View source for Decimal

{{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]]
{{Numeral systems|expand=Place-value notation|expand2=By radix/base}}

The '''decimal''' [[numeral system]] (also called the '''base-ten''' [[positional numeral system]] and '''denary''' {{IPAc-en|d|iː|n|ər|i}}<ref>{{OED|denary}}</ref> or '''decanary''') is the standard system for denoting [[integer]] and non-integer [[number]]s. 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>{{Cite book |last1=Yong |first1=Lam Lay |url=http://dx.doi.org/10.1142/5425 |title=Fleeting Footsteps |last2=Se |first2=Ang Tian |date=April 2004 |publisher=[[World Scientific]] |isbn=978-981-238-696-0 |at=268 |doi=10.1142/5425 |access-date=March 17, 2022 |archive-date=April 1, 2023 |archive-url=https://web.archive.org/web/20230401132256/https://www.worldscientific.com/worldscibooks/10.1142/5425 |url-status=live }}</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 {{math|25.9703}} or {{math|3,1415}}).<ref name=":1">{{Cite web |last=Weisstein |first=Eric W. |date=March 10, 2022 |title=Decimal Point |url=https://mathworld.wolfram.com/DecimalPoint.html |url-status=live |access-date=March 17, 2022 |website=Wolfram MathWorld |language=en |archive-date=March 21, 2022 |archive-url=https://web.archive.org/web/20220321195047/https://mathworld.wolfram.com/DecimalPoint.html }}</ref>
''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''".

The numbers that may be represented exactly by a decimal of finite length are the [[#Decimal fractions|'''decimal fractions''']]. That is, [[fraction (mathematics)|fractions]] of the form {{math|''a''/10<sup>''n''</sup>}}, 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 [[approximation (mathematics)|approximate]] real numbers. By increasing the number of digits after the decimal separator, one can make the [[approximation error]]s as small as one wants, when one has a method for computing the new digits.  In the sciences, the number of decimal places given generally gives an indication of the precision to which a quantity is known; for example, if a mass is given as 1.32 milligrams, it usually means there is reasonable confidence that the true mass is somewhere between 1.315 milligrams and 1.325 milligrams, whereas if it is given as 1.320 milligrams, then it is likely between 1.3195 and 1.3205 milligrams.  The same holds in pure mathematics; for example, if one computes the square root of 22 to two digits past the decimal point, the answer is 4.69, whereas computing it to three digits, the answer is 4.690.  The extra 0 at the end is meaningful, in spite of the fact that 4.69 and 4.690 are the same real number.

{{anchor|terminating decimal}}
In principle, the decimal expansion of any [[real number]] can be carried out as far as desired past the decimal point. If the expansion reaches a point where all remaining digits are zero, then the remainder can be omitted, and such an expansion is called a '''terminating decimal'''. 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}}}}).<ref>The [[Vinculum (symbol)|vinculum (overline)]] in 5.123<span style="text-decoration: overline;">144</span> indicates that the '144' sequence repeats indefinitely, i.e. {{val|5.123144144144144|s=...}}.</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.

==Origin==
[[File:Two hand, ten fingers.jpg|thumb|right|Ten digits on two hands, the possible origin of decimal counting|upright=1.2]]
Many [[numeral system]]s 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">{{Cite book |last=Lockhart |first=Paul |title=Arithmetic |date=2017 |publisher=The Belknap Press of Harvard University Press |isbn=978-0-674-97223-0 |location=Cambridge, Massachusetts London, England}}</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 [[integer]]s. This system has been extended to represent some non-integer numbers, called ''[[#Decimal fractions|decimal fractions]]'' or ''decimal numbers'', for forming the ''decimal numeral system''.<ref name=":0" />

