Decimal
This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation.
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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 base-10 positional notation such as the Hindu-Arabic numeral system; however, it can also be used more generally to refer to non-positional 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 'tenth-metre', 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.

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Decimal notation

Decimal notation is the writing of numbers in a base-10 numeral system. Examples are Roman numerals, Brahmi 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 10, which in modern usage reach 1044.

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 Hindu-Arabic 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 whose denominator is a power of ten.

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 English-speaking and many Asian countries, a period (.) or raised period (·) is used as the decimal separator; in many other countries, 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; if considered as a separate number, a zero is often written in front. Especially for negative numbers, we have to distinguish between the fractional part of the notation and the fractional part of the number itself, because the latter gets its own decimal sign. It is usual for a decimal number whose absolute value is less than one to have a leading zero.

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.[3]

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,[3] and has a unique infinite decimal expansion ending with recurring decimals.

1/3 = 0.333333… (with 3 repeating)
1/9 = 0.111111… (with 1 repeating)

100-1=99=9×11

1/11 = 0.090909… (with 09 repeating)

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)

Other prime factors in the denominator will give longer recurring sequences; see for instance 1/7, and 1/13.

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 only 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 r 2
       2 0
       1 4                       20/7 = 2 r 6
         6 0
         5 6                     60/7 = 8 r 4
           4 0
           3 5                   40/7 = 5 r 5
             5 0
             4 9                 50/7 = 7 r 1
               1 0
                 7               10/7 = 1 r 3
                 3 0
                 2 8             30/7 = 4 r 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\cdots = \frac{123}{10000} \sum_{k=0}^\infty 0.001^k = \frac{123}{10000}\ \frac{1}{1-0.001} = \frac{123}{9990} = \frac{41}{3330}

Real numbers

Every real number has a (possibly infinite) decimal representation; i.e., it can be written as

 x = \mathop{\rm sign}(x) \sum_{i\in\mathbb Z} a_i\,10^i

where

  • sign() is the sign function, and
  • ai ∈ { 0,1,…,9 } for all iZ 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 i increases, even if there are infinitely many non-zero 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.

Non-uniqueness of decimal representation

Consider those rational numbers which have only the factors 2 and 5 in the denominator, i.e., which can be written as p/(2a5b). 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 nk
  • Recurring representation: other rational
  • Non-terminating, non-recurring representation: irrational

A version of this even holds for irrational-base numeration systems, such as golden mean base representation.

Decimal computation

Decimal computation was carried out in ancient times in many ways, typically on sand tables 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).[4] 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,[5] especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for Floating-Point Arithmetic).[6]

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.[7]

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,[8][9] just as the Cretan hieroglyphs (ca. 1625−1500 BC) of the Minoans whose numerals are closely based on the Egyptian model.[10][11] 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.[12] Notably, the polymath Archimedes (c. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 108[12] 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.[13] The Hittites hieroglyphs (since 15th century BC), just like the Egyptian and early numerals in Greece, was strictly decimal.[14]

The Egyptian herratic numerals, the Greek 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, through 90, 100, 200, through 900, 1000, 2000, 3000, 4000, to 10,000.[15]

History of decimal fractions

counting rod decimal fraction 1/7

According to Joseph Needham, 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.[16] The written Chinese decimal fractions were non-positional.[16] However, counting rod fractions were positional.

Qin Jiushao in his book Mathematical Treatise in Nine Sections (1247) denoted 0.96644 by

Counting rod 0.pngCounting rod h9 num.pngCounting rod v6.pngCounting rod h6.pngCounting rod v4.pngCounting rod h4.png, meaning
096644

[17]

Immanuel Bonfils invented decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them.[18]

The Persian mathematician Jamshīd al-Kāshī 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'l-Hasan al-Uqlidisi as early as the 10th century.[19]

Khwarizmi introduced fractions to Islamic countries in the early 9th century. His representation of fractions was taken from traditional Chinese mathematical fractions. This form of fraction with the numerator on top and the denominator on the bottom, without a horizontal bar, was also used in the 10th century by Abu'l-Hasan al-Uqlidisi and again in the 15th century work "Arithmetic Key" by Jamshīd al-Kāshī.

