Positional decimal Writing

The positional decimal writing of a number is its representation in a Numbering system of Base 10 employing a positional Notation. One calls sometimes also a notation answering this definition “numbering system indo-Arabic”, because this system developed in India and arrived in Europe by the means of the Arab. The symbols (or glyphes) used can be varied. However, the majority are figures of origin brahmi, of which most widespread are the European figures. With ten digits from 0 to 9, this writing corresponds to the decimal Développement.

Notations

In worked out forms, the positional decimal writing uses:

ten digits (corresponding zero to nine),
a separator of position enters of the groups of figures of the whole part of the number (example: 1.000.000),
of the symbols positive and negative for the signed numbers (example: -6),
a decimal marker enters the units and the fractional part of the number (example: 12,5),
a symbol to indicate that figures repeat (example ad infinitum: 1,309 for 1,3090909…).

Such a writing thus makes it possible to represent the natural entireties using 10 symbols only (ten digits) and the rational numbers using 13 (ten digits, a sign for the negative numbers, a decimal marker, an ad infinitum being repeated symbol for the figures).

Symbols

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Colombian numeration is mixed, at the same time positional and multiplicative, because it uses the symbol of ten instead of the zero positional one:

History

In China

In India

Symbol zero

First symbols in bond with a decimal writing corresponding to the zero positional one, in the form of round or point, replacing the space which preceded, were found in current Kampuchea into 683 and in Sumatra the same year and in 684. These areas are, at the time, under Chinese influence, and the first profits then from many exchanges with India. In India, the first inscription distinctly comprising this zero date of 876. In China, the oldest printed text utilizing the zero positional one with the marking system of the rods is the Su Chu Jin Chang , of 1247, but an allusion to the use of a point appears in the Kai Yuan Chan Jing , appeared between 718 and 729.

Adoption by the Arabs

In order to support the rise of sciences, the Arabs develop a scientific patronage starting from second half of the 8th century, they invite foreign scientists, build libraries, translate old texts, generally from the syriaque one or of the pehlévi (means-Persan). A numeration alphabetical, additive, is then of use, as at the majority of the people whose alphabet derives from the phenician.

With the visit of an Indian astronomer at the court of the caliph Al-Mansour, with Baghdad, this last realizes the importance of Indian sciences. It charges Al-Fazari then, into 772, to translate into Arabic Indian astronomical tables. Into this period are translated Aryabhata and Brahmagupta. Towards 820, the caliph Al-Mam' a founds in Baghdad the “House of Wisdom”, in order to allow the scientists to work around a large library, released from the material constraints. In this context, at the beginning of the 9th century, Al-Khwarizmi describes the Indian notations in a disappeared work, named today Kit ab Al-jam' wal tafriq Bi hisab Al-Hind (Book on the addition and the subtraction according to the method of the Indians), but if it makes use of the zero positional one, it does not regard the zero as a number. In its work Kit ab Al-fusul fi-l-hisab Al-Hindi (Book on the chapters of the arithmetic Indian), written in the middle of the 10th century in Damas, Abu L-Hassan Al-Uqlidisi praises the merits of the new numbering system.

Adoption in Europe

This system, in Europe, was confined for a long time with the Arab part of current the Spain, Roman numeration being of use beyond. Its first appearance in the Christian world, without the zero, is in a manuscript, the Codex Vigilanus writes into 976 in a monastery of the north of Spain, the convent of Albelda. Between 967 and 969, Gerbert d' Aurillac discovers Arab science in the Catalan abbeys. Later, thanks to its correspondents, it obtains the work multiplicatione and divisione (On the multiplication and division). He proposes then, in the neighborhoods of the year millet, a simplification of the abacus by adopting tokens comprising the Arab numerals from one to nine. Become pope into 999, it tries to introduce its computational tool, the abacus of Gerbert, within Christendom, but its company is put at evil by the opposition of the clerks.

Towards the end of the 11th century, with the territories taken to the Arabs, the Christians discover scientific manuscripts. Thus Tolède, taken in 1085, becomes, under the impulse of évèque Raimond, the principal center of translation of Arab Latin works, and the Liber Algorismi of number Indorum (Al-Khwarizmi Book on the Indian figures) there is translated. In 1130, the English Adélard de Bath publishes Algoritmi of number indorum and a translation of Al-Khwarizmi. The new system is not long in being called “algorism” (of the Latinized noun of Al-Khwarizmi, Algorizmi, and modified later in algorithm), its partisans “algorists”, and its opponents “abacists” (in favor of the Abaque).