== Decimal notation ==

For writing numbers, the decimal system uses ten [[decimal digit]]s, a [[decimal mark]], and, for [[negative number]]s, 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 [[glyph]]s are used for the digits</ref> the [[decimal separator]] is the dot "{{math|.}}" in many countries (mostly English-speaking),<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Decimal|url=https://mathworld.wolfram.com/Decimal.html|access-date=2020-08-22|website=mathworld.wolfram.com|language=en|archive-date=2020-03-18|archive-url=https://web.archive.org/web/20200318204545/https://mathworld.wolfram.com/Decimal.html|url-status=live}}</ref> and a comma "{{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 {{math|''m'' > 0}}, that is, if the first sequence contains at least two digits, it is generally assumed that the first digit {{math|''a''<sub>''m''</sub>}} 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, {{math|1=3.14 = 03.14 = 003.14}}. Similarly, if the final digit on the right of the decimal mark is zero—that is, if {{math|1=''b''<sub>''n''</sub> = 0}}—it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; {{NoteTag|text=Sometimes, the extra zeros are used for indicating the [[accuracy and precision|accuracy]] of a measurement. For example, "15.00&nbsp;m" may indicate that the measurement error is less than one centimetre (0.01&nbsp;m), while "15&nbsp;m" may mean that the length is roughly fifteen metres and that the error may exceed 10&nbsp;centimetres.}} for example, {{math|1=15 = 15.0 = 15.00}} and {{math|1=5.2 = 5.20 = 5.200}}.

For representing a [[negative number]], a minus sign is placed before {{math|''a''<sub>''m''</sub>}}.

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, {{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 number]]s that may be expressed as a [[Fraction (mathematics)|fraction]] whose [[denominator]] is a [[exponentiation|power]] of ten.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Decimal_fraction|title=Decimal Fraction|encyclopedia=[[Encyclopedia of Mathematics]]|access-date=2013-06-18|archive-date=2013-12-11|archive-url=https://web.archive.org/web/20131211035917/http://www.encyclopediaofmath.org/index.php/Decimal_fraction|url-status=live}}</ref> For example, the decimal expressions <math>0.8, 14.89, 0.00079, 1.618, 3.14159</math> represent the fractions {{math|{{sfrac|8|10}}}}, {{math|{{sfrac|1489|100}}}}, {{math|{{sfrac|79|100000}}}}, {{Math|{{sfrac||1618|1000}}}} 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 [[Decimal separator|separator]] (a point or comma) represents the fraction with denominator {{math|10<sup>''n''</sup>}}, 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 fraction]]s, 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 numbers===
Decimal numerals do not allow an exact representation for all [[real number]]s. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates {{pi}}, being less than 10<sup>−5</sup> 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'' − ''L'') = 10<sup>−''n''</sup>}}.

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|10<sup>−''n''</sup>}}. 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==
{{main|Decimal representation}}

For a [[real number]] {{Mvar|x}} and an integer {{Math|''n'' ≥ 0}}, let {{Math|[''x'']<sub>''n''</sub>}} 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''<sub>''i''</sub>}} denote the last digit of {{Math|[''x'']<sub>''i''</sub>}}. It is straightforward to see that {{Math|[''x'']<sub>''n''</sub>}} may be obtained by appending {{Math|''d''<sub>''n''</sub>}} to the right of {{Math|[''x'']<sub>''n''−1</sub>}}. This way one has
:{{Math|1=[''x'']<sub>''n''</sub> = [''x'']<sub>0</sub>.''d''<sub>1</sub>''d''<sub>2</sub>...''d''<sub>''n''−1</sub>''d''<sub>''n''</sub>}},

and the difference of {{Math|[''x'']<sub>''n''−1</sub>}} and {{Math|[''x'']<sub>''n''</sub>}} 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 {{Math|1=''d''<sub>''n''</sub> = 0}}, or gets arbitrarily small as ''{{Mvar|n}}'' tends to infinity. According to the definition of a [[limit (mathematics)|limit]], ''{{Mvar|x}}'' is the limit of {{Math|[''x'']<sub>''n''</sub>}} when ''{{Mvar|n}}'' tends to [[infinity]]. This is written as<math display="inline">\; x = \lim_{n\rightarrow\infty} [x]_n \;</math>or
: {{Math|1=''x'' = [''x'']<sub>0</sub>.''d''<sub>1</sub>''d''<sub>2</sub>...''d''<sub>''n''</sub>...}},
which is called an '''infinite decimal expansion''' of ''{{Mvar|x}}''.

