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decimal
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{{Short description|Number in base-10 numeral system}}{{Other uses}}(File:Decimal_digit.png|thumb|upright=1.2|Place value of number in decimal system)The decimal numeral system (also called the base-ten positional numeral system and denary {{IPAc-en|Ë|d|iË|n|Ér|i}}{{OED|denary}} or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (decimal fractions) of the HinduâArabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.BOOK, Yong, Lam Lay,weblink Fleeting Footsteps, Se, Ang Tian, April 2004, World Scientific, 978-981-238-696-0, 268, 10.1142/5425, March 17, 2022, April 1, 2023,weblink live, A decimal numeral (also often just decimal or, less correctly, decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in {{math|25.9703}} or {{math|3,1415}}).WEB, Weisstein, Eric W., March 10, 2022, Decimal Point,weblink live, March 17, 2022, Wolfram MathWorld, en, March 21, 2022,weblink Decimal may also refer specifically to the digits after the decimal separator, such as in "{{math|3.14}} is the approximation of {{pi}} to two 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 the decimal fractions. That is, fractions of the form {{math|a/10n}}, where {{math|a}} is an integer, and {{math|n}} is a non-negative integer. Decimal fractions also result from the addition of an integer and a fractional part; the resulting sum sometimes is called a fractional number.Decimals are commonly used to approximate real numbers. By increasing the number of digits after the decimal separator, one can make the approximation errors as small as one wants, when one has a method for computing the new digits. {{anchor|terminating decimal}}Originally and in most uses, a decimal has only a finite number of digits after the decimal seperator. However, 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, with a finite number of non-zero digits after the decimal separator, are sometimes called terminating decimals. A repeating decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., {{math|1=5.123144144144144... = 5.123{{overline|144}}}}).The vinculum (overline) in 5.123144 indicates that the '144' sequence repeats indefinitely, i.e. {{val|5.123144144144144|s=...}}. An infinite decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
Origin
{{unreferenced section|date=May 2022}}(File:Two hand, ten fingers.jpg|thumb|right|Ten digits on two hands, the possible origin of decimal counting|upright=1.2)Many numeral 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 the Egyptian numerals, then the 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 HinduâArabic numeral system for representing integers. This system has been extended to represent some non-integer 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 "â". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;In some countries, such as Arabic-speaking ones, other glyphs are used for the digits the decimal separator is the dot "{{math|.}}" in many countries (mostly English-speaking),WEB, Weisstein, Eric W., Decimal,weblink 2020-08-22, mathworld.wolfram.com, en, 2020-03-18,weblink live, and a comma "{{math|,}}" in other countries.For representing a non-negative number, a decimal numeral consists of- either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer:
- :a_ma_{m-1}ldots a_0
- or a decimal mark separating two sequences of digits (such as "20.70828")
a_ma_{m-1}ldots a_0.b_1b_2ldots b_n.
a_m10^m+a_{m-1}10^{m-1}+cdots+a_{0}10^0+frac{b_1}{10^1}+frac{b_2}{10^2}+cdots+frac{b_n}{10^n}.
The integer part or integral part of a decimal numeral is the integer written to the left of the decimal separator (see also truncation). 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 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, {{math|.1234}}, instead of {{math|0.1234}}). 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 a positional numeral system.Decimal fractions
{{Table Numeral Systems}}Decimal fractions (sometimes called decimal numbers, especially in contexts involving explicit fractions) are the rational numbers that may be expressed as a fraction whose denominator is a power of ten.ENCYCLOPEDIA,weblink Decimal Fraction, Encyclopedia of Mathematics, 2013-06-18, 2013-12-11,weblink" title="web.archive.org/web/20131211035917weblink">weblink live, For example, the decimal expressions 0.8, 14.89, 0.00079, 1.618, 3.14159 represent the fractions {{math|{{sfrac|4|5}}}}, {{math|{{sfrac|1489|100}}}}, {{math|{{sfrac|79|100000}}}}, {{Math|{{sfrac||809|500}}}} and {{Math|{{sfrac||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) is {{math|{{sfrac|1|3}}}}, 3 not being a power of 10. More generally, a decimal with {{math|n}} digits after the separator (a point or comma) represents the fraction with denominator {{math|10n}}, whose numerator is the integer obtained by removing the separator.It follows that a number is a decimal fraction if and only if it has a finite decimal representation.Expressed as fully 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
