Decimal

For other uses, seeDecimal (disambiguation).

Place value of number in decimal system

Thedecimalnumeral system(also called thebase-tenpositional numeral systemanddenary/ˈdiːnəri/ordecanary) is the standard system for denotingintegerand non-integernumbers.It is the extension to non-integer numbers (decimal fractions) of theHindu–Arabic numeral system.The way of denoting numbers in the decimal system is often referred to asdecimal notation.

Adecimal numeral(also often justdecimalor, less correctly,decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by adecimal separator(usually "." or "," as in25.9703or3,1415). Decimalmay also refer specifically to the digits after the decimal separator, such as in "3.14is the approximation ofπtotwo 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 thedecimal fractions.That is,fractionsof the forma/10ⁿ,whereais an integer, andnis anon-negative integer.Decimal fractions also result from the addition of an integer and afractional part;the resulting sum sometimes is called afractional number.

Decimals are commonly used toapproximatereal numbers. By increasing the number of digits after the decimal separator, one can make theapproximation errorsas small as one wants, when one has a method for computing the new digits.

Originally and in most uses, a decimal has only a finite number of digits after the decimal seperator. However, the decimal system has been extended toinfinite decimalsfor representing anyreal number,by using aninfinite sequenceof digits after the decimal separator (seedecimal representation). In this context, the usual decimals, with a finite number of non-zero digits after the decimal separator, are sometimes calledterminating decimals.Arepeating decimalis an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g.,5.123144144144144... = 5.123144).An infinite decimal represents arational number,thequotientof two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.

Origin

Ten digits on two hands, the possible origin of decimal counting

Manynumeral systemsof 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 theEgyptian numerals,then theBrahmi numerals,Greek numerals,Hebrew numerals,Roman numerals,andChinese numerals.Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of theHindu–Arabic numeral systemfor representingintegers.This system has been extended to represent some non-integer numbers, calleddecimal fractionsordecimal numbers,for forming thedecimal numeral system.

Decimal notation

For writing numbers, the decimal system uses tendecimal digits,adecimal mark,and, fornegative numbers,aminus sign"−". The decimal digits are0,1,2, 3, 4, 5, 6, 7, 8, 9;thedecimal separatoris the dot "."in many countries (mostly English-speaking),and a comma ","in other countries.

For representing anon-negative number,a decimal numeral consists of

• either a (finite) sequence of digits (such as "2017" ), where the entire sequence represents an integer:

  $a_{m}a_{m-1}\ldots a_{0}$

• or a decimal mark separating two sequences of digits (such as "20.70828" )

$a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}$.

Ifm> 0,that is, if the first sequence contains at least two digits, it is generally assumed that the first digita_mis 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,3.14 = 03.14 = 003.14.Similarly, if the final digit on the right of the decimal mark is zero—that is, ifb_n= 0—it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number;for example,15 = 15.0 = 15.00and5.2 = 5.20 = 5.200.

For representing anegative number,a minus sign is placed beforea_m.

The numeral$a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}$represents the number

$a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots +a_{0}10^{0}+{\frac {b_{1}}{10^{1}}}+{\frac {b_{2}}{10^{2}}}+\cdots +{\frac {b_{n}}{10^{n}}}$.

Theinteger partorintegral partof a decimal numeral is the integer written to the left of the decimal separator (see alsotruncation). 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 thefractional 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 incomputing,that the integer part is not written (for example,.1234,instead of0.1234). 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 apositional numeral system.

Decimal fractions

Decimal fractions(sometimes calleddecimal numbers,especially in contexts involving explicit fractions) are therational numbersthat may be expressed as afractionwhosedenominatoris apowerof ten.For example, the decimal expressions$0.8,14.89,0.00079,1.618,3.14159$represent the fractions⁠4/5⁠,⁠1489/100⁠,⁠79/100000⁠,⁠+809/500⁠and⁠+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) is⁠1/3⁠,3 not being a power of 10.

More generally, a decimal withndigits after theseparator(a point or comma) represents the fraction with denominator10ⁿ,whose numerator is the integer obtained by removing the separator.

It follows that a number is a decimal fractionif and only ifit has a finite decimal representation.

Expressed asfully reduced fractions,the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are

$1=2^{0}\cdot 5^{0},2=2^{1}\cdot 5^{0},4=2^{2}\cdot 5^{0},5=2^{0}\cdot 5^{1},8=2^{3}\cdot 5^{0},10=2^{1}\cdot 5^{1},16=2^{4}\cdot 5^{0},20=2^{2}\cdot 5^{1},25=2^{0}\cdot 5^{2},\ldots$

Approximation using decimal numbers

Decimal numerals do not allow an exact representation for allreal numbers.Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximatesπ,being less than 10⁻⁵off; so decimals are widely used inscience,engineeringand everyday life.