Conversely, for any integer {{Math|[''x'']<sub>0</sub>}} and any sequence of digits<math display="inline">\;(d_n)_{n=1}^{\infty}</math> the (infinite) expression {{Math|[''x'']<sub>0</sub>.''d''<sub>1</sub>''d''<sub>2</sub>...''d''<sub>''n''</sub>...}} is an ''infinite decimal expansion'' of a real number ''{{Mvar|x}}''. This expansion is unique if neither all {{Math|''d''<sub>''n''</sub>}} are equal to 9 nor all {{Math|''d''<sub>''n''</sub>}} are equal to 0 for ''{{Mvar|n}}'' large enough (for all ''{{Mvar|n}}'' greater than some natural number {{Mvar|N}}).

If all {{Math|''d''<sub>''n''</sub>}} for {{Math|''n'' > ''N''}} equal to 9 and {{Math|1=[''x'']<sub>''n''</sub> = [''x'']<sub>0</sub>.''d''<sub>1</sub>''d''<sub>2</sub>...''d''<sub>''n''</sub>}}, 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.: {{Math|''d''<sub>''N''</sub>}}, by {{Math|''d''<sub>''N''</sub> + 1}}, and replacing all subsequent 9s by 0s (see [[0.999...]]).

Any such decimal fraction, i.e.: {{Math|1=''d''<sub>''n''</sub> = 0}} for {{Math|''n'' > ''N''}}, may be converted to its equivalent infinite decimal expansion by replacing {{Math|''d''<sub>''N''</sub>}} by  {{Math|''d''<sub>''N''</sub> − 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'']<sub>''n''</sub>}}, and the other containing only 9s after some place, which is obtained by defining {{Math|[''x'']<sub>''n''</sub>}} as the greatest number that is ''less'' than {{Mvar|x}}, having exactly ''{{Mvar|n}}'' digits after the decimal mark.

=== Rational numbers ===
{{main|Repeating decimal}}

[[Long division]] allows computing the infinite decimal expansion of a [[rational number]]. If the rational number is a [[#decimal fraction|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.
{|
|-
|For example, if ''x'' is || {{figure space|6}}0.4156156156...
|-
|then 10,000''x'' is || {{figure space|3}}4156.156156156...
|-
|and 10''x'' is|| {{figure space|6}}4.156156156...
|-
|so 10,000''x'' − 10''x'', i.e. 9,990''x'', is||{{figure space|3}}4152.000000000...
|-
|and ''x'' is|| {{figure space|3}}{{sfrac|4152|9990}}
|}
or, dividing both numerator and denominator by 6, {{sfrac|692|1665}}.

== Decimal computation ==

[[File:Decimal multiplication table.JPG|thumb|right|360px|Diagram of the world's earliest known multiplication table ({{circa|305 BCE}}) from the [[Warring States period]]]]

