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The decimal numeral system (also called the base-ten positional numeral system and denary Template:IPAc-en<ref>Template:OED</ref> 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 Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.<ref>Template:Cite book</ref>
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 Template:Math or Template:Math).<ref name=":1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Decimal may also refer specifically to the digits after the decimal separator, such as in "Template:Math is the approximation of Template: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 Template:Math, where Template:Math is an integer, and Template:Math 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.
Template:Anchor Originally and in most uses, a decimal has only a finite number of digits after the decimal separator. 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., Template:Math).<ref>The vinculum (overline) in 5.123144 indicates that the '144' sequence repeats indefinitely, i.e. Template:Val.</ref> 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.
OriginEdit
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.<ref name=":0">Template:Cite book</ref> Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers, for forming the decimal numeral system.<ref name=":0" />
Decimal notationEdit
For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign "−". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;<ref>In some countries, such as Arabic-speaking ones, other glyphs are used for the digits</ref> the decimal separator is the dot "Template:Math" in many countries (mostly English-speaking),<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and a comma "Template:Math" in other countries.<ref name=":1" />
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:
- <math>a_ma_{m-1}\ldots a_0</math>
- or a decimal mark separating two sequences of digits (such as "20.70828")
- <math>a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n</math>.
If Template:Math, that is, if the first sequence contains at least two digits, it is generally assumed that the first digit Template:Math is not zero. In some circumstances it may be useful to have one or more 0's on the left; this does not change the value represented by the decimal: for example, Template:Math. Similarly, if the final digit on the right of the decimal mark is zero—that is, if Template:Math—it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; Template:NoteTag for example, Template:Math and Template:Math.
For representing a negative number, a minus sign is placed before Template:Math.
The numeral <math>a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n</math> represents the number
- <math>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}</math>.
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, Template:Math, instead of Template:Math). 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 fractionsEdit
Template:Sidebar with collapsible groups 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.<ref>Template:Cite encyclopedia</ref> For example, the decimal expressions <math>0.8, 14.89, 0.00079, 1.618, 3.14159</math> represent the fractions Template:Math, Template:Math, Template:Math, Template:Math and Template:Math, 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 Template:Math, 3 not being a power of 10.
More generally, a decimal with Template:Math digits after the separator (a point or comma) represents the fraction with denominator Template:Math, 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
- <math>1=2^0\cdot 5^0, 2=2^1\cdot 5^0, 4=2^2\cdot 5^0, 5=2^0\cdot 5^1, 8=2^3\cdot 5^0, 10=2^1\cdot 5^1, 16=2^4\cdot 5^0, 20=2^2\cdot5^1, 25=2^0\cdot 5^2, \ldots</math>
Approximation using decimal numbersEdit
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 Template:Pi, being less than 10−5 off; so decimals are widely used in science, engineering and everyday life.
More precisely, for every real number Template:Mvar and every positive integer Template:Mvar, there are two decimals Template:Mvar and Template:Mvar with at most Template:Mvar digits after the decimal mark such that Template:Math and Template:Math.
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 Template:Math digits after the decimal mark, as soon as the absolute measurement error is bounded from above by Template:Math. 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 expansionEdit
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For a real number Template:Mvar and an integer Template:Math, let Template:Math denote the (finite) decimal expansion of the greatest number that is not greater than Template:Mvar that has exactly Template:Mvar digits after the decimal mark. Let Template:Math denote the last digit of Template:Math. It is straightforward to see that Template:Math may be obtained by appending Template:Math to the right of Template:Math. This way one has
and the difference of Template:Math and Template:Math amounts to
- <math>\left\vert \left [ x \right ]_n-\left [ x \right ]_{n-1} \right\vert=d_n\cdot10^{-n}<10^{-n+1}</math>,
which is either 0, if Template:Math, or gets arbitrarily small as Template:Mvar tends to infinity. According to the definition of a limit, Template:Mvar is the limit of Template:Math when Template:Mvar tends to infinity. This is written as<math display="inline">\; x = \lim_{n\rightarrow\infty} [x]_n \;</math>or
which is called an infinite decimal expansion of Template:Mvar.