More precisely, for every real numberxand every positive integern,there are two decimalsLanduwith at mostndigits after the decimal mark such thatL≤x≤uand(u−L) = 10⁻ⁿ.

Numbers are very often obtained as the result ofmeasurement.As measurements are subject tomeasurement uncertaintywith a knownupper bound,the result of a measurement is well-represented by a decimal withndigits after the decimal mark, as soon as the absolute measurement error is bounded from above by10⁻ⁿ.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 alsosignificant figures).

Infinite decimal expansion

Main article:Decimal representation

For areal numberxand an integern≥ 0,let[x]_ndenote the (finite) decimal expansion of the greatest number that is not greater thanxthat has exactlyndigits after the decimal mark. Letd_idenote the last digit of[x]_i.It is straightforward to see that[x]_nmay be obtained by appendingd_nto the right of[x]_n−1.This way one has

[x]_n= [x]₀.d₁d₂...d_n−1d_n,

and the difference of[x]_n−1and[x]_namounts to

$\left\vert \left[x\right]_{n}-\left[x\right]_{n-1}\right\vert =d_{n}\cdot 10^{-n}<10^{-n+1}$,

which is either 0, ifd_n= 0,or gets arbitrarily small asntends to infinity. According to the definition of alimit,xis the limit of[x]_nwhenntends toinfinity.This is written as${\textstyle \;x=\lim _{n\rightarrow \infty }[x]_{n}\;}$or

x= [x]₀.d₁d₂...d_n...,

which is called aninfinite decimal expansionofx.

Conversely, for any integer[x]₀and any sequence of digits${\textstyle \;(d_{n})_{n=1}^{\infty }}$the (infinite) expression[x]₀.d₁d₂...d_n...is aninfinite decimal expansionof a real numberx.This expansion is unique if neither alld_nare equal to 9 nor alld_nare equal to 0 fornlarge enough (for allngreater than some natural numberN).

If alld_nforn>Nequal to 9 and[x]_n= [x]₀.d₁d₂...d_n,the limit of the sequence${\textstyle \;([x]_{n})_{n=1}^{\infty }}$is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.:d_N,byd_N+ 1,and replacing all subsequent 9s by 0s (see0.999...).

Any such decimal fraction, i.e.:d_n= 0forn>N,may be converted to its equivalent infinite decimal expansion by replacingd_Nbyd_N− 1and replacing all subsequent 0s by 9s (see0.999...).

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[x]_n,and the other containing only 9s after some place, which is obtained by defining[x]_nas the greatest number that islessthanx,having exactlyndigits after the decimal mark.

Rational numbers

Main article:Repeating decimal

Long divisionallows computing the infinite decimal expansion of arational number.If the rational number is adecimal 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 arepeating decimal.For example,

⁠1/81⁠= 0. 012345679 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, ifxis	0.4156156156...
then 10,000xis	4156.156156156...
and 10xis	4.156156156...
so 10,000x− 10x,i.e. 9,990x,is	4152.000000000...
andxis	⁠4152/9990⁠

or, dividing both numerator and denominator by 6,⁠692/1665⁠.

Decimal computation

Diagram of the world's earliest known multiplica­tion table (c. 305 BCE) from theWarring States period

Most moderncomputerhardware and software systems commonly use abinary representationinternally (although many early computers, such as theENIACor theIBM 650,used decimal representation internally). For external use by computer specialists, this binary representation is sometimes presented in the relatedoctalorhexadecimalsystems.

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 ofbinary-coded decimal,especially in database implementations, but there are other decimal representations in use (includingdecimal floating pointsuch as in newer revisions of theIEEE 754 Standard for Floating-Point Arithmetic).

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$10$have no finite binary fractional representation; and is generally impossible for multiplication (or division).SeeArbitrary-precision arithmeticfor exact calculations.

History

The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BCE, during theWarring Statesperiod in China.

