The DECIMAL numeral system (also called BASETEN and occasionally called DENARY) has ten as its base , which, in decimal, is written 10, as is the base in every positional numeral system. It is the numerical base most widely used by modern civilizations. _ Decimal notation_ often refers to a base10 positional notation such as the HinduArabic numeral system ; 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. A _decimal number_, or just _decimal_, refers to any number written in decimal notation , although it is more commonly used to refer to numbers that have a fractional part separated from the integer part with a decimal separator (e.g. 11.25). A decimal may be a terminating decimal, which has a finite fractional part (e.g. 15.600); a repeating decimal , which has an infinite (nonterminating) fractional part made up of a repeating sequence of digits (e.g. 5.123144 ); or an infinite decimal, which has a fractional part that neither terminates nor has an infinitely repeating pattern (e.g. 3.14159265...). Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, whereas irrational numbers have infinite nonrepeating decimal representations. CONTENTS * 1 Decimal notation * 1.1 Decimal fractions * 1.2 Other rational numbers * 1.3 Real numbers * 1.4 Nonuniqueness of decimal representation * 2 Decimal computation * 3 History * 3.1 History of decimal fractions * 3.2 Natural languages * 3.3 Other bases * 4 See also * 5 References * 6 External links DECIMAL NOTATION Decimal notation is the writing of numbers in a baseten numeral system . Examples are Brahmi numerals , Greek numerals , Hebrew numerals , Roman numerals , and Chinese numerals , as well as the Hindu Arabic 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 ten, which in modern usage reach 1072. However, when people who use Hindu Arabic 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 fingers on two hands, the possible starting point of the decimal counting. 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 Numeral systems , bits and Gray code HEX DEC OCT 3 2 1 0 STEP 0hex 00dec 00oct 0 0 0 0 g0 1hex 01dec 01oct 0 0 0 1 h1 2hex 02dec 02oct 0 0 1 0 j3 3hex 03dec 03oct 0 0 1 1 i2 4hex 04dec 04oct 0 1 0 0 n7 5hex 05dec 05oct 0 1 0 1 m6 6hex 06dec 06oct 0 1 1 0 k4 7hex 07dec 07oct 0 1 1 1 l5 8hex 08dec 10oct 1 0 0 0 vF 9hex 09dec 11oct 1 0 0 1 uE Ahex 10dec 12oct 1 0 1 0 sC Bhex 11dec 13oct 1 0 1 1 tD Chex 12dec 14oct 1 1 0 0 o8 Dhex 13dec 15oct 1 1 0 1 p9 Ehex 14dec 16oct 1 1 1 0 rB Fhex 15dec 17oct 1 1 1 1 qA A DECIMAL FRACTION is a fraction the denominator of which is a power of ten. Decimal fractions are commonly expressed in decimal notation rather than fraction notation by discarding the denominator and inserting the decimal separator into the numerator at the position from the right corresponding to the power of ten of the denominator and filling the gap with leading zeros if needed, e.g. decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8, 14.89, 0.00024, 5.8900 respectively. 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. 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 a fully reduced rational number's denominator has any prime factors other than 2 or 5, it cannot be expressed as a finite decimal fraction, 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 _q_ − 1 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, 0.0123123123 = 123 10000 k = 0 0.001 k = 123 10000 1 1 0.001 = 123 9990 = 41 3330 {displaystyle 0.0123123123cdots ={frac {123}{10000}}sum _{k=0}^{infty }0.001^{k}={frac {123}{10000}} {frac {1}{10.001}}={frac {123}{9990}}={frac {41}{3330}}} REAL NUMBERS Further information: Decimal representation Every real number has a (possibly infinite) decimal representation; i.e., it can be written as x = s i g n i Z a i 10 i {displaystyle x={mathop {rm {sign}}}sum _{iin mathbb {Z} }a_{i},10^{i}} where * sign ∈ {+,−}, which is related to the sign function , * ℤ is the set of all integers (positive, negative, and zero), and * _ai_ ∈ { 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 _ai_. 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 _ This section NEEDS ADDITIONAL CITATIONS FOR VERIFICATION . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed. (March 2012)_ _(Learn how and when to remove this template message )_ 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 base_n_ 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 * v * t * e Multiples of bytes DECIMAL VALUE METRIC 1000 kB kilobyte 10002 MB megabyte 10003 GB gigabyte 10004 TB terabyte 10005 PB petabyte 10006 EB exabyte 10007 ZB zettabyte 10008 YB yottabyte BINARY VALUE IEC JEDEC 1024 KiB kibibyte KB kilobyte 10242 MiB mebibyte MB megabyte 10243 GiB gibibyte GB gigabyte 10244 TiB tebibyte – 10245 PiB pebibyte – 10246 EiB exbibyte – 10247 ZiB zebibyte – 10248 YiB yobibyte – Orders of magnitude of data Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period 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). 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 , especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for FloatingPoint Arithmetic ). 