1=2^0cdot 5^0, 2=2^1cdot 5^0, 4=2^2cdot 5^0, 5=2^0cdot 5^1, 8=2^3cdot 5^0, 10=2^1cdot 5^1, 16=2^4cdot 5^0, 20=2^2cdot5^1, 25=2^0cdot 5^2, ldots
Approximation using decimal numbers
Decimal numerals do not allow an exact representation for all real numbers. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates {{pi}}, being less than 10â5 off; so decimals are widely used in science, engineering and everyday life.More precisely, for every real number {{Mvar|x}} and every positive integer {{Mvar|n}}, there are two decimals {{Mvar|L}} and {{Mvar|u}} with at most {{Mvar|n}} digits after the decimal mark such that {{Math|L ⤠x ⤠u}} and {{Math|1=(u â L) = 10ân}}.Numbers are very often obtained as the result of measurement. As measurements are subject to measurement uncertainty with a known upper bound, the result of a measurement is well-represented by a decimal with {{math|n}} digits after the decimal mark, as soon as the absolute measurement error is bounded from above by {{Math|10ân}}. 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 also significant figures).Infinite decimal expansion
{{unreferenced section|date=April 2020}}For a real number {{Mvar|x}} and an integer {{Math|n ⥠0}}, let {{Math|[x]n}} denote the (finite) decimal expansion of the greatest number that is not greater than {{Mvar|x}} that has exactly {{Mvar|n}} digits after the decimal mark. Let {{Math|d'i}} denote the last digit of {{Math|[x]i}}. It is straightforward to see that {{Math|[x]n}} may be obtained by appending {{Math|d'n}} to the right of {{Math|[x]nâ1}}. This way one has
{{Math|1=[x]n = [x]0.d1d2...d'nâ1d'n}},
and the difference of {{Math|[x]nâ1}} and {{Math|[x]n}} amounts to
leftvert left [ x right ]_n-left [ x right ]_{n-1} rightvert=d_ncdot10^{-n} N}} equal to 9 and {{Math|1=[x]n = [x]0.d1d2...d'n}}, the limit of the sequence;([x]_n)_{n=1}^{infty} is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: {{Math|d'N}}, by {{Math|dN + 1}}, and replacing all subsequent 9s by 0s (see 0.999...).
Any such decimal fraction, i.e.: {{Math|1=d'n = 0}} for {{Math|n > N}}, may be converted to its equivalent infinite decimal expansion by replacing {{Math|d'N}} by {{Math|dN â 1}} and replacing all subsequent 0s by 9s (see 0.999...).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 {{Math|[x]n}}, and the other containing only 9s after some place, which is obtained by defining {{Math|[x]n}} as the greatest number that is less than {{Mvar|x}}, having exactly {{Mvar|n}} digits after the decimal mark.Rational numbers
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 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 a repeating decimal. For example,
{{sfrac|81}} = 0.{{thin space}}012345679{{thin space}}012... (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.{|x is > | 6}}0.4156156156... |
x is > | 3}}4156.156156156... |
x is> | 6}}4.156156156... |
x â 10x, i.e. 9,990x, is> | 3}}4152.000000000... |
x is> | 3}}{{sfrac | 9990}} |
Decimal computation
File:Decimal multiplication table.JPG|thumb|right|300px|Diagram of the world's earliest known multiplication table ({{circa|305 BCE}}) from the Warring States periodWarring States periodMost 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)."Fingers or Fists? (The Choice of Decimal or Binary Representation)", Werner Buchholz, Communications of the ACM, Vol. 2 #12, pp. 3â11, ACM Press, December 1959.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 binary-coded decimal,BOOK, Decimal Computation, Hermann, Schmid, Hermann Schmid (computer scientist), 1974, 1983, 1 (reprint), Robert E. Krieger Publishing Company, Malabar, Florida, 0-89874-318-4, BOOK, Decimal Computation, Hermann, Schmid, Hermann Schmid (computer scientist), 1974, 1st, John Wiley & Sons, Binghamton, New York, 0-471-76180-X, registration,weblink especially in database implementations, but there are other decimal representations in use (including decimal floating point such as in newer revisions of the IEEE 754 Standard for Floating-Point Arithmetic).Decimal Floating-Point: Algorism for Computers, Cowlishaw, Mike F., Proceedings 16th IEEE Symposium on Computer Arithmetic, {{isbn|0-7695-1894-X}}, pp. 104â11, IEEE Comp. Soc., 2003Decimal 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 10 have no finite binary fractional representation; and is generally impossible for multiplication (or division).WEB,weblink Decimal Arithmetic â FAQ, 2008-08-15, 2009-04-29,weblink" title="web.archive.org/web/20090429044324weblink">weblink live, Decimal Floating-Point: Algorism for Computers {{Webarchive|url=https://web.archive.org/web/20031116234555weblink |date=2003-11-16 }}, Cowlishaw, M. F., Proceedings 16th IEEE Symposium on Computer Arithmetic (ARITH 16 {{Webarchive|url=https://web.archive.org/web/20100819005155weblink |date=2010-08-19 }}), {{isbn|0-7695-1894-X}}, pp. 104â11, IEEE Comp. Soc., June 2003 See Arbitrary-precision arithmetic for exact calculations.History