Most modern [[computer]] hardware and software systems commonly use a [[Binary numeral system|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">{{cite book |title=Decimal Computation |first=Hermann |author=Schmid<!-- General Electric Company, Binghamton, New York, US --> |author-link=Hermann Schmid (computer scientist) |orig-year=1974 |date=1983 |edition=1 (reprint) |publisher=Robert E. Krieger Publishing Company |location=Malabar, Florida |isbn=0-89874-318-4}}</ref><ref name="Schmid_1974">{{cite book |title=Decimal Computation |first=Hermann |author=Schmid<!-- General Electric Company, Binghamton, New York, USA --> |author-link=Hermann Schmid (computer scientist) |date=1974 |edition=1st |publisher=[[John Wiley & Sons]] |location=Binghamton, New York|isbn=0-471-76180-X |url-access=registration |url=https://archive.org/details/decimalcomputati0000schm }}</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|IEEE 754 Standard for Floating-Point Arithmetic]]).<ref>''Decimal Floating-Point: Algorism for Computers'', [[Mike Cowlishaw|Cowlishaw, Mike F.]], Proceedings [[16th IEEE Symposium on Computer Arithmetic]], {{isbn|0-7695-1894-X}}, 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>{{Cite web |url=http://speleotrove.com/decimal/decifaq.html |title=Decimal Arithmetic – FAQ<!-- Bot generated title --> |access-date=2008-08-15 |archive-date=2009-04-29 |archive-url=https://web.archive.org/web/20090429044324/http://speleotrove.com/decimal/decifaq.html |url-status=live }}</ref><ref>[http://www.dec.usc.es/arith16/papers/paper-107.pdf Decimal Floating-Point: Algorism for Computers] {{Webarchive|url=https://web.archive.org/web/20031116234555/http://www.dec.usc.es/arith16/papers/paper-107.pdf |date=2003-11-16 }}, [[Cowlishaw]], M. F., ''Proceedings [[16th IEEE Symposium on Computer Arithmetic]]'' ([http://www.dec.usc.es/arith16/ ARITH&nbsp;16] {{Webarchive|url=https://web.archive.org/web/20100819005155/http://www.dec.usc.es/arith16/ |date=2010-08-19 }}), {{isbn|0-7695-1894-X}}, pp.&nbsp;104–11, IEEE Comp. Soc., June 2003</ref> 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 States]] period in China.]]
Many ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers.<ref>{{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}}</ref> Standardized weights used in the [[Indus Valley Civilisation]] ({{circa|3300–1300 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 – the ''Mohenjo-daro ruler'' – was divided into ten equal parts.<ref>Sergent, Bernard (1997), ''Genèse de l'Inde'' (in French), Paris: Payot, p. 113, {{ISBN|2-228-89116-9}}</ref><ref>{{cite journal | last1 = Coppa | first1 = A. | display-authors = etal | year = 2006 | title = Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population | bibcode = 2006Natur.440..755C | journal = Nature | volume = 440 | issue = 7085| pages = 755–56 | doi = 10.1038/440755a | pmid = 16598247 | s2cid = 6787162 }}</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, {{isbn|0-14-009919-0}}, pp.&nbsp;200–13 (Egyptian Numerals)</ref> as did the [[Linear A]] script ({{circa|1800–1450 BCE}}) of the [[Minoan civilization|Minoans]]<ref>Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, {{isbn|978-0-486-42165-0}}, p.&nbsp;50</ref><ref>Georges Ifrah: ''From One to Zero. A Universal History of Numbers'', Penguin Books, 1988, {{isbn|0-14-009919-0}}, pp. 213–18 (Cretan numerals)</ref> and the [[Linear B]] script (c. 1400–1200 BCE) of the [[Mycenaean Greece|Mycenaeans]]. The [[Únětice culture]] in central Europe (2300-1600 BC) used standardised weights and a decimal system in trade.<ref>{{cite book |last1=Krause |first1=Harald |url=https://www.academia.edu/34550316 |title=Spangenbarrenhort Oberding |last2=Kutscher |first2=Sabrina |date=2017 |publisher=Museum Erding |isbn=978-3-9817606-5-1 |pages=238–243 |chapter=Spangenbarrenhort Oberding: Zusammenfassung und Ausblick}}</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">{{Cite web |url=http://www-history.mcs.st-and.ac.uk/HistTopics/Greek_numbers.html |title=Greek numbers |access-date=2019-07-21 |archive-date=2019-07-21 |archive-url=https://web.archive.org/web/20190721085640/http://www-history.mcs.st-and.ac.uk/HistTopics/Greek_numbers.html |url-status=live }}</ref> Notably, the polymath [[Archimedes]] (c. 287–212 BCE) invented a decimal positional system in his [[The Sand Reckoner|Sand Reckoner]] which was based on 10<sup>8</sup>.<ref name="Greek numerals"/><ref>[[Karl Menninger (mathematics)|Menninger, Karl]]: ''Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl'', Vandenhoeck und Ruprecht, 3rd. ed., 1979, {{isbn|3-525-40725-4}}, pp.&nbsp;150–53</ref> [[Hittites|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, {{isbn|0-14-009919-0}}, 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/>
[[File:Chounumerals.svg|thumb|right|240px|The world's earliest positional decimal system<br/> Upper row vertical form<br/> Lower row horizontal form]]

=== History of decimal fractions ===
[[File:Rod fraction.jpg|thumb|right|120px|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.<ref name=jnfractn1>{{Cite book | author=Joseph Needham | author-link=Joseph Needham | chapter = 19.2 Decimals, Metrology, and the Handling of Large Numbers |pages=82–90 | title = Science and Civilisation in China |volume=III, "Mathematics and the Sciences of the Heavens and the Earth" | title-link=Science and Civilisation in China | year = 1959 | publisher = Cambridge University Press}}</ref> Calculations with decimal fractions of lengths were [[Rod calculus#Decimal fraction|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 {{mvar|π}}]]. [[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  {{isbn|3-540-33782-2}}</ref> The number 0.96644 is denoted

:{{lang|zh|寸}}
:[[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]].

