Hindu Contribution to Mathematics

johnee

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Does no one remember the Hindu contribution to Mathematics?


Whenever I read about the great ìArabicî contribution to Mathematics and Science (often in an apologetic tone of ìhow could these great people come to such a pass?î) the thing that really upsets me is the complete omission of any reference to the Hindu contribution to mathematics and numbers.

Slightly more than a year ago (Aug í04), in an article in the Sunday Times, Michael Portillo, eminent Conservative party leader in the UK and a one-time aspirant to the leadership of the Tory Party, wrote that, ìIslam brought back to the West knowledge of architecture, mathematics and astronomy that had been lost during the Dark Ages.î

In response, I wrote,

ìÖThe phrase ìbrought backî is at best, condescending and at worse, historically inaccurate.

For this knowledge, which Arab traders brought to Europe (typified in the Arabic numeral system – itself a misnomer, since the Arabs did not invent it but merely acted as the purveyors of this knowledge) was not Islamic or Arabic. In fact much of this knowledge was originally derived from ancient Vedic literature from India and passed through Arab traders and conquests to Middle East and eventually reaching Europe.

To quote from Carl B. Boyer in his “History of Mathematics”, ì…Mohammed ibn-Musa al-Khwarizmi, …, who died sometime before 850, wrote more than a half dozen astronomical and mathematical works, of which the earliest were probably based on the Sindhind derived from India. Besides … [he] wrote two books on arithmetic and algebra which played very important roles in the history of mathematics. … In this work, based presumably on an Arabic translation of Brahmagupta, al-Khwarizmi gave so full an account of the Hindu numerals that he probably is responsible for the widespread but false impression that our system of numeration is Arabic in origin. … [pages 227-228]…î.

In a translation of Alberuni ës ìIndicaî, a seminal work of this period (c.1030 AD), Edward Sachau, writes this in his introduction, ìMany Arab authors took up the subjects communicated to them by the Hindus and worked them out in original compositions , commentaries and extracts. A favourite subject of theirs was Indian mathematics…” etc.

Needless to say, the letter never got published.

Then, more recently, while reading the ìThe World is Flatî by Thomas L. Friedman , I came across this text in Chapter 11, “The Unflat World” (Pg 405), “As Nayan Chanda, the editor of YaleGlobal Online pointed out to me, it was the Arab-Muslim world that gave birth to algebra and algorithms, terms both derived form Arabic words. In other words, noted Chanda, “The entire modern information revolution, which is built to a large degree on algorithms, can trace its roots all the way back to Arab-Muslim civilization and the great learning centres of Baghdad and Alexandria,” which first introduced these concepts, then transferred them to Europe through Muslim Spain.

Dismayed, I wrote the following email to Nayan:

ìMay I respectfully point out that is not historically accurate and continuing research is providing evidence that the roots of the so-called Arab contribution to Mathematics and Science were further east in the lands of India and in the works of Indian mathematicians and scholars from several centuries ago.î

I then gave a couple of examples and concluded by saying:

ìI hope that you will re-consider your views in the light of these excerpts and a significant body of research that is now publicly available on this subject. I would be more than happy to provide more details if you wish.î

No acknowledgement was expected and none was received. I wanted to copy Thomas Friedman on it but could not find his contact details on his website ñ the only email address was that of his literary agent and PR agency.

This apparently widespread misunderstanding and ignorance – about the Hindu contribution to the number system and sciences – prompted me to dig deeper. Here is what I found:

From an online research piece on Al-Khwarizmi and his work (by Shawn Overbay, Jimmy Schorer, and Heather Conger)

ì Al-Khwarizmi wrote numerous books that played important roles in arithematic and algebra. In his work, De numero indorum (Concerning the Hindu Art of Reckoning), it was based presumably on an Arabic translation of Brahmagupta where he gave a full account of the Hindu numerals which was the first to expound the system with its digits 0,1,2,3,….,9 and decimal place value which was a fairly recent arrival from India. Because of this book with the Latin translations made a false inquiry that our system of numeration is arabic in origin. The new notation came to be known as that of al-Khwarizmi, or more carelessly, algorismi; ultimately the scheme of numeration making use of the Hindu numerals came to be called simply algorism or algorithm, a word that, originally derived from the name al-Khwarizmi, now means, more generally, any peculiar rule of procedure or operation.

Interestingly, as the article notes, ìThe Hindu numerals like much new mathematics were not welcomed by all. In 1299 there was a law in the commercial center of Florence forbidding their use; to this day this law is respected when we write the amount on a check in longhand (ernie.bgsu.edu).î From a very well-researched online article, ìNumbers: Their History and Meaningî

ìIt is now universally accepted that our decimal numbers derive from forms, which were invented in India and transmitted via Arab culture to Europe, undergoing a number of changes on the way. We also know that several different ways of writing numbers evolved in India before it became possible for existing decimal numerals to be marred with the place-value principle of the Babylonians to give birth to the system which eventually became the one which we use today.

Because of lack of authentic records, very little is known of the development of ancient Hindu mathematics. The earliest history is preserved in the 5000-year-old ruins of a city at Mohenjo Daro, located Northeast of present-day Karachi in Pakistan. Evidence of wide streets, brick dwellings an apartment houses with tiled bathrooms, covered city drains, and community swimming pools indicates a civilisation as advanced as that found anywhere else in the ancient Orient.

These early peoples had systems of writing, counting, weighing, and measuring, and they dug canals for irrigation. All this required basic mathematics and engineering.
And later in the article, ìThe special interest of the Indian system is that it is the earliest form of the one, which we use today. Two and three were represented by repetitions of the horizontal stroke for one. There were distinct symbols for four to nine and also for ten and multiples of ten up to ninety, and for hundred and thousand.î

and further ìÖKnowledge of the Hindu system spread through the Arab world, reaching the Arabs of the West in Spain before the end of the tenth century. The earliest European manuscript, which came from the Hindu numerals were modified in north-Spain from the year 976.î And finally an important point for those who maintain that the concept of zero was also evident in some other civilisations: ìOnly the Hindus within the context of Indo-European civilisations have consistently used zero.î

Fortunately, online encyclopaedias came across as less biased and more open in acknowledging the true source of the ìArabicî number system. For example, from MSN Encarta

ìThe system of numbers that we use today, with each number having an absolute value and a place value (units, tens, hundreds, and so forth) originated in India. Mathematicians in India also were the first to recognize zero as both an integer and a placeholder. When the Indian numeration system was developed is not known, but digits similar to the Arabic numerals used today have been found in a Hindu temple built about 250 bc.

In the 5th century Hindu mathematician and astronomer Aryabhata studied many of the same problems as Diophantus but went beyond the Greek mathematician in his use of fractions as opposed to whole numbers to solve indeterminate equations (equations that have no unique solutions). Aryabhata also figured the value of ìPî (pi) accurately to eight places, thus coming closer to its value than any other mathematician of ancient times. In astronomy, he proposed that Earth orbited the sun and correctly explained eclipses of the Sun and Moon.

The