Monday, 11 August 2014


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For different utilization, see Decimal (disambiguation).

The world's most punctual decimal increase table was produced using bamboo slips, dating from 305 BC, amid the Warring States period

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This article means to be an open presentation. For the numerical definition, see Decimal representation.

Numeral frameworks

by society

Hindu–arabic beginnings

Indian Bengali Tamil Telugu

Eastern Arabic Western Arabic

Burmese Khmer Lao Mongolian

Sinhala Thai

East Asian

Chinese Suzhou Japanese Korean Vietnamese

Tallying poles


Abjad Armenian Āryabhaṭa Cyrillic

Ge'ez Georgian Greek Hebrew Roman


Aegean Attic Babylonian Brahmi

Egyptian Etruscan Inuit Kharosthi

Mayan Quipu


Positional frameworks by base

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 20 24 26 27 32 36 60

Non-standard positional frameworks

Arrangement of numeral frameworks

v t e

The decimal numeral framework (likewise called base ten or every so often denary) has ten as its base. It is the numerical base most broadly utilized by advanced civilizations.[1][2]

Decimal documentation regularly alludes to a base-10 positional documentation, for example, the Hindu-Arabic numeral framework; then again, it can additionally be utilized all the more for the most part to allude to non-positional frameworks, for example, Roman or Chinese numerals which are likewise focused around forces of ten.

Decimals likewise allude to decimal parts, either independently or rather than foul portions. In this connection, a decimal is a tenth part, and decimals turn into an arrangement of settled tenths. There was a documentation being used like 'tenth-meter', significance the tenth decimal of the meter, at present an Angstrom. The difference here is in the middle of decimals and revolting divisions, and decimal divisions and different divisions of measures, in the same way as the inch. It is conceivable to take after a decimal extension with a revolting division; this is finished with the late divisions of the troy ounce, which has three spots of decimals, emulated by a trinary place.

Substance  [hide]

1 Decimal documentation

1.1 Decimal divisions

1.2 Other objective numbers

1.3 Real numbers

1.4 Non-uniqueness of decimal representation

2 Decimal reckoning

3 History

3.1 History of decimal divisions

3.2 Natural dialects

3.3 Other bases

4 See likewise

5 References

6 External connec

Decimal notation

Decimal documentation is the composition of numbers in a base-10 numeral framework. Illustrations are Greek numerals, Roman numerals, Brahmi numerals, and Chinese numerals, and additionally the Hindu-Arabic numerals utilized by speakers of numerous European dialects. Roman numerals have images for the decimal powers (1, 10, 100, 1000) and optional images for a large portion of these qualities (5, 50, 500). Brahmi numerals have images for the nine numbers 1–9, the nine decades 10–90, or more an image for 100 and an alternate for 1000. Chinese numerals have images for 1–9, and extra images for forces of 10, which in current use achieve 1044.

Nonetheless, when individuals who use Hindu-Arabic numerals discuss decimal documentation, they frequently mean decimal numeration, as above, as well as decimal parts, all passed on as a major aspect of a positional framework. Positional decimal frameworks incorporate a zero and utilization images (called digits) for the ten values (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to speak to any number, regardless of how extensive or how little. These digits are regularly utilized with a decimal separator which demonstrates the begin of a fragmentary part, and with an image, for example, the in addition to sign + (for positive) or less sign − (for negative) nearby the numeral to demonstrate whether it is more prominent or short of what zero, individually.

Positional documentation utilization positions for each one force of ten: units, tens, hundreds, thousands, and so forth. The position of every digit inside a number indicates the multiplier (force of ten) reproduced with that digit—each one position has an esteem ten times that of the position to its correct. There were no less than two probably free wellsprings of positional decimal frameworks in aged progress: the Chinese checking pole framework and the Hindu-Arabic numeral framework (the recent slipped from Brahmi numerals).

Ten fingers on two hands, the conceivable beginning stage of the decimal tallying.