A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century.[20]

Natural languages

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 ten-one and 23 as two-ten-three, and 89345 is expressed as 8万9千3百4十5 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 "ten-one".

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.[21]

Other bases

Some cultures do, or did, use other bases of numbers.

  • Pre-Columbian Mesoamerican cultures such as the Maya used a base-20 system (using all twenty fingers and toes).
  • The Babylonians used a combination of decimal with base 60.
  • 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.[22]
  • Many languages[23] use quinary number systems, including Gumatj, Nunggubuyu,[24] Kuurn Kopan Noot[25] 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 base-12 systems[citation needed]
  • The Huli language of Papua New Guinea is reported to have base-15 numbers.[26] Ngui means 15, ngui ki means 15×2 = 30, and ngui ngui means 15×15 = 225.
  • Umbu-Ungu, also known as Kakoli, is reported to have base-24 numbers.[27] Tokapu means 24, tokapu talu means 24×2 = 48, and tokapu tokapu means 24×24 = 576.
  • Ngiti is reported to have a base-32 number system with base-4 cycles.[28]

See also

References

  1. ^ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
  2. ^ 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)
  3. ^ a b Math Made Nice-n-Easy. Piscataway, N.J.: Research Education Association. 1999. p. 141. ISBN 0-87891-200-2. http://books.google.com/books?id=ebx9StilsqIC&pg=PA141#v=onepage&q&f=false. 
  4. ^ 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.
  5. ^ Decimal Computation, Hermann Schmid, John Wiley & Sons 1974 (ISBN 047176180X); reprinted in 1983 by Robert E. Krieger Publishing Company (ISBN 0898743184)
  6. ^ Decimal Floating-Point: Algorism for Computers, Cowlishaw, M. F., Proceedings 16th IEEE Symposium on Computer Arithmetic, ISBN 0-7695-1894-X, pp104-111, IEEE Comp. Soc., June 2003
  7. ^ Decimal Arithmetic - FAQ
  8. ^ Egyptian numerals
  9. ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0140099190, pp. 200-213 (Egyptian Numerals)
  10. ^ Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, ISBN 9780486421650, p. 50
  11. ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0140099190, pp.213-218 (Cretan numerals)
  12. ^ a b Greek numerals
  13. ^ Menninger, Karl: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, ISBN 3-525-40725-4, pp. 150-153
  14. ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0140099190, pp. 218f. (The Hittite hieroglyphic system)
  15. ^ Lam Lay Yong et al The Fleeting Footsteps p 137-139
  16. ^ 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. 
  17. ^ Jean-Claude Martzloff, A History of Chinesse Mathematics, Springer 1997 ISBN 3-540-33782-2
  18. ^ 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.
  19. ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 9780691114859. 
  20. ^ B. L. van der Waerden (1985). A History of Algebra. From Khwarizmi to Emmy Noether. Berlin: Springer-Verlag. 
  21. ^ Azar, Beth (1999). "English words may hinder math skills development". American Psychology Association Monitor 30 (4). Archived from the original on 2007-10-21. http://web.archive.org/web/20071021015527/http://www.apa.org/monitor/apr99/english.html. 
  22. ^ 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.
  23. ^ Harald Hammarström, Rarities in Numeral Systems: "Bases 5, 10, and 20 are omnipresent."
  24. ^ Harris, John (1982). Hargrave, Susanne. ed. Facts and fallacies of aboriginal number systems. 8. 153–181. http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf 
  25. ^ Dawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.
  26. ^ Cheetham, Brian (1978). "Counting and Number in Huli". Papua New Guinea Journal of Education 14: 16–35. http://www.uog.ac.pg/PUB08-Oct-03/cheetham.htm 
  27. ^ Bowers, Nancy; Lepi, Pundia (1975). "Kaugel Valley systems of reckoning". Journal of the Polynesian Society 84 (3): 309–324. http://www.ethnomath.org/resources/bowers-lepi1975.pdf 
  28. ^ Hammarström, Harald (2006). "Proceedings of Rara & Rarissima Conference". http://www.cs.chalmers.se/~harald2/rarapaper.pdf 