File:Qinghuajian, Suan Biao.jpg|thumb|upright|The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BCE, during the Warring StatesWarring StatesMany ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers.{{citation|first=Tobias|last=Dantzig|title=Number / The Language of Science |edition=4th |year=1954|publisher=The Free Press (Macmillan Publishing Co.) |isbn=0-02-906990-4|page=12}} Standardized weights used in the Indus Valley Civilisation ({{circa|3300â1300 BCE}}) were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler â the Mohenjo-daro ruler â was divided into ten equal parts.Sergent, Bernard (1997), Genèse de l'Inde (in French), Paris: Payot, p. 113, {{ISBN|2-228-89116-9}}JOURNAL, Coppa, A., etal, 2006, Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population, 2006Natur.440..755C, Nature, 440, 7085, 755â56, 10.1038/440755a, 16598247, 6787162, Bisht, R. S. (1982), "Excavations at Banawali: 1974â77", in Possehl, Gregory L. (ed.), Harappan Civilisation: A Contemporary Perspective, New Delhi: Oxford and IBH Publishing Co., pp. 113â24 Egyptian hieroglyphs, in evidence since around 3000 BCE, used a purely decimal system,Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, {{isbn|0-14-009919-0}}, pp. 200â13 (Egyptian Numerals) as did the Linear A script ({{circa|1800â1450 BCE}}) of the MinoansGraham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, {{isbn|978-0-486-42165-0}}, p. 50Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, {{isbn|0-14-009919-0}}, pp. 213â18 (Cretan numerals) and the Linear B script (c. 1400â1200 BCE) of the Mycenaeans. The ÃnÄtice culture in central Europe (2300-1600 BC) used standardised weights and a decimal system in trade.BOOK, Krause, Harald,weblink Spangenbarrenhort Oberding, Kutscher, Sabrina, 2017, Museum Erding, 978-3-9817606-5-1, 238â243, Spangenbarrenhort Oberding: Zusammenfassung und Ausblick, The number system of classical Greece also used powers of ten, including an intermediate base of 5, as did Roman numerals.WEB,weblink Greek numbers, 2019-07-21, 2019-07-21,weblink" title="web.archive.org/web/20190721085640weblink">weblink live, Notably, the polymath Archimedes (c. 287â212 BCE) invented a decimal positional system in his Sand Reckoner which was based on 108.Menninger, Karl: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, {{isbn|3-525-40725-4}}, pp. 150â53 Hittite hieroglyphs (since 15th century BCE) were also strictly decimal.Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, {{isbn|0-14-009919-0}}, pp. 218f. (The Hittite hieroglyphic system)Some non-mathematical ancient texts such as the Vedas, dating back to 1700â900 BCE make use of decimals and mathematical decimal fractions.(Atharva Veda 5.15, 1â11)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, 1000, 2000, 3000, 4000, to 10,000.Lam Lay Yong et al. The Fleeting Footsteps pp. 137â39The world's earliest positional decimal system was the Chinese rod calculus.(File:Chounumerals.svg|thumb|right|280px|The world's earliest positional decimal system Upper row vertical form Lower row horizontal form)History of decimal fractions
(File:Rod fraction.jpg|thumb|right|150px|counting rod decimal fraction 1/7)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.BOOK, Joseph Needham, Joseph Needham, 19.2 Decimals, Metrology, and the Handling of Large Numbers, 82â90, Science and Civilisation in China, III, "Mathematics and the Sciences of the Heavens and the Earth", Science and Civilisation in China, 1959, Cambridge University Press, Calculations with decimal fractions of lengths were performed using positional counting rods, as described in the 3rdâ5th century CE Sunzi Suanjing. The 5th century CE mathematician Zu Chongzhi calculated a 7-digit approximation of {{mvar|Ï}}. Qin Jiushao's book Mathematical Treatise in Nine Sections (1247) explicitly writes a decimal fraction representing a number rather than a measurement, using counting rods.Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997 {{isbn|3-540-33782-2}} The number 0.96644 is denoted
(File:Counting rod 0.png|frameless|18px) (File:Counting rod h9 num.png|frameless|18px) (File:Counting rod v6.png|frameless|18px) (File:Counting rod h6.png|frameless|18px) (File:Counting rod v4.png|frameless|18px) (File:Counting rod h4.png|frameless|18px).