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>{{cite journal | last1 = Lay Yong | first1 = Lam | author-link = Lam Lay Yong | title = A Chinese Genesis, Rewriting the history of our numeral system | journal = Archive for History of Exact Sciences | volume = 38 | pages = 101–08 }}</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>{{cite book | first=J. Lennart | last=Berggren | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam |editor-first=Victor J.|editor-last=Katz|publisher=Princeton University Press | year=2007 | isbn=978-0-691-11485-9 | page=530 }}</ref> The Jewish mathematician [[Immanuel Bonfils]] used decimal fractions around 1350 but did not develop any notation to represent them.<ref>[[Solomon Gandz|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 />

<div style="float: right;">[[File:Stevin-decimal notation.svg]]</div>
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>{{Cite book | author = B. L. van der Waerden | author-link = Bartel Leendert van der Waerden | year = 1985 | title = A History of Algebra. From Khwarizmi to Emmy Noether | publisher = Springer-Verlag | place = Berlin}}</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>{{cite book|title=[[Commons:File:The Construction of the Wonderful Canon of Logarithms.djvu|The Construction of the Wonderful Canon of Logarithms]]|first=John|last=Napier|translator-last1=Macdonald|translator-first1= William Rae|date=1889|orig-date=1620|publisher=Blackwood & Sons|publication-place=Edinburgh|via=Internet Archive|quote=In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many cyphers after it as there are figures after the period.}}</ref>{{rp|p. 8, archive p. 32}}

=== Natural languages ===
A method of expressing every possible [[natural number]] using a set of ten symbols emerged in India.<ref>{{cite web |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/ |title=Indian numerals|work=Ancient Indian mathematics }}</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>{{Citation |title=Appendix:Cognate sets for Dravidian languages |date=2024-09-25 |work=Wiktionary, the free dictionary |url=https://en.wiktionary.org/wiki/Appendix:Cognate_sets_for_Dravidian_languages |access-date=2024-11-09 |language=en}}</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 {{lang|zh|十}}, 100 {{lang|zh|百}}, 1000 {{lang|zh|千}}, 10,000 {{lang|zh|万}}), and in which 11 is expressed as ''ten-one'' and 23 as ''two-ten-three'', and 89,345 is expressed as 8 (ten thousands) {{lang|zh|万}} 9 (thousand) {{lang|zh|千}} 3 (hundred) {{lang|zh|百}} 4 (tens) {{lang|zh|十}} 5 is found in [[Chinese language|Chinese]], and in [[Vietnamese language|Vietnamese]] with a few irregularities.  [[Japanese language|Japanese]], [[Korean language|Korean]], and [[Thai language|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 [[Quechuan languages|Quechua]] and [[Aymara language|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>{{Cite journal| last=Azar| first=Beth| year=1999| title=English words may hinder math skills development| url=http://www.apa.org/monitor/apr99/english.html  |journal=APA Monitor| volume=30| issue=4 |archive-url = https://web.archive.org/web/20071021015527/http://www.apa.org/monitor/apr99/english.html |archive-date = 2007-10-21}}</ref>