Ten is the number which is the tally of fingers and thumbs on both hands (or toes on the feet). The English word digit and also its interpretation in numerous dialects is additionally the anatomical term for fingers and toes. In English, decimal (decimus < Lat.) implies tenth, annihilate means diminish by a tenth, and denary (denarius < Lat.) implies the unit of ten.

The images for the digits in as something to be shared use around the globe today are called Arabic numerals by Europeans and Indian numerals by Arabs, the two bunches' terms both alluding to the society from which they took in the framework. Notwithstanding, the images utilized within distinctive territories are not indistinguishable; for example, Western Arabic numerals (from which the European numerals are inferred) vary from the structures utilized by other Arab societies.

Decimal fractions[edit]

A decimal portion is a small amount of the denominator of which is a force of ten.[3]

Decimal portions are regularly communicated without a denominator, the decimal separator being embedded into the numerator (with heading zeros included if necessary) at the position from the right relating to the force of ten of the denominator; e.g., 8/10, 83/100, 83/1000, and 8/10000 are communicated as 0.8, 0.83, 0.083, and 0.0008. In English-talking, some Latin American and numerous Asian nations, a period (.) or raised period (·) is utilized as the decimal separator; in numerous different nations, especially in Europe, a comma (,) is utilized.

The whole number part, or essential piece of a decimal number is the part to the left of the decimal separator. (See additionally truncation.) The part from the decimal separator to the right is the fragmentary part. It is typical for a decimal number that comprises just of a fragmentary part (numerically, a fitting portion) to have a heading focus in its documentation (its numeral). This helps disambiguation between a decimal sign and other accentuation, and particularly when the negative number sign is demonstrated, it aides picture the indication of the numeral all in all.

Trailing zeros after the decimal point are not essential, albeit in science, designing and facts they might be held to demonstrate an obliged accuracy or to demonstrate a level of trust in the precision of the number: Although 0.080 and 0.08 are numerically equivalent, in building 0.080 recommends an estimation with a blunder of up to one section in two thousand (±0.0005), while 0.08 proposes an estimation with a mistake of up to one in two hundred (see noteworthy figures).

Other balanced numbers[edit]

Any balanced number with a denominator whose just prime components are 2 and/or 5 may be exactly communicated as a decimal portion and has a limited decimal expansion.[4]

1/2 = 0.5

1/20 = 0.05

1/5 = 0.2

1/50 = 0.02

1/4 = 0.25

1/40 = 0.025

1/25 = 0.04

1/8 = 0.125

1/125 = 0.008

1/10 = 0.1

In the event that the levelheaded number's denominator has any prime variables other than 2 or 5, it can't be communicated as a limited decimal fraction,[4] and has a special in the long run rehashing endless decimal extension.

1/3 = 0.333333…  (with 3 rehashing)

1/9 = 0.111111…  (with 1 rehashing)

100 − 1 = 99 = 9 × 11:

1/11 = 0.090909…

1000 − 1 = 9 × 111 = 27 × 37:

1/27 = 0.037037037…

1/37 = 0.027027027…

1/111 = 0 .009009009…


1/81 = 0.012345679012…  (with 012345679 rehashing)

That a balanced number must have a limited or repeating decimal extension might be seen to be a result of the long division calculation, in that there are just q-1 conceivable nonzero remnants on division by q, so the repeating example will have a period short of what q. For example, to discover 3/7 by long division:

0.4 2 8 5 7 1 4 ...

7 ) 3.0 0

2 8  30/7 = 4 with a rest of 2

2 0

1 4  20/7 = 2 with a rest of 6

6 0

5 6  60/7 = 8 with a rest of 4

4 0

3 5  40/7 = 5 with a rest of 5

5 0

4 9  50/7 = 7 with a rest of 1


Decimal computation

Decimal computation was carried out in ancient times in lots of ways, usually in rod calculus, with decimal multiplication table used in ancient China & with sand tables in India & Middle East or with a variety of abaci.

