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Matemática

HISTORY OF THE NUMERATION

THE ORIGIN OF THE NUMERATION INDO-ARABIC

Author: Prof. Luiz Márcio Imenes

The origin of our system of numeration is sufficiently old. It appeared in Asia, has many centuries, in the valley of Indo river , where today is the Pakistan .

There are s ome important informations so that you it can better understand this history.

In the valley of Indo river , it has four a thousand years more than, it developed one of the first Indian civilizations, that arrived to implant a net of about one hundred villages, including some cities.

The ruins of one of these cities, today known as Mohenjo Daro, disclose to the existence of streets sidewalk, houses with adobe bricks, swimming pools for public baths and until systems of water supply and canalization of sewer.

In these ruins some written registers had been also uncovered,but until the moment that system of writing have even so not been deciphered.

All these indications allow to see indistinctly that this civilization has reached one high degree of organization. However, for return of 1.500 b.C., this culture disappeared, possibly due to the invasions of the Aryan peoples. The civilizations that had blossomed later in this region also had developed its proper writing, beyond a numerical system that was the base for the ours.

(Indian numeration not-positional , as a register of the I century)

Note: that system is not positional, so, there is no need of

the symbol of zero.

 Note of the IEJU-SA : the system is positional when, placed one of the symbols to the right or the left of another one, the value of the number thus gotten is modified. The current system used in Ocident is positional: if you place the number 1 to the left side of number 2 , we will have 12 (twelve) and if you invert this position, we will have 21 (twenty and one).

Some centuries would be transferred until the decimal positional numeration was developed . It seems that the Indian system of positional numeration was configured only for return of the V century .

Decimal positional indian numeration (under register of the IX century)

In this system that is positional, we use the zero.

For the creation of its system positional decimal the Indians had received influences from many of the peoples with which they had had contact.

The positional principle already appeared in the system of the Mesopotamians. The base ten was used for the Egyptians and Chinese. About to the zero, indications exist of that already was used for the Mesopotamians in the final phase of its civilization. The great merit of the Indians was to congregate these different characteristics in one same numerical system.

And so why that system is called Indo-Arabic?

The diffusion of the Indian system is credited to the Arabs. Special circumstances l had favored this fact.

Even for return of the VI century , Arabia was inhabited mainly for nomadic tribes of the desert. At this time, few cities functioned as commerce centers. In the VII century , the Islamic religion had beginning, established by Mohamed , that obtained to join the tribes of the desert.

The followers of Mohamed, invading numerous neighboring territories, had started to control, in little more than a century, an immense empire, that extended itself from Spain to the valley of the Indo river.

Limits of the Muslim expansion at the VIII century.

In the contact with the Indians, the Arabs had assimilated the positional decimal numeration system. When invading the Europe , for return of the VIII century , they had taken this representation of the numbers there.

For having the Arabs, of this form, spread out the Indian numerical system, it passed to be known as Indo-Arabic.

Other systems of counting and numeration

The counting by marks is a very old practical one. In archaeological hollowings bones and pieces of wood had been found with marks that, probably, related the countings.

In England , until century XVIII, it was custom to register amounts in bars wooden known as carvings :

English carvings of the XIII century. To carv means " to cut". Different cuts represented

different quantities

In the remain of the Europe :

Carvings formely used at Dalmacia ( actually Yugoslavia )

The counting with marks discloses that, many times, is necessary to register one definitive value.

The necessity to keep register of values is bigger or lesser depending on certain factors as, for example, the degree of development of agriculture and the commerce, the ownership of good, the organization of the society, the production of merchandises, etc..

The necessity to register amounts gave origin to the written numeration. Diverse old civilizations had created its proper numerical systems.

Examples of formely systems :

 Egyptian system               Roman system             Chinese system

NOTE OF IEJU-SA : Still in the IX century a.D., in the American continent, the Mayans already possessed a written numeration, including the zero:

The Mayans had representation by teeny circles filled in until the number four. The number five was a trace.Of six to nine, mixing representation { 1(5)+ 1 = 6, 1(5)+2 = 7, 1(5)+3 = 8, 1(5)+ 4 = 9 }. The zero, or emptiness, was represented for a symbol, that appeared with varied forms, similar to an half-open eye.

This notation was used primarily to count days in a calendar of 360 days in one year and used an impure vigesimal system, having, in the third position, the substitution of 20 for 18 as multiple of 20, from there in ahead coming back to take advantage multiple 20 .

Examples:

A break up of the Codex de Dresden,

from the Mayans, showing numbers.

.

Source: History of the Mathematics, Carl B. Boyer, Tr. Elza F. Gomide. Ed. Edgard Blücher Ltda., 2 nd . ed.,pp.145-146

Changes in the Writing Numbers

Before the invention of the press, in the XI century , the books were copied manually, one by one. As each copist had his calligraphy, the letters and the symbols to represent numbers had been suffering to many modifications during all these centuries from manual copiagem.