Conversely, for any integer Template:Math and any sequence of digits<math display="inline">\;(d_n)_{n=1}^{\infty}</math> the (infinite) expression Template:Math is an infinite decimal expansion of a real number Template:Mvar. This expansion is unique if neither all Template:Math are equal to 9 nor all Template:Math are equal to 0 for Template:Mvar large enough (for all Template:Mvar greater than some natural number Template:Mvar).
If all Template:Math for Template:Math equal to 9 and Template:Math, the limit of the sequence<math display="inline">\;([x]_n)_{n=1}^{\infty}</math> is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: Template:Math, by Template:Math, and replacing all subsequent 9s by 0s (see 0.999...).
Any such decimal fraction, i.e.: Template:Math for Template:Math, may be converted to its equivalent infinite decimal expansion by replacing Template:Math by Template:Math 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 Template:Math, and the other containing only 9s after some place, which is obtained by defining Template:Math as the greatest number that is less than Template:Mvar, having exactly Template:Mvar digits after the decimal mark.
Rational numbersEdit
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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,
- Template:Sfrac = 0.Template:Thin space012345679Template:Thin space012... (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.
| For example, if x is | Template:Figure space0.4156156156... |
| then 10,000x is | Template:Figure space4156.156156156... |
| and 10x is | Template:Figure space4.156156156... |
| so 10,000x − 10x, i.e. 9,990x, is | Template:Figure space4152.000000000... |
| and x is | Template:Figure spaceTemplate:Sfrac |
or, dividing both numerator and denominator by 6, Template:Sfrac.
Decimal computationEdit
Most 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).<ref>"Fingers or Fists? (The Choice of Decimal or Binary Representation)", Werner Buchholz, Communications of the ACM, Vol. 2 #12, pp. 3–11, ACM Press, December 1959.</ref> 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,<ref name="Schmid_1983">Template:Cite book</ref><ref name="Schmid_1974">Template:Cite book</ref> 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).<ref>Decimal Floating-Point: Algorism for Computers, Cowlishaw, Mike F., Proceedings 16th IEEE Symposium on Computer Arithmetic, Template:Isbn, pp. 104–11, IEEE Comp. Soc., 2003</ref>
Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This is not possible in binary, because the negative powers of <math>10</math> have no finite binary fractional representation; and is generally impossible for multiplication (or division).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Decimal Floating-Point: Algorism for Computers Template:Webarchive, Cowlishaw, M. F., Proceedings 16th IEEE Symposium on Computer Arithmetic (ARITH 16 Template:Webarchive), Template:Isbn, pp. 104–11, IEEE Comp. Soc., June 2003</ref> See Arbitrary-precision arithmetic for exact calculations.
HistoryEdit
Many ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers.<ref>Template:Citation</ref> Standardized weights used in the Indus Valley Civilisation (Template:Circa) 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 – the Mohenjo-daro ruler – was divided into ten equal parts.<ref>Sergent, Bernard (1997), Genèse de l'Inde (in French), Paris: Payot, p. 113, Template:ISBN</ref><ref>Template:Cite journal</ref><ref>Bisht, R. S. (1982), "Excavations at Banawali: 1974–77", in Possehl, Gregory L. (ed.), Harappan Civilisation: A Contemporary Perspective, New Delhi: Oxford and IBH Publishing Co., pp. 113–24</ref> Egyptian hieroglyphs, in evidence since around 3000 BCE, used a purely decimal system,<ref>Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, Template:Isbn, pp. 200–13 (Egyptian Numerals)</ref> as did the Linear A script (Template:Circa) of the Minoans<ref>Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, Template:Isbn, p. 50</ref><ref>Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, Template:Isbn, pp. 213–18 (Cretan numerals)</ref> and the Linear B script (c. 1400–1200 BCE) of the Mycenaeans. The Únětice culture in central Europe (2300-1600 BC) used standardised weights and a decimal system in trade.<ref>Template:Cite book</ref> The number system of classical Greece also used powers of ten, including an intermediate base of 5, as did Roman numerals.<ref name="Greek numerals">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Notably, the polymath Archimedes (c. 287–212 BCE) invented a decimal positional system in his Sand Reckoner which was based on 108.<ref name="Greek numerals"/><ref>Menninger, Karl: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, Template:Isbn, pp. 150–53</ref> Hittite hieroglyphs (since 15th century BCE) were also strictly decimal.<ref>Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, Template:Isbn, pp. 218f. (The Hittite hieroglyphic system)</ref>
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, 1,000, 2,000, 3,000, 4,000, to 10,000.<ref>Lam Lay Yong et al. The Fleeting Footsteps pp. 137–39</ref> The world's earliest positional decimal system was the Chinese rod calculus.<ref name=Lam/>
Upper row vertical form
Lower row horizontal form
History of decimal fractionsEdit
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.<ref name=jnfractn1>Template:Cite book</ref> Calculations with decimal fractions of lengths were performed using positional counting rods, as described in the 3rd–5th century CE Sunzi Suanjing. The 5th century CE mathematician Zu Chongzhi calculated a 7-digit [[approximations of π|approximation of Template:Mvar]]. 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.<ref>Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997 Template:Isbn</ref> The number 0.96644 is denoted
- {{#invoke:Lang|lang}}
- File:Counting rod 0.png File:Counting rod h9 num.png File:Counting rod v6.png File:Counting rod h6.png File:Counting rod v4.png File:Counting rod h4.png.
Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East.<ref name=Lam>Lam Lay Yong, "The Development of Hindu–Arabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p. 38, Kurt Vogel notation</ref>
Al-Khwarizmi introduced fractions to Islamic countries in the early 9th century CE, written with a numerator above and denominator below, without a horizontal bar. This form of fraction remained in use for centuries.<ref name=Lam/><ref>Template:Cite journal</ref>
Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century.<ref name=Berggren>Template:Cite book</ref> The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350 but did not develop any notation to represent them.<ref>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.</ref> The Persian mathematician Jamshid al-Kashi used, and claimed to have discovered, decimal fractions in the 15th century.<ref name=Berggren />
A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. Stevin's influential booklet De Thiende ("the art of tenths") was first published in Dutch in 1585 and translated into French as La Disme.<ref name=van>Template:Cite book</ref>
John Napier introduced using the period (.) to separate the integer part of a decimal number from the fractional part in his book on constructing tables of logarithms, published posthumously in 1620.<ref name=constructionIA>Template:Cite book</ref>Template:Rp
Natural languagesEdit
A method of expressing every possible natural number using a set of ten symbols emerged in India.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.<ref>Template:Citation</ref>
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 {{#invoke:Lang|lang}}, 100 {{#invoke:Lang|lang}}, 1000 {{#invoke:Lang|lang}}, 10,000 {{#invoke:Lang|lang}}), and in which 11 is expressed as ten-one and 23 as two-ten-three, and 89,345 is expressed as 8 (ten thousands) {{#invoke:Lang|lang}} 9 (thousand) {{#invoke:Lang|lang}} 3 (hundred) {{#invoke:Lang|lang}} 4 (tens) {{#invoke:Lang|lang}} 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.<ref>Template:Cite journal</ref>
Other basesEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template: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 languages<ref>Template:Cite journal</ref> in Mexico have octal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.<ref>Template:Cite news</ref>
- 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.<ref>Template:Citation.</ref><ref>Template:Citation.</ref> 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<ref>Gordon's Introduction to Old Norse Template:Webarchive p. 293</ref> 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<ref>Template:Cite journal</ref> 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'.<ref>Template:Cite journal</ref><ref>Template:Cite book</ref>
- 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.<ref>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), Template:Isbn.</ref>
- Many languages<ref name="Hammarstrom 2010">Template:Cite book</ref> use quinary (base-5) number systems, including Gumatj, Nunggubuyu,<ref>Template:Cite journal</ref> Kuurn Kopan Noot<ref>Dawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.</ref> 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.<ref>Template:Cite conference</ref> So did some small communities in India and Nepal, as indicated by their languages.<ref>Template:Cite book</ref>
- The Huli language of Papua New Guinea is reported to have base-15 numbers.<ref>Template:Cite journal</ref> 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.<ref>Template:Cite journal</ref> 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.<ref name="Hammarstrom 2010"/>
- The Ndom language of Papua New Guinea is reported to have base-6 numerals.<ref>Template:Citation</ref> Mer means 6, mer an thef means 6 × 2 = 12, nif means 36, and nif thef means 36×2 = 72.