Modern computer hardware & application systems often use a binary representation internally (although lots of early computers, such as the ENIAC or the IBM 650, used decimal representation internally).[5] For outside use by computer specialists, this binary representation is sometimes introduced 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, although lots of computer languages are unable to encode that number exactly.)

Both computer hardware & application also use internal representations which are effectively decimal for storing decimal values & doing arithmetic. Often this arithmetic is done on information which are encoded using some variant of binary-coded decimal,[6] in database implementations, but there's other decimal representations in use (such as in the new IEEE 754 Standard for Floating-Point Arithmetic).[7]

Decimal arithmetic is used in computers so that decimal fractional results can be computed exactly, which is impossible using a binary fractional representation. This is often important for financial & other calculations.[8]


Many ancient cultures calculated from early on with numerals based on ten: Egyptian hieroglyphs, in evidence since around 3000 BC, used a purely decimal system,[9][10] just as the Cretan hieroglyphs (ca. 1625−1500 BC) of the Minoans whose numerals are closely based on the Egyptian model.[11][12] The decimal system was handed down to the consecutive Bronze Age cultures of Greece, including Linear A (ca. 18th century BC−1450 BC) and Linear B (ca. 1375−1200 BC) — the number system of classical Greece also used powers of ten, including, like the Roman numerals did, an intermediate base of 5.[13] Notably, the polymath Archimedes (c. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 108[13] 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.[14] The Hittites hieroglyphs (since 15th century BC), just like the Egyptian and early numerals in Greece, was strictly decimal.[15]

The Egyptian hieratic numerals, the Greek alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20, to 90, 100, 200, to 900, 1000, 2000, 3000, 4000, to 10,000.[16]

History of decimal fractions[edit]

counting rod decimal fraction 1/7
According to Joseph Needham, decimal fractions were first developed and used by the Chinese in the 1st century BC, and then spread to the Middle East and from there to Europe.[17] The written Chinese decimal fractions were non-positional.[17] However, counting rod fractions were positional.

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

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


The Jewish mathematician Immanuel Bonfils invented decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them.[19]

The Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, though J. Lennart Berggren notes that positional decimal fractions were used five centuries before him by Arab mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.[20]

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

Stevin-decimal notation.svg
A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century.[21]

Natural languages[edit]
Telugu language uses a straightforward decimal system. Other Dravidian languages such as Tamil and Malayalam have replaced the number nine tondu with 'onpattu' ("one to ten") during the early Middle Ages, while Telugu preserved the number nine as tommidi.

The Hungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tízenegy" literally "one on ten"), as with those between 20-100 (23 as "huszonhárom" = "three on twenty").

A straightforward decimal rank system with a word for each order 10十,100百,1000千,10000万, and in which 11 is expressed as ten-one and 23 as two-ten-three, and 89345 is expressed as 8 (ten thousands) 万9 (thousand) 千3 (hundred) 百4 (tens) 十 5 is found in Chinese languages, and in Vietnamese with a few irregularities. Japanese, Korean, and Thai have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades. For example in English 11 is "eleven" not "ten-one".

Incan languages such as Quechua and Aymara have an almost straightforward decimal system, in which 11 is expressed as ten with one and 23 as two-ten with three.

Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.[22]

Other bases[edit]
Some cultures do, or did, use other bases of numbers.

Pre-Columbian Mesoamerican cultures such as the Maya used a base-20 system (using all twenty fingers and toes).
The Yuki language in California and the Pamean languages[23] in Mexico have octal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.[24]
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.[improper synthesis?] 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'.[citation needed]
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.[25]
Many languages[26] use quinary (base-5) number systems, including Gumatj, Nunggubuyu,[27] Kuurn Kopan Noot[28] and Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
Some Nigerians use base-12 systems.[29]
The Huli language of Papua New Guinea is reported to have base-15 numbers.[30] 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.[31] 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.[26]