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# Decimal  ( Elementary Arithmetic )

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The decimal (also called base-ten positional numeral system, and occasionally called denary) is the standard system for denoting and non-integer . It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925. The way of denoting numbers in the decimal system is often referred to as Decimal notation.Lam Lay Yong & Ang Tian Se (2004) Fleeting Footsteps. Tracing the Conception of Arithmetic and Algebra in Ancient China, Revised Edition, World Scientific, Singapore.

A decimal numeral, or just decimal, or, improperly decimal number, refers generally to the notation of a number in the decimal system, which contains a (for example 10.00 or 3.14159). Sometimes these terms are used for any numeral in the decimal system. A decimal may also refer to any digit after the decimal separator, such as in "3.14 is the approximation of to two decimals".

The numbers that may be represented in the decimal system are the decimal fractions, that is the fractions of the form , where is an integer, and is a nonnegative integer.

The decimal system has been extended to infinite decimals, for representing any , by using an infinite sequence of digits after the decimal separator (see Decimal representation). In this context, the usual decimals are sometimes called terminating decimals. A repeating decimal, is an infinite decimal, that, after some place repeats indefinitely the same sequence of digits (for example ).The viniculum (overline) in 5.123144 indicates that the '144' sequence repeats itself indefinitely, i.e. . An infinite decimal represents a if and only if it is a repeating decimal or is a finite decimal.

Origin
Many of ancient civilisations use ten and its powers for representing numbers, probably because there are ten fingers on two hands and people started counting by using their fingers. Examples are Armenian numerals, , , , , and . 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 . This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers for forming the decimal numeral system.

Decimal notation
For writing numbers, the decimal system uses ten , a , and, for , a "−". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;In some countries, such as ones, other are used for the digits the decimal separator is the dot "" in many countries (including English speaking ones), but may be a comma "" in other countries (mainly in ).

For representing a nonnegative number, a decimal consists of

• either a (finite) sequence of digits such as 2017, or in full generality,
:$a_ma_\left\{m-1\right\}\ldots a_0$
(in this case, the (entire) decimal represents an integer)
• or two sequence of digits separated by a decimal mark such as 3.14159, 15.00, or in full generality
:$a_ma_\left\{m-1\right\}\ldots a_0.b_1b_2\ldots b_n$
It is generally assumed that, if , the first digit is not zero, but, 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, . Similarly, if , it may be removed, and conversely, trailing zeros may be added without changing the represented number. For example, and . sometimes, the unnecessary zeros are used for indicating the accuracy of a measurement. For example, 15.00m may indicate that the measurement error is less than one centimeter, while 15m may mean that the length is roughly fifteen meters, and that the error may exceed 10 cm.

For representing a , a minus sign is placed before .

The numeral $a_ma_\left\{m-1\right\}\ldots a_0.b_1b_2\ldots b_n$ represents the number

$a_m10^m+a_\left\{m-1\right\}10^\left\{m-1\right\}+\cdots+a_\left\{0\right\}10^0+\frac\left\{b_1\right\}\left\{10^1\right\}+\frac\left\{b_2\right\}\left\{10^2\right\}+\cdots+\frac\left\{b_n\right\}\left\{10^n\right\}$
Therefore, 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
The numbers that are represented by decimal numeral are the decimal fractions (sometimes called decimal numbers), that is, the that may be expressed as a fraction, the of which is a of ten. For example, the numerals $0.8, 14.89, 0.00024$ represent the fractions , , . More generally, a decimal with digits after the separator represents the fraction with denominator , whose numerator is the integer obtained by removing the separator.

Expressed as a fully reduced fraction, the decimal numbers are those, whose denominator is a product of a power of 2 by a power of 5. Thus the smallest denominators of decimal numbers are

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

The , or integral part of a decimal is the integer written to the left of the decimal separator (see also ). For a nonnegative decimal, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the , 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 , that the integer part is not written (for example , instead of ). In normal writing, this is generally avoided because of the risk of confusion between the decimal mark and other punctuation.

Real number approximation
Decimal numerals do not allow an exact representation for all , so e.g. for the real number . Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates the real , being less than 10−5 off; and so decimals are widely used in , and everyday life.