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. HISTORY The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China. Many ancient cultures calculated with numerals based on ten, sometimes argued due to human hands typically having ten digits. Egyptian hieroglyphs , in evidence since around 3000 BC, used a purely decimal system, just as the Cretan hieroglyphs (ca. 1625−1500 BC) of the Minoans whose numerals are closely based on the Egyptian model. 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. Notably, the polymath Archimedes (ca. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 108 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. The Hittites hieroglyphs (since 15th century BC), just like the Egyptian and early numerals in Greece, was strictly decimal. Some ancient texts like the Vedas dating back to 19001700 BCE mention decimals and mathematical decimal fractions. 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. The world's earliest positional decimal system was the Chinese rod calculus The world's earliest positional decimal system Upper row vertical form Lower row horizontal form HISTORY OF DECIMAL FRACTIONS counting rod decimal fraction 1/7 Decimal fractions were first developed and used by the Chinese in the end of 4th century BC, and then spread to the Middle East and from there to Europe. The written Chinese decimal fractions were nonpositional. However, counting rod fractions were positional. Qin Jiushao in his book Mathematical Treatise in Nine Sections (1247) denoted 0.96644 by 寸 , meaning 寸 096644 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. The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin , but did not develop any notation to represent them. The Persian mathematician Jamshīd alKāshī claimed to have discovered decimal fractions himself in the 15th century. 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 . 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". A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. 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. 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. OTHER BASES Main article: positional notation Units of information * bit or shannon (base 2 ) * nat (base _e_ ) * ban , dit or hartley (base 10) * qubit (quantum ) * v * t * e 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 in Mexico have octal (base8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves. * 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. 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'. * 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 . * Many languages use quinary (base5) number systems, including Gumatj , Nunggubuyu , Kuurn Kopan Noot 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. So did some small communities in India and Nepal, as indicated by their languages. * The Huli language of Papua New Guinea is reported to have base15 numbers. _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. _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. * The Ndom language of Papua New Guinea is reported to have base6 numerals. _Mer_ means 6, _mer an thef_ means 6 × 2 = 12, _nif_ means 36, and _nif thef_ means 36×2 = 72. SEE ALSO * 0.999... * 10 (number) * Algorism * Binarycoded decimal * Decimal computer * Decimal representation * Decimal separator * Dewey Decimal Classification * Duodecimal * HinduArabic numeral system * Numeral system * Octal * Scientific notation * SI prefix REFERENCES * ^ _The History of Arithmetic_, Louis Charles Karpinski , 200pp, Rand McNally & Company, 1925. * ^ Lam Lay Yong ">144 indicates that the '144' sequence repeats itself indefinitely, i.e. 7000512314414414414♠5.123144144144144.... * ^ " Decimal Fraction". _ Encyclopedia of Mathematics _. Retrieved 20130618. * ^ _A_ _B_ _Math Made NicenEasy_. Piscataway, N.J.: Research Education Association. 1999. p. 141. ISBN 0878912002 . * ^ _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) . _ Decimal Computation_ (1 (reprint) ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company. ISBN 0898743184 . * ^ Schmid, Hermann (1974). _ Decimal Computation_ (1 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.213218 (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 p 137139 * ^ _A_ _B_ _C_ _D_ _E_ Lam Lay Yong , "The Development of HinduArabic 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 SIL AAB Series B_. 8. pp. 153–181. * ^ 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. * ^ Mazaudon, Martine (2002). "Les principes de construction du nombre dans les langues tibétobirmanes". In François, Jacques. _La Pluralité_ (PDF). Leuven: Peeters. pp. 91–119. ISBN 9042912952 * ^ Cheetham, Brian (1978). "Counting and Number in Huli". _Papua New Guinea Journal of Education_. 14: 16–35. Archived from the original on 20070928. * ^ Bowers, Nancy; Lepi, Pundia (1975). "Kaugel Valley systems of reckoning" (PDF). _Journal of the Polynesian Society_. 84 (3): 309–324. * ^ Owens, Kay (2001), "The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania", _Mathematics Education Research Journal_, 13 (1): 47–71, doi :10.1007/BF03217098 EXTERNAL LINKS * Decimal arithmetic FAQ * Cultural Aspects of Young Children\'s Mathematics Knowledge AUTHORITY CONTROL