=== Other bases ===
{{main | Positional notation}}
{{Fundamental info units}}
Some cultures do, or did, use other bases of numbers.
* [[Pre-Columbian]] [[Mesoamerica]]n cultures such as the [[Maya numerals|Maya]] used a [[vigesimal|base-20]] system (perhaps based on using all twenty fingers and [[toe]]s).
* The [[Yuki tribe|Yuki]] language in [[California]] and the Pamean languages<ref>{{Cite journal
 | last=Avelino
 | first=Heriberto
 | title=The typology of Pame number systems and the limits of Mesoamerica as a linguistic area
 | journal=Linguistic Typology
 | year=2006
 | volume=10
 | issue=1
 | pages=41–60
 | url=http://linguistics.berkeley.edu/~avelino/Avelino_2006.pdf |archive-url=https://web.archive.org/web/20060712201924/http://www.linguistics.berkeley.edu/~avelino/Avelino_2006.pdf |archive-date=2006-07-12 |url-status=live
 | doi=10.1515/LINGTY.2006.002
 | s2cid=20412558
 }}</ref> in [[Mexico]] have [[octal]] ([[radix|base]]-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.<ref>{{cite news|jstor=2686959|title=Ethnomathematics: A Multicultural View of Mathematical Ideas|author=Marcia Ascher|author-link= Marcia Ascher |publisher=The College Mathematics Journal}}</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>{{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 = 293–99
 | title = Observations on the Germanic numerals
 | volume = 11}}.</ref><ref>{{citation
 | last = Voyles | first = Joseph
 | date = October 1987
 | issue = 4
 | journal = The Journal of English and Germanic Philology
 | jstor = 27709904
 | pages = 487–95
 | title = The cardinal numerals in pre-and proto-Germanic
 | volume = 86}}.</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 [https://www.scribd.com/doc/49127454/Introduction-to-Old-Norse-by-E-V-Gordon Introduction to Old Norse] {{Webarchive|url=https://web.archive.org/web/20160415205641/https://www.scribd.com/doc/49127454/Introduction-to-Old-Norse-by-E-V-Gordon |date=2016-04-15 }} p.&nbsp;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>{{cite journal
 | last = Goodare | first = Julian
 | date = November 1994
 | doi = 10.9750/psas.123.395.418
 | journal = Proceedings of the Society of Antiquaries of Scotland
 | pages = 395–418
 | title = The long hundred in medieval and early modern Scotland
 | volume = 123}}</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>{{Cite journal|last=Stevenson|first=W.H.|date=1890|title=The Long Hundred and its uses in England|journal=Archaeological Review|volume=December 1889|pages=313–22}}</ref><ref>{{Cite book|last=Poole, Reginald Lane|title=The Exchequer in the twelfth century : the Ford lectures delivered in the University of Oxford in Michaelmas term, 1911|date=2006|publisher=Lawbook Exchange|isbn=1-58477-658-7|location=Clark, NJ|oclc=76960942}}</ref>
* Many or all of the [[Chumashan languages]] originally used a [[quaternary numeral system|base-4]] counting system, in which the names for numbers were structured according to multiples of 4 and [[hexadecimal|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), {{isbn|0-292-75531-7}}.</ref>
* Many languages<ref name="Hammarstrom 2010">{{Cite book
| contribution=Rarities in Numeral Systems
| first=Harald
| last=Hammarström
| editor1-first=Jan
| editor1-last=Wohlgemuth
| editor2-first=Michael
| editor2-last=Cysouw
| title=Rethinking Universals: How rarities affect linguistic theory
| date=17 May 2007
| location=Berlin
| publisher=Mouton de Gruyter
| series=Empirical Approaches to Language Typology
| volume=45
| publication-date=2010
| url=http://www.cs.chalmers.se/~harald2/rarapaper.pdf
| url-status=dead
| archive-url=https://web.archive.org/web/20070819214057/http://www.cs.chalmers.se/~harald2/rarapaper.pdf
| archive-date=19 August 2007
}}</ref> use [[quinary|quinary (base-5)]] number systems, including [[Gumatj language|Gumatj]], [[Nunggubuyu language|Nunggubuyu]],<ref>{{Cite journal
 |title        = Facts and fallacies of aboriginal number systems
 |last         = Harris
 |first        = John
 |editor-last  = Hargrave
 |editor-first = Susanne
 |pages        = 153–81