More precisely, for every real number x, and every positive integer n, there are two decimals l and u, with at most n digits after the decimal mark, such that lxu and ( ul) = 10n.

Numbers are very often obtained as the result of a . As measurements are generally afflicted with some measurement error with a known , the result of a measurement is well represented by a decimal with digits after the decimal mark, as soon as the absolute measurement error is bounded from above by 10n. 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 decimal number, the numeral 0.080 is to suggest 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
For a x and an integer n ≥ 0, let x n denote the (finite) decimal expansion of the greatest number that is not greater than x, which has exactly n digits after the decimal mark. Let d i denote the last digit of x i. It is straightforward to see that x n may be obtained by appending d n to the right of x n–1. This way one has
x n = x0. d1 d2... d n−1 d n,

and the difference of x n–1 and x n amounts to

| x nx n–1| = d n ⋅ 10n < 10n+1,

which is either 0, if d n = 0, or gets arbitrarily small, when n tends to infinity. According to the definition of a limit, x is the limit of x n when n tends to . This is written as $\; x = \lim_{n\rightarrow\infty} x_n \;$ or

x = x0. d1 d2... d n...,
which is called an infinite decimal expansion of x.

Conversely, for any integer x0 and any sequence of digits $\;(d_n)_{n=1}^{\infty}$ the (infinite) expression is an infinite decimal expansion of a real number x. This expansion is unique if neither all d n are equal to 9 nor all d n are equal to 0 for n large enough (for all n greater than some natural number N).

If all d n for n > N equal to 9 and the limit of the sequence $\;(x_n)_{n=1}^{\infty}$ is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: d N, by d N + 1, and replacing all subsequent 9s by 0s (see 0.999...).

Any such decimal fraction, i.e., d n = 0 for n > N, may be converted to its equivalent infinite decimal expansion by replacing d N by d N − 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 x n, and the other containing only 9s after some place, which is obtained by defining x n as the greatest number that is less than x, having exactly n digits after the decimal mark.

Rational numbers
The allows computing the infinite decimal expansion of a . 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 0. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainder 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,
= 0.012345679012... (with 012345679 repeating).

Conversely, every eventually repeating sequence of digit is the infinite decimal expansion of a rational number. This is a consequence of the fact that the recurring part of a decimal representation is, in fact, an infinite which will sum to a rational number. For example,

$0.0123123123\ldots =\frac\left\{123\right\}\left\{10000\right\} \sum_\left\{k=0\right\}^\infty 0.001^k = \frac\left\{123\right\}\left\{10000\right\}\ \frac\left\{1\right\}\left\{1-0.001\right\} = \frac\left\{123\right\}\left\{9990\right\} = \frac\left\{41\right\}\left\{3330\right\}$

Decimal computation

Modern hardware and software systems commonly use a binary representation internally (although many early computers, such as the or the IBM 650, used decimal representation internally). Fingers or Fists? (The Choice of Decimal or Binary Representation), , Communications of the ACM, Vol. 2 #12, pp3–11, ACM Press, December 1959. For external use by computer specialists, this binary representation is sometimes presented in the related or 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,

(1983). 9780898743180, Robert E. Krieger Publishing Company.
(1974). 047176180X, John Wiley & Sons. 047176180X
especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for Floating-Point Arithmetic). Decimal Floating-Point: Algorism for Computers, , Proceedings 16th IEEE Symposium on Computer Arithmetic, , pp104-111, IEEE Comp. Soc., June 2003

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). Decimal Arithmetic – FAQ Decimal Floating-Point: Algorism for Computers, , M. F., Proceedings 16th IEEE Symposium on Computer Arithmetic ( ARITH 16), , pp. 104–111, IEEE Comp. Soc., June 2003 See Arbitrary-precision arithmetic for exact calculations.