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  • décimal — décimal, ale, aux [ desimal, o ] adj. et n. f. • 1746; de décime ♦ Qui procède par dix; qui a pour base le nombre dix. Numération décimale, à base dix. Nombre décimal, pouvant s écrire sous la forme d une fraction dont le dénominateur est une… …   Encyclopédie Universelle

  • decimal — DECIMÁL, Ă, decimali, e, adj., s.n. 1. adj. (Rar) Zecimal. 2. s.n. Basculă în care echilibrarea greutăţii corpurilor se realizează cu greutăţi etalon de zece ori mai mici. – Din fr. décimal. Trimis de dante, 14.07.2004. Sursa: DEX 98  DECIMÁL… …   Dicționar Român

  • decimal — (De décimo). 1. adj. Se dice de cada una de las diez partes iguales en que se divide una cantidad. 2. Perteneciente o relativo al diezmo. 3. Se dice del sistema métrico de pesas y medidas, cuyas unidades son múltiplos o divisores de diez respecto …   Diccionario de la lengua española

  • Decimāl — Decimāl, ein Beywort aus dem Latein. decimalis, welches nur in einigen Zusammensetzungen in der Rechenkunst und Geometrie gebraucht wird. Der Decimal Bruch, oder eine Decimal Zahl, ein Bruch, dessen Nenner aus 10, 100, 1.000 u.s.f. bestehet; die… …   Grammatisch-kritisches Wörterbuch der Hochdeutschen Mundart

  • Decimal — Dec i*mal, a. [F. d[ e]cimal (cf. LL. decimalis), fr. L. decimus tenth, fr. decem ten. See {Ten}, and cf. {Dime}.] Of or pertaining to decimals; numbered or proceeding by tens; having a tenfold increase or decrease, each unit being ten times the… …   The Collaborative International Dictionary of English

  • Decimal — Dec i*mal, n. A number expressed in the scale of tens; specifically, and almost exclusively, used as synonymous with a decimal fraction. [1913 Webster] {Circulating decimal}, or {Circulatory decimal}, a decimal fraction in which the same figure,… …   The Collaborative International Dictionary of English

  • decimal — Ⅰ. decimal UK US /ˈdesɪməl/ adjective ► relating to a system of counting based on the number ten: »Rises in adjusted prices are expressed to three decimal places, as in 0.246% Ⅱ. decimal UK US /ˈdesɪməl/ noun [C] also SPECIALIZED decimal… …   Financial and business terms

  • decimal — [des′ə məl] adj. [OFr < ML decimalis < L decimus, tenth < decem, TEN] of or based on the number 10; progressing by tens n. 1. a fraction with an unwritten denominator of 10 or some power of 10, indicated by a point (decimal point) before …   English World dictionary

  • décimal — décimal, ale (dé si mal, ma l ) adj. 1°   Qui regarde les dîmes. Une matière décimale. 2°   Terme d arithmétique. Qui procède par dix. Calcul décimal. Nombres décimaux. •   On peut croire qu à la longue la division décimale du jour remplacera sa… …   Dictionnaire de la Langue Française d'Émile Littré

  • decimal — adjetivo 1. [Sistema de numeración] que se basa en el 10. numeración decimal. 2. [Sistema métrico de pesos y medidas] que tiene como unidades múltiplos o divisores de diez con respecto a una cantidad que se toma como base. sistema* métrico… …   Diccionario Salamanca de la Lengua Española

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