Natural languages
A method of expressing every possible natural number using a set of ten symbols emerged in India.WEB,weblink Indian numerals, Ancient Indian mathematics, 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.{{cn|date=March 2024}}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 , 100 , 1000 , 10,000 ), and in which 11 is expressed as ten-one and 23 as two-ten-three, and 89,345 is expressed as 8 (ten thousands) 9 (thousand) 3 (hundred) 4 (tens) 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 "ten-one" or "one-teen".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 two-ten with three.Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.JOURNAL, Azar, Beth, 1999, English words may hinder math skills development,weblink American Psychological Association Monitor, 30, 4,weblink" title="web.archive.org/web/20071021015527weblink">weblink 2007-10-21,Other bases
{{Fundamental info units}}Some cultures do, or did, use other bases of numbers.- Pre-Columbian Mesoamerican cultures such as the Maya used a base-20 system (perhaps based on using all twenty fingers and toes).
- The Yuki language in California and the Pamean languagesJOURNAL
, Avelino
, Heriberto
, The typology of Pame number systems and the limits of Mesoamerica as a linguistic area
, Linguistic Typology
, 2006
, 10
, 1
, 41â60
,weblinkweblink" title="web.archive.org/web/20060712201924weblink">weblink 2006-07-12, live
, 10.1515/LINGTY.2006.002
, 20412558
, in Mexico have octal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.NEWS, 2686959, Ethnomathematics: A Multicultural View of Mathematical Ideas, Marcia Ascher, Marcia Ascher, The College Mathematics Journal,
, Heriberto
, The typology of Pame number systems and the limits of Mesoamerica as a linguistic area
, Linguistic Typology
, 2006
, 10
, 1
, 41â60
,weblinkweblink" title="web.archive.org/web/20060712201924weblink">weblink 2006-07-12, live
, 10.1515/LINGTY.2006.002
, 20412558
, in Mexico have octal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.NEWS, 2686959, Ethnomathematics: A Multicultural View of Mathematical Ideas, Marcia Ascher, Marcia Ascher, The College Mathematics Journal,
- 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.{{citation
| last = McClean | first = R. J.
| date = July 1958
| doi = 10.1111/j.1468-0483.1958.tb00018.x
| issue = 4
| journal = German Life and Letters
| quote = Some of the Germanic languages appear to show traces of an ancient blending of the decimal with the vigesimal system
| pages = 293â99
| title = Observations on the Germanic numerals
| volume = 11}}.{{citation
| last = Voyles | first = Joseph
| date = October 1987
| issue = 4
| journal = The Journal of English and Germanic Philology
| jstor = 27709904
| pages = 487â95
| title = The cardinal numerals in pre-and proto-Germanic
| volume = 86}}. 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's Introduction to Old Norse {{Webarchive|url=https://web.archive.org/web/20160415205641weblink |date=2016-04-15 }} 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{{Dead link|date=January 2024 |bot=InternetArchiveBot |fix-attempted=yes }} 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'.JOURNAL, Stevenson, W.H., 1890, The Long Hundred and its uses in England, Archaeological Review, December 1889, 313â22, BOOK, Poole, Reginald Lane, The Exchequer in the twelfth century : the Ford lectures delivered in the University of Oxford in Michaelmas term, 1911, 2006, Lawbook Exchange, 1-58477-658-7, Clark, NJ, 76960942,
| date = July 1958
| doi = 10.1111/j.1468-0483.1958.tb00018.x
| issue = 4
| journal = German Life and Letters
| quote = Some of the Germanic languages appear to show traces of an ancient blending of the decimal with the vigesimal system
| pages = 293â99
| title = Observations on the Germanic numerals
| volume = 11}}.{{citation
| last = Voyles | first = Joseph
| date = October 1987
| issue = 4
| journal = The Journal of English and Germanic Philology
| jstor = 27709904
| pages = 487â95
| title = The cardinal numerals in pre-and proto-Germanic
| volume = 86}}. 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's Introduction to Old Norse {{Webarchive|url=https://web.archive.org/web/20160415205641weblink |date=2016-04-15 }} 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{{Dead link|date=January 2024 |bot=InternetArchiveBot |fix-attempted=yes }} 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'.JOURNAL, Stevenson, W.H., 1890, The Long Hundred and its uses in England, Archaeological Review, December 1889, 313â22, BOOK, Poole, Reginald Lane, The Exchequer in the twelfth century : the Ford lectures delivered in the University of Oxford in Michaelmas term, 1911, 2006, Lawbook Exchange, 1-58477-658-7, Clark, NJ, 76960942,
- Many or all of the Chumashan languages originally used a base-4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.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|0-292-75531-7}}.