Moreover, as the system of numeration created in India was adopted by the Arabs and passed the Europeans, it is natural that the form to write the ten numbers was suffering alterations.

Between the Arabs, the symbols had finished for taking the following configuration, that is used currently by them:

In India , the symbols had also continued to be modified. One of the forms today used there is this:

And we, nowadays, represent the ten numbers thus:

At last, a very practical system

When the Indian numeration, brought by the Arabs, reached the Europe , the Roman system was used there.

The Roman numeration was remained in use in the Europe during many centuries, all over to the great power of the Catholic Church during all the Average Age (from the V century to the XV century , approximately).

For us, because we know the two systems, it is easy to perceive the enormous advantages of our system with relation to the roman. That fact can make you to assume that the numeration Indo-Arabic was readily accepted for the Europeans.

In the truth, it was not well thus. Some centuries had been necessary so that the new ideas won definitively, what only occurred in century XVI.

During much time, a true battle was stopped enters the adepts of the new numeration and the defenders of the Roman system. The numerals Indo-Arabic had arrived to be forbidden in official documents, but they were used in secrecy.

The persecution, however, did not obtain to hinder the dissemination it new system, that finished imposing itself for its qualities.

But what qualities are these?

It is easy you yourself to prove the practicalness for the calculations. Trie, for example, to make an addition using the Roman or the Egyptian system.

Another aspect still exists that well discloses to the practicalness of the Indo-Arabic numeration.Observe that, in the Egyptian system, there are different symbols to represent the numbers one, ten, one hundred, a thousand and a million.

 Following this rule, other symbols would be necessary to represent bigger numbers. And this would not have end.   Perhaps, at the time where such system was invented, the practical necessities did not involve so intense amounts. However, in the current world, we frequently come across with the necessity to register very great numbers. Thus, as much the Roman numerical system as the Egyptian would not be really practical nowadays.   The questions to relate numbers with symbols inspired Jorge Luis Borges, a famous Argentine writer, to write the story that we summarize follow.

Once upon a time there was a man who possessed a fantastic memory. He obtained to remember everything,for example, of that he had even eaten in a lunch of twenty years ago.

One day he decided to invent a system of an well simple numeration, simplest as possible. For each number he created a name and a symbol, and thus he was making until the million, the billion, the trillion, following always ahead.

Thanks to its great memory, he did not forget anything that he had invented. Exactly thus, poor thing, he died without finishing the system.

And it is clearly that he never would finish, therefore he would go to need infinite names, infinite symbols and... infinite time!

Although we are making a comparison with a so fantastic history, see that, to be practical, a numeration system cannot be like that. It must, in contrast, make use of few symbols that, combined, can represent any amounts.

It is what happens with the Indo-Arabic numeration, that uses only ten symbols:

Living the Mathematics

The Egyptian System of numeration is not-positional. As much makes to write eleven thus In or thus In. What it counts is the addition of the values of each symbol.

Using only these two symbols ? or n , write all the possible sequences where appear three symbols in different positions. For example: IIn     In     III .

In the total, there are 8 different sequences. In these eight, how many distinct numbers are there

and which are they?

Now, write numbers of three numerals only using the 1 and the 2. For example: 111,221,212. How many different numbers you obtained? Which are they?

Author : Professor Luiz Marcio Imenes

In : Coleção Vivendo a Matemática - A Numeração Indo-Arábica.Ed. Scipione - 7ª. edição, 1995 - São Paulo, Brazil - pp. 11-12, 34-44.

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_numerals.html

http://www.matemática.br/historia/egito.html

http://www.suapesquisa.com/historia/

http://www.mat.ufrgs.br/~portosil/passa7a.html

Know more (some of these works are depleted and is more easily found them in libraries or used book stores):

•  KARLSON, Paul . A Magia dos Números : a Matemática ao alcance de todos. Porto Alegre, Editora Globo, 1961.

•  DANTZIG, Tobias. Número : A Linguagem da Ciência. Rio de Janeiro, Zahar Editores, 1970.

•  BOYER, Carl B. História da Matemática. São Paulo, Editora Edgard Blücher Ltda. e EDUSP, 1974.

•  IFRAH,Georges. Histoire Universelle des Chiffres. Paris, Éditions Seghers, 1981.

•  BERGAMINI, David. As Matemáticas. Biblioteca Científica LIFE. Rio de Janeiro, Livraria José Olympio Editora.

•  IMENES, Luiz Márcio. Os Números na História da Civilização. Coleção Vivendo a Matemática. São Paulo, Editora Scipione,1988.

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