase numeration systems, such as golden mean base representation.
Decimal computation
Decimal computation was carried out in ancient times in many ways, typically in rod calculus, with decimal multiplication table used in ancient China and with sand tables in India and Middle East or with a variety of abaci.
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]} especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for FloatingPoint Arithmetic).^{[8]}
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.^{[9]}
History
Many ancient cultures calculated from early on with numerals based on ten: Egyptian hieroglyphs, in evidence since around 3000 BC, used a purely decimal system,^{[10]}^{[11]} just as the Cretan hieroglyphs (ca. 1625−1500 BC) of the Minoans whose numerals are closely based on the Egyptian model.^{[12]}^{[13]} 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.^{[14]} Notably, the polymath Archimedes (c. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 10^{8}^{[14]} 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.^{[15]} The Hittites hieroglyphs (since 15th century BC), just like the Egyptian and early numerals in Greece, was strictly decimal.^{[16]}
The Egyptian hieratic numerals, the Greek 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.^{[17]} The world's earliest positional decimal system was the Chinese rod calculus^{[18]}
History of decimal fractions
According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, and then spread to the Middle East and from there to Europe.^{[18]}^{[19]} The written Chinese decimal fractions were nonpositional.^{[19]} However, counting rod fractions were positional.^{[18]}
Qin Jiushao in his book Mathematical Treatise in Nine Sections (1247) denoted 0.96644 by
 寸
 096644
^{[20]}
The Jewish mathematician Immanuel Bonfils invented decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them.^{[21]}
The Persian mathematician Jamshīd alKāshī claimed to have discovered decimal fractions himself in the 15th century, though J. Lennart Berggren notes that positional decimal fractions were used five centuries before him by Arab mathematician Abu'lHasan alUqlidisi as early as the 10th century.^{[22]}Al Khwarizmi introduced fraction to Islamic countries in the early 9th century, his fraction presentation was an exact copy of traditional Chinese mathematical fraction. 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".^{[18]}^{[23]}
A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century.^{[24]}
Natural languages
Telugu language uses a straightforward decimal system. Other Dravidian languages such as Tamil and Malayalam have replaced the number nine tondu with 'onpattu' ("one to ten") during the early Middle Ages, while Telugu preserved the number nine as tommidi.
The Hungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tízenegy" literally "one on ten"), as with those between 20100 (23 as "huszonhárom" = "three on twenty").
A straightforward decimal rank system with a word for each order 10十,100百,1000千,10000万, and in which 11 is expressed as tenone and 23 as twotenthree, and 89345 is expressed as 8 (ten thousands) 万9 (thousand) 千3 (hundred) 百4 (tens) 十 5 is found in Chinese languages, 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".
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.^{[25]}
Other bases
Some cultures do, or did, use other bases of numbers.
 PreColumbian Mesoamerican cultures such as the Maya used a base20 system (presumably using all twenty fingers and toes).
 The Yuki language in California and the Pamean languages^{[26]} in Mexico have octal (base8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.^{[27]}
 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.Template:Synthesisinline 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'. {{ safesubst:#invoke:Unsubstdate=__DATE__ $B=
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 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.^{[28]}
 Many languages^{[29]} use quinary (base5) number systems, including Gumatj, Nunggubuyu,^{[30]} Kuurn Kopan Noot^{[31]} 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 a duodecimal (base12) systems.^{[32]} So did some small communities in India and Nepal, as indicated by their languages.^{[33]}
 The Huli language of Papua New Guinea is reported to have base15 numbers.^{[34]} 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.^{[35]} 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.^{[29]}
 The Ndom language of Papua New Guinea is reported to have base6 numerals.^{[36]} Mer means 6, mer an thef means 6×2 = 12, nif means 36, and nif thef means 36×2 = 72.
See also
References
 ↑ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
 ↑ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994 (Also: The Universal History of Numbers: From prehistory to the invention of the computer, Georges Ifrah, ISBN 0471393401, John Wiley and Sons Inc., New York, 2000. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk)
 ↑ Lam Lay Yong & Ang Tian Se (2004) Fleeting Footsteps. Tracing the Conception of Arithmetic and Algebra in Ancient China, Revised Edition, World Scientific, Singapore.
 ↑ Template:Cite web
 ↑ ^{5.0} ^{5.1} {{#invoke:citation/CS1citation CitationClass=book }}
 ↑ 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.
 ↑ Decimal Computation, Hermann Schmid, John Wiley & Sons 1974 (ISBN 047176180X); reprinted in 1983 by Robert E. Krieger Publishing Company (ISBN 0898743184)
 ↑ 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
 ↑ Egyptian numerals
 ↑ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0140099190, pp. 200213 (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.213218 (Cretan numerals)
 ↑ ^{14.0} ^{14.1} Greek numerals
 ↑ Menninger, Karl: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, ISBN 3525407254, pp. 150153
 ↑ 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 p 137139
 ↑ ^{18.0} ^{18.1} ^{18.2} ^{18.3} Lam Lay Yong, "The Development of HinduArabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p38, Kurt Vogel notation
 ↑ ^{19.0} ^{19.1} {{#invoke:citation/CS1citation CitationClass=book }}
 ↑ JeanClaude Martzloff, A History of Chinese Mathematics, Springer 1997 ISBN 3540337822
 ↑ 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.
 ↑ {{#invoke:citation/CS1citation CitationClass=book }}
 ↑ Lam Lay Yong, "A Chinese Genesis, Rewriting the history of our numeral system", Archive for History of Exact Science 38: 101–108.
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 ↑ Template:Cite web
 ↑ 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.
 ↑ ^{29.0} ^{29.1} {{#invoke:citation/CS1citation CitationClass=book }}
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 ↑ Dawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.
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