 |year         = 1982
 |journal      = Work Papers of SIL-AAB Series B
 |volume       = 8
 |url          = http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf
 |url-status      = dead
 |archive-url   = https://web.archive.org/web/20070831202737/http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf
 |archive-date  = 2007-08-31
}}</ref> [[Kuurn Kopan Noot language|Kuurn Kopan Noot]]<ref>Dawson, J. "[https://archive.org/details/australianabori00dawsgoog ''Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria''] (1881), p.&nbsp;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 [[Nigeria]]ns use [[duodecimal]] systems.<ref>{{Cite conference
| title=Decimal vs. Duodecimal: An interaction between two systems of numeration
| last=Matsushita
| first=Shuji
| conference=2nd Meeting of the AFLANG, October 1998, Tokyo
| year=1998
| url=http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html
| archive-url=https://web.archive.org/web/20081005230737/http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html
| archive-date=2008-10-05
| access-date=2011-05-29
}}</ref>  So did some small communities in India and Nepal, as indicated by their languages.<ref>{{Cite book
| contribution=Les principes de construction du nombre dans les langues tibéto-birmanes
| first=Martine
| last=Mazaudon
| title=La Pluralité
| editor-first=Jacques
| editor-last=François
| year=2002
| pages=91–119
| publisher=Peeters
| place=Leuven
| isbn=90-429-1295-2
| url=http://lacito.vjf.cnrs.fr/documents/publi/num_WEB.pdf
| access-date=2014-09-12
| archive-date=2016-03-28
| archive-url=https://web.archive.org/web/20160328145817/http://lacito.vjf.cnrs.fr/documents/publi/num_WEB.pdf
| url-status=dead
}}</ref>
* The [[Huli language]] of [[Papua New Guinea]] is reported to have [[pentadecimal|base-15]] numbers.<ref>{{Cite journal
 | last=Cheetham
 | first=Brian
 | title=Counting and Number in Huli
 | journal=Papua New Guinea Journal of Education
 | year=1978
 | volume=14
 | pages=16–35
 | url=http://www.uog.ac.pg/PUB08-Oct-03/cheetham.htm
 | archive-url=https://web.archive.org/web/20070928061238/http://www.uog.ac.pg/PUB08-Oct-03/cheetham.htm
 | archive-date=2007-09-28
}}</ref>  ''Ngui'' means 15, ''ngui ki'' means 15 × 2 = 30, and ''ngui ngui'' means 15 × 15 = 225.
* [[Umbu-Ungu language|Umbu-Ungu]], also known as Kakoli, is reported to have [[base 24|base-24]] numbers.<ref>{{Cite journal
 |last1        = Bowers
 |first1       = Nancy
 |last2       = Lepi
 |first2      = Pundia
 |title       = Kaugel Valley systems of reckoning
 |year        = 1975
 |journal     = Journal of the Polynesian Society
 |volume      = 84
 |issue       = 3
 |pages       = 309–24
 |url         = http://www.ethnomath.org/resources/bowers-lepi1975.pdf
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20110604091351/http://www.ethnomath.org/resources/bowers-lepi1975.pdf
 |archive-date = 2011-06-04
}}</ref> ''Tokapu'' means 24, ''tokapu talu'' means 24 × 2 = 48, and ''tokapu tokapu'' means 24 × 24 = 576.
* [[Ngiti language|Ngiti]] is reported to have a [[base 32|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>{{ Citation | last=Owens | first=Kay | title=The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania | journal=Mathematics Education Research Journal | year=2001 | volume=13 | issue=1 | pages=47–71 | url=http://www.uog.ac.pg/glec/Key/Kay/owens131.htm | doi=10.1007/BF03217098 | bibcode=2001MEdRJ..13...47O | s2cid=161535519 | url-status=dead | archive-url=https://web.archive.org/web/20150926003303/http://www.uog.ac.pg/glec/Key/Kay/owens131.htm | archive-date=2015-09-26 }}</ref>  ''Mer'' means 6, ''mer an thef'' means 6 × 2 = 12, ''nif'' means 36, and ''nif thef'' means 36×2 = 72.

== See also ==
{{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]]
* [[Metric prefix]]
* [[Octal]]
* [[Scientific notation]]
* [[Serial decimal]]
}}

==Notes==
{{reflist|group=note}}

== References ==
{{Reflist}}

{{Orders of magnitude}}
{{Authority control}}

[[Category:Elementary arithmetic]]
[[Category:Fractions (mathematics)]]
[[Category:Positional numeral systems]]