Monday, August 14, 2006

Indian Mathematics and Its Contribution to the World

Here is a cool articles that describes some history of Indian system of Mathematics. Read if you wish:

History of the Hindu-Arabic numeral system
From Wikipedia, the free encyclopedia

Numeral systems

Hindu-Arabic numerals
Western ArabicEastern Arabic
Indian familyBrahmi
East Asian numerals
Alphabetic numerals
Other systems
Numeral system topics

Positional systems
Decimal base,
Binaries: 2, 4, 8,16, 32, 64, 128,
other: 3, 9, 12, 24, 30, 36, 60, more.
The Hindu-Arabic numeral system originated from the Hindu numeral system, which is a pure place value system, that requires a zero.[1] Though it is, in actuality, a purely Hindu numeral system, it is know to the Western world as Hindu-Arabic because of its introduction to Europe through the Islamic middle east.
1 Origins
2 Positional notation
3 Adoption by the Arabs
4 Adoption in Europe
5 Impact on Mathematics
6 References
7 External link
In Sanskrit literature number words for 1-9, 10, 100 and further powers of 10 - up to 10 - were used (similar to decimal system).[2]. The most widely used place value symbols belong to the Nagari script numerals, very similar to the Brahmi numerals, which form the basis of the modern Arabic numerals. [3]
Historians trace many modern numerals to the Brahmi numerals, which were in use around the middle of the third century BC.[4] The place value system, however, evolved later. The Brahmi numerals have been found in inscriptions in caves and on coins in regions near Pune, Mumbai, and Uttar Pradesh. Dating these numerals tells us that they were in use over quite a long time span up to the 4th century AD.[5]
During the Gupta period (early 4th century AD to the late 6th century AD), the Gupta numerals developed from the Brahmi numerals and were spread over large areas by the Gupta empire as they conquered territory. [6] Beginning around 7th century, the Gupta numerals evolved into the Nagari numerals.
Positional notation
There is evidence that the Babylonians had a place value system as early as the 19th century BC. However, the Babylonian systems were to base 60. Babylonians used a separator mark to separate various place values. This separator mark never was used at the end of a number, and it was not possible to tell the difference between 2 and 20. This innovation was brought about by Brahmagupta of India. Further, the Babylonian place value marker did not stand alone, as per the Indian "0". There is unsure evidence that the Indians developed a positional number system as early as the first century CE [7]. However, the oldest dated Indian document showing use of the modern place value form is a legal document dated 346 in the Chedii calendar, which translates to 594 CE. [8], but some historians claim that the date has been added as a later forgery. Despite such doubts, historians are fairly certain that an early place-value system was in use in India by the end of the 5th century. [9] Indian books dated to this period are able to denote numbers in the hundred thousands using a place value system. [10] Many other inscriptions have been found which are dated and make use of the place-value system for either the date or some other numbers within the text [11], although some historians claim these to also be forgeries.[12] In around 500 CE Aryabhata devised a positional number system without a zero digit. He used the word "kha" for the zero position.[13] Evidence suggests that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. [14]. The same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it.
The use of zero in these positional systems are the final step to the system of numerals we are familiar with today. The first inscription showing the use of zero which is dated and is not disputed by any historian is the inscription at Gwalior dated 933 in the Vikrama calendar (876 CE.) [15] [16].
This 9th century date is the scientific consensus on the earliest acceptable evidence for the use of positional zero in India. According to Lam Lay Yong,
"the earliest appearance in India of a symbol for zero in the Hindu-Arabic numeral system is found in an inscription at Gwalior which is dated 870 AD".[17].
Professor EF Robertson and DR JJ O'Connor report:
"The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876" on the Gwalior tablet stone[18].
According to Menninger (p. 400):
"This long journey begins with the Indian inscription which contains the earliest true zero known thus far (Fig. 226). This famous text, inscribed on the wall of a small temple in the vicinity of Gvalior (near Lashkar in Central India) first gives the date 933 (A.D. 870 in our reckoning) in words and in Brahmi numerals. Then it goes on to list four gifts to a temple, including a tract of land "270 royal hastas long and 187 wide, for a flower-garden." Here, in the number 270 the zero first appears as a small circle (fourth line in the Figure); in the twentieth line of the inscription it appears once more in the expression "50 wreaths of flowers" which the gardeners promise to give in perpetuity to honor the divinity." The Encyclopaedia Britannica says, "Hindu literature gives evidence that the zero may have been known before the birth of Christ, but no inscription has been found with such a symbol before the 9th century."[19].
The earliest Arabic accounts of the Indian numerals, dating to the 7th century, describe them as a system of nine symbols. It is, therefore, uncertain whether the crucial inclusion of zero as the tenth symbol of the system should be attributed to the Indians, or if it is due to Al-Khwarizmi's 825 On the Calculation with Hindu Numerals.
Adoption by the Arabs
Before the rise of the Arab empire, the Hindu-Arabic numeral system was already moving West and was mentioned in Syria in 662 AD by the Syrian-Orthodox scholar Severus Sebokht who wrote:
"I will omit all discussion of the science of the Indians, ... , of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians, and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe, because they speak Greek, that they have arrived at the limits of science, would read the Indian texts, they would be convinced, even if a little late in the day, that there are others who know something of value."[20]
According to al-Qifti's chronology of the scholars[21]:
"... a person from India presented himself before the Caliph al-Mansur in the year [776 AD] who was well versed in the siddhanta method of calculation related to the movement of the heavenly bodies, and having ways of calculating equations based on the half-chord [essentially the sine] calculated in half-degrees ... Thiinto Arabic, and a work to be written, based on the translation, to give the Arabs a solid base for calculating the movements of the planets ..."