Starting from the end of the 12th century, the trade takes an international dimension, and Italy occupies a central place there. In 1202, Léonard of Pisa, known as Fibonacci, having learned Bougie Arabic and calculation (Béjaïa), in Algeria, publishes the Liber Abaci (Book of the abacus), a treaty of calculation and accountancy (the professors of calculation are then called “Masters of abacus”) widely diffused, in which it exposes the Arab numerals. During the 13th century, the Arab numerals start to be taught in the schools of accountancy in Italy. However, with the War One hundred Year old and the epidemic of Black Death, which is spread in Europe as from 1438, the trade crumbles.

Thereafter, the economy begins again of more beautiful, the banks and the insurances accompanying the marketing activity. The treaties of arithmetic with the use of the merchants, taking as a starting point the Liber abaci of Fibonacci, multiply in Italy, starting from the middle of the 14th century. About 1480, with the Printing works, developed by Gutenberg in 1434, of the works of this type are published in several towns of Europe. The last ecclesiastical vetoes concerning the use of the new system are raised at the 15th century, but, until the end of the century, the principle of the decimal notation of position requires to be explained. With and 16th centuries, various notations, in particular for the decimal numbers, supplement the system. This one is essential definitively in France at the end of the 18th century with the Révolution. Lastly, in 1889, with the axiomatic definition of the natural entireties, appeared in the Arithmetices principia nova methodo exposita , of Giuseppe Peano, the zero acquires the statute of number definitively.

In the world

Under the influence of Europe, many countries adopt the positional decimal marking system, with the European figures. The few writings of the south of India which do not profit adapt it to their figures during the 20th century, by taking for model the dominant cultures of the North of India or other countries.

Does the Indian positional system have a Chinese origin?

With the diffusion of the Buddhism, bonds are woven between India and China, since the kingdom Koushan, with the borders of the the Himalayas, until its collapse into 320. Many translations between Chinese and Sanskrit are undertaken in Hotan from 260. Then, a large hearth of translation, functioning from 344 to 413, is created in Kuqa. Within sight of the important exchanges between China and India preceding the publication by the Lokavibhaga , it is possible that the Indian positional principle is, in fact, of Chinese origin. Besides the term “sunya” points out the vacuum left by the Chinese with the rods calculate, even if it is then assimilated to one zero positional owing to the fact that the vacuum is not really any more one since it is written.

However, it is also possible that the Jaïn S know the numeration of position before this publication. Indeed, in India, there for a long time exist terms for all the powers of ten, until 10¹², none them not being privileged; and, Jaïns frequently handling the great numbers, it could be practical for them to omit them. In addition, the term of “place” to indicate the order of magnitude of a number is used in the Anuyogadvara-sutra , text jaïn writes at the 5th century, but probably composed at first century BC

However, no argument makes it possible to solve the question definitively. Indeed, on the one hand, the use of the term “sunya” does not certify filiation, because it is rather natural to indicate by “vacuum” an absence of quantity; in addition, the remarks on the Anuyogadvara-sutra do not cancel either this assumption, because the number of “places” of which it is question there can refer to the number of powers of ten settings to contribution instead of the number of figures employed, and the written version of this text necessarily in conformity with original, itself is not gone back to dubious manner.

At all events, the contribution of India is not negligible: the Indians are the first to use a positional marking system employing of the figures of which the layout is independent of the value represented. Lastly, if India played a great part in the development of the principle of position, it seems that the Sanskrit is a language which lent itself to it well.

History of Chinese Mathematical mathematics

Mathematical Indians
Mathematical Arabic

Numeration

Figure arabo-Indian
0 (number)
decimal Development

Webographic resources

Numbers, computational tools and expressions mathematical in old China
diffusion of the decimal notation of position
the contribution of India to Arab mathematics
History of mathematics
History of the numeration and arithmetic the Indians of the origins to the twelfth century
The arithmetic invention off nothing
Calculation and religion

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