- Many languagesBOOK, Rarities in Numeral Systems, Harald, Hammarström, Jan, Wohlgemuth, Michael, Cysouw, Rethinking Universals: How rarities affect linguistic theory, 17 May 2007, Berlin, Mouton de Gruyter, Empirical Approaches to Language Typology, 45, 2010,weblink dead,weblink" title="web.archive.org/web/20070819214057weblink">weblink 19 August 2007
, Facts and fallacies of aboriginal number systems
, Harris
, John
, Hargrave
, Susanne
, 153â81
, 1982
, Work Papers of SIL-AAB Series B
, 8
,weblink
, dead
,weblink" title="web.archive.org/web/20070831202737weblink">weblink
, 2007-08-31
, Kuurn Kopan NootDawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii. and Saraveca. Of these, Gumatj is the only true 5â25 language known, in which 25 is the higher group of 5. , Harris
, John
, Hargrave
, Susanne
, 153â81
, 1982
, Work Papers of SIL-AAB Series B
, 8
,weblink
, dead
,weblink" title="web.archive.org/web/20070831202737weblink">weblink
, 2007-08-31
- Some Nigerians use duodecimal systems.CONFERENCE, Decimal vs. Duodecimal: An interaction between two systems of numeration, Matsushita, Shuji, 2nd Meeting of the AFLANG, October 1998, Tokyo, 1998,weblinkweblink" title="web.archive.org/web/20081005230737weblink">weblink 2008-10-05, 2011-05-29
- The Huli language of Papua New Guinea is reported to have base-15 numbers.JOURNAL
, Cheetham
, Brian
, Counting and Number in Huli
, Papua New Guinea Journal of Education
, 1978
, 14
, 16â35
,weblink
,weblink" title="web.archive.org/web/20070928061238weblink">weblink
, 2007-09-28
, Ngui means 15, ngui ki means 15 Ã 2 = 30, and ngui ngui means 15 Ã 15 = 225.
, Brian
, Counting and Number in Huli
, Papua New Guinea Journal of Education
, 1978
, 14
, 16â35
,weblink
,weblink" title="web.archive.org/web/20070928061238weblink">weblink
, 2007-09-28
, Bowers
, Nancy
, Lepi
, Pundia
, Kaugel Valley systems of reckoning
, 1975
, Journal of the Polynesian Society
, 84
, 3
, 309â24
,weblink
, dead
,weblink" title="web.archive.org/web/20110604091351weblink">weblink
, 2011-06-04
, Tokapu means 24, tokapu talu means 24 Ã 2 = 48, and tokapu tokapu means 24 Ã 24 = 576. , Nancy
, Lepi
, Pundia
, Kaugel Valley systems of reckoning
, 1975
, Journal of the Polynesian Society
, 84
, 3
, 309â24
,weblink
, dead
,weblink" title="web.archive.org/web/20110604091351weblink">weblink
, 2011-06-04
- Ngiti is reported to have a base-32 number system with base-4 cycles.
- The Ndom language of Papua New Guinea is reported to have base-6 numerals. Mer means 6, mer an thef means 6 Ã 2 = 12, nif means 36, and nif thef means 36Ã2 = 72.
See also
{hide}columns-list|colwidth=30em|- Algorism
- Binary-coded decimal (BCD)
- Decimal classification
- Decimal computer
- Decimal time
- Decimal representation
- Decimal section numbering
- Decimal separator
- Decimalisation
- Densely packed decimal (DPD)
- Duodecimal
- Octal
- Scientific notation
- Serial decimal
- Metric prefix
Notes
{{reflist|group=note}}References
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