The work was most likely to have been Brahmagupta's Brahmasphutasiddhanta (Ifrah) [22] (The Opening of the Universe) which was written in 628[23]. Irrespective of whether Ifrah is right, since all Indian texts after Aryabhata's Aryabhatiya used the Indian number system, certainly from this time the Arabs had a translation of a text written in the Indian number system. [24]
In his text The Arithmetic of Al-Uqlîdisî (Dordrecht: D. Reidel, 1978), A.S. Saidan's studies were unable to answer in full how the numerals reached the Arab world:
"It seems plausible that it drifted gradually, probably before the seventh century, through two channels, one starting from Sind, undergoing Persian filtration and spreading in what is now known as the Middle East, and the other starting from the coasts of the Indian Ocean and extending to the southern coasts of the Mediterranean."[25]
He notes, however, that Al-Uqlidisi's work, Kitâb al-FusÞl fî al-Hisâb al-Hindî, "the earliest extant Arabic work of Hindu-Arabic arithmetic", written in Damascus in AD 952–953, showed “this system at its earliest stages and the first steps in its development.” (ibid, p. xi.), especially so that "The manuscript claimed to have a collection of all past knowledge on arithmetic" and "a clear exposition of what was currently known about the subject". Saidan also writes:
Whatever the case may be, it should be pointed out that Arabic works give no reference whatsoever to any Sanskrit text or Hindu arithmetician, nor do they quote any Sanskrit term or statement.
Until Al-Uglidisi's work, the Indian numerals and arithmetics required the use of a sand board, which was an obstacle to their use in official manuscripts. As-Suli in the first half of the tenth Century:
Official scribes nevertheless avoid using [the Indian system] because it requires equipment [like a dust board] and they consider that a system that requires nothing but the members of the body is more secure and more fitting to the dignity of a leader.[27]
In his work cited above, Al-Uglidisi showed required modification to the numerals and arithmetics to make them suitable for use by pen and paper, which was a major improvement.
Al-Uqlidisi book was the earliest known text to offer treatment of decimal fraction.[28][29]
The numerals though were already in wide use throughout the Arab empire, as Avicenna who was born in 980 tells in his autobiography that he learnt them, as a child, from a vegetable seller. He also tells that when his father, in Bukhara, was visited by scholars from Egypt in 997, including Abu Abdullah al-Natili, they taught him more about them. J J O'Connor and E F Robertson point out:
He also tells of being taught Indian calculation and algebra by a seller of vegetables. All this shows that by the beginning of the eleventh century calculation with the Indian symbols was fairly widespread and, quite significantly, was known to a vegetable trader.[30]
The numerals came to fame due to their use in the pivotal work of the Persian mathematician Al-Khwarizmi, whose book On the Calculation with Hindu Numerals was written about 825, and the Arab mathematician Al-Kindi, who wrote four volumes (see [2]) "On the Use of the Indian Numerals" (Ketab fi Isti'mal al-'Adad al-Hindi) about 830. They, amongst other works, contributed to the diffusion of the Indian system of numeration in the Middle-East and the West.
Adoption in Europe
Main article: Arabic numerals
Fibonacci, an Italian mathematician who had studied in Bejaia (Bougie), Algeria, promoted the Arabic numeral system in Europe with his book Liber Abaci, which was published in 1202. The system did not come into wide use in Europe, however, until the invention of printing (See, for example, the 1482 Ptolemaeus map of the world printed by Lienhart Holle in Ulm, and other examples in the Gutenberg Museum in Mainz, Germany.)
In the last few centuries, the European variety of Arabic numbers was spread around the world and gradually became the most commonly used numeral system in the world. Even in many countries in languages which have their own numeral systems, the European Arabic numerals are widely used in commerce and mathematics.
Impact on Mathematics
The significance of the development of the positional number system is probably best described by the French mathematician Pierre Simon Laplace (1749 - 1827) who wrote:
"It is India that gave us the ingenuous method of expressing all numbers by the means of ten symbols, each symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity."
Tobias Dantzig, the father of George Dantzig had this to say in Number:
"This long period of nearly five thousand years saw the rise and fall of many a civilization, each leaving behind it a heritage of literature, art, philosophy, and religion. But what was the net achievement in the field of reckoning, the earliest art practiced by man? An inflexible numeration so crude as to make progress well nigh impossible, and a calculating device so limited in scope that even elementary calculations called for the services of an expert [...] Man used these devices for thousands of years without contributing a single important idea to the system [...] Even when compared with the slow growth of ideas during the dark ages, the history of reckoning presents a peculiar picture of desolate stagnation. When viewed in this light, the achievements of the unknown Hindu, who some time in the first centuries of our era discovered the principle of position, assumes the importance of a world event."
"The Development of Hindu-Arabic and Traditional Chinese Arithmetic" by Professor Lam Lay Yon, member of the International Academy of the History of Science
Indian numerals by J J O'Connor and E F Robertson
Arabic numerals by J J O'Connor and E F Robertson
Hindu-Arabic numerals
The Arabic numeral system by: J J O'Connor and E F Robertson
External link
Laputan Logic - The Evolution of Numbers
Retrieved from ""


Blogger Drongomala said...

excellent article. It was Nehru's book 'Glimpsed of World History' that explained that the Arabs used to call mathematics 'hindisat' - the indian art. Most people in the west think that the Arabs invented it - they were mereley the excellent brokers.

The genesis of the decimal system was traced to and Indian father and daughter team. The father frenetically wanted a marriage arranged with a certain fellow but existing numeracy systems didn't allow it to work old - legend has it that he developed this new mathematics so that it would fit.

There would have been no moon landing with XIIV....;)

All the best

3:50 PM, August 22, 2006  
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