History
Many ancient cultures calculated with numerals based on ten, sometimes argued due to human hands typically having ten digits. Standardized weights used in Indus Valley Civilization (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 – the Mohenjo-daro ruler – was divided into ten equal parts.Sergent, Bernard (1997), Genèse de l'Inde (in French), Paris: Payot, p. 113, Coppa, A.; et al. (6 April 2006), "Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population" (PDF), Nature, 440 (7085): 755–6, Bibcode:2006Natur.440..755C, doi:10.1038/440755a, 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. 113–124 Egyptian hieroglyphs, in evidence since around 3000 BC, used a purely decimal system,Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, , pp. 200–213 (Egyptian Numerals) just as the Cretan hieroglyphs (ca. 1625−1500 BC) of the whose numerals are closely based on the Egyptian model.Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, , p. 50Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, , pp. 213–218 (Cretan numerals) The decimal system was handed down to the consecutive Bronze Age cultures of Greece, including (ca. 18th century BC−1450 BC) and (ca. 1375−1200 BC) — the number system of also used powers of ten, including, like the did, an intermediate base of 5. Greek numerals Notably, the polymath (ca. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 108 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.Menninger, Karl: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, , pp. 150–153 The hieroglyphs (since 15th century BC), just like the Egyptian and early numerals in Greece, was strictly decimal.Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, , pp. 218f. (The Hittite hieroglyphic system)

Some non-mathematical ancient texts like the dating back to 1900–1700 BCE make use of decimals and mathematical decimal fractions.(Atharva Veda 5.15, 1-11)

The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20, to 90, 100, 200, to 900, 1000, 2000, 3000, 4000, to 10,000.Lam Lay Yong et al. The Fleeting Footsteps pp 137–139 The world's earliest positional decimal system was the Chinese .

History of decimal fractions
Decimal fractions were first developed and used by the Chinese in the end of 4th century BC, and then spread to the Middle East and from there to Europe. The written Chinese decimal fractions were non-positional. However, counting rod fractions were positional.Lam Lay Yong, "The Development of Hindu–Arabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p38, Kurt Vogel notation

in his book Mathematical Treatise in Nine Sections (1247Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997 ) denoted 0.96644 by

::::寸
::::
, meaning
::::寸
::::096644

J. Lennart Berggren notes that 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.

(2018). 9780691114859, Princeton University Press.
The Jewish mathematician used decimal fractions around 1350, anticipating , but did not develop any notation to represent them.: 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. The Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century. introduced fraction to Islamic countries in the early 9th century, his fraction presentation was an exact copy of traditional Chinese mathematical fraction from . This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century Abu'l-Hasan al-Uqlidisi and 15th century Jamshīd al-Kāshī's work "Arithmetic Key".Lam Lay Yong, "A Chinese Genesis, Rewriting the history of our numeral system", Archive for History of Exact Science 38: 101–108.
A forerunner of modern European decimal notation was introduced by in the 16th century.

Natural languages
The ingenious method of expressing every possible number using a set of ten symbols emerged in India. Several Indian languages show a straightforward decimal system. Many Indo-Aryan and Dravidian languages have numbers between 10 and 20 expressed in a regular pattern of addition to 10.

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 十, 100 百, 1000 千, 10,000 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) 万 9 (thousand) 千 3 (hundred) 百 4 (tens) 十 5 is found in , and in Vietnamese with a few irregularities. Japanese, , and 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 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.

Other bases
Some cultures do, or did, use other bases of numbers.
• cultures such as the used a system (perhaps based on using all twenty fingers and ).
• The language in and the Pamean languages in have 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 of 120 in number, and a long thousand of 1200 in number. The descriptions like 'long' only appear after the small hundred of 100 in number appeared with the Christians. Gordon's Introduction to Old Norse p 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' is translated to 200, and the cognate to 'two hundred' is translated at 240. Goodare details the use of the long hundred in Scotland in the Middle Ages, giving examples, 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 the Chumashan languages originally used a base-4 counting system, in which the names for numbers were structured according to multiples of 4 and .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), .
• Many languages use number systems, including , Nunggubuyu, Kuurn Kopan NootDawson, J. " Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii. and . Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
• Some use systems. So did some small communities in India and Nepal, as indicated by their languages.
(2018). 9789042912953, Peeters. .
• The of Papua New Guinea is reported to have numbers. 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. Tokapu means 24, tokapu talu means 24 × 2 = 48, and tokapu tokapu means 24 × 24 = 576.
• is reported to have a base-32 number system with base-4 cycles.
• The of Papua New Guinea is reported to have base-6 numerals. Mer means 6, mer an thef means 6 × 2 = 12, nif means 36, and nif thef means 36×2 = 72.

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