Many ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers.Standardized weights used in theIndus Valley Civilisation(c. 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 – theMohenjo-daro ruler– was divided into ten equal parts.Egyptian hieroglyphs,in evidence since around 3000 BCE, used a purely decimal system,as did theLinear Ascript (c. 1800–1450 BCE) of theMinoansand theLinear Bscript (c. 1400–1200 BCE) of theMycenaeans.TheÚnětice culturein central Europe (2300-1600 BC) used standardised weights and a decimal system in trade.The number system ofclassical Greecealso used powers of ten, including an intermediate base of 5, as didRoman numerals.Notably, the polymathArchimedes(c. 287–212 BCE) invented a decimal positional system in hisSand Reckonerwhich was based on 10⁸.Hittitehieroglyphs (since 15th century BCE) were also strictly decimal.

The Egyptian hieratic numerals, the Greek Alpha bet numerals, the Hebrew Alpha bet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1000, 2000, 3000, 4000, to 10,000. The world's earliest positional decimal system was the Chineserod calculus.

The world's earliest positional decimal system
Upper row vertical form
Lower row horizontal form

History of decimal fractions

counting rod decimal fraction 1/7

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.Calculations with decimal fractions of lengths wereperformed using positional counting rods,as described in the 3rd–5th century CESunzi Suanjing.The 5th century CE mathematicianZu Chongzhicalculated a 7-digitapproximation ofπ.Qin Jiushao's bookMathematical Treatise in Nine Sections(1247) explicitly writes a decimal fraction representing a number rather than a measurement, using counting rods.The number 0.96644 is denoted

Tấc

.

Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East.

Al-Khwarizmiintroduced 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.

Positional decimal fractions appear for the first time in a book by the Arab mathematicianAbu'l-Hasan al-Uqlidisiwritten in the 10th century.The Jewish mathematicianImmanuel Bonfilsused decimal fractions around 1350 but did not develop any notation to represent them.The Persian mathematicianJamshid al-Kashiused, and claimed to have discovered, decimal fractions in the 15th century.

A forerunner of modern European decimal notation was introduced bySimon Stevinin the 16th century. Stevin's influential bookletDe Thiende( "the art of tenths" ) was first published in Dutch in 1585 and translated into French asLa Disme.

John Napierintroduced 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.

Natural languages

A method of expressing every possiblenatural numberusing a set of ten symbols emerged in India.Several Indian languages show a straightforward decimal system.Dravidian languageshave numbers between 10 and 20 expressed in a regular pattern of addition to 10.

TheHungarian languagealso 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 (10Mười,100Trăm,1000Ngàn,10,000Vạn), and in which 11 is expressed asten-oneand 23 astwo-ten-three,and 89,345 is expressed as 8 (ten thousands)Vạn9 (thousand)Ngàn3 (hundred)Trăm4 (tens)Mười5 is found inChinese,and inVietnamesewith a few irregularities.Japanese,Korean,andThaihave 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 asQuechuaandAymarahave an almost straightforward decimal system, in which 11 is expressed asten with oneand 23 astwo-ten with three.

Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.

Other bases

Main article:Positional notation

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

• Pre-ColumbianMesoamericancultures such as theMayaused abase-20system (perhaps based on using all twenty fingers andtoes).
• TheYukilanguage inCaliforniaand the Pamean languagesinMexicohaveoctal(base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.
• 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.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'sIntroduction to Old NorseArchived2016-04-15 at theWayback Machinep. 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240.Goodaredetails 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'.
• Many or all of theChumashan languagesoriginally used abase-4counting system, in which the names for numbers were structured according to multiples of 4 and16.
• Many languagesusequinary (base-5)number systems, includingGumatj,Nunggubuyu,Kuurn Kopan NootandSaraveca.Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
• SomeNigeriansuseduodecimalsystems.So did some small communities in India and Nepal, as indicated by their languages.
• TheHuli languageofPapua New Guineais reported to havebase-15numbers.Nguimeans 15,ngui kimeans 15 × 2 = 30, andngui nguimeans 15 × 15 = 225.
• Umbu-Ungu,also known as Kakoli, is reported to havebase-24numbers.Tokapumeans 24,tokapu talumeans 24 × 2 = 48, andtokapu tokapumeans 24 × 24 = 576.
• Ngitiis reported to have abase-32number system with base-4 cycles.
• TheNdom languageofPapua New Guineais reported to havebase-6numerals.Mermeans 6,mer an thefmeans 6 × 2 = 12,nifmeans 36, andnif thefmeans 36×2 = 72.

See also

• Algorism
• Binary-coded decimal(BCD)
• Decimal classification
• Decimal computer
• Decimal time
• Decimal representation
• Decimal section numbering
• Decimal separator
• Decimalisation
• Densely packed decimal(DPD)
• Duodecimal
• Octal
• Scientific notation
• Serial decimal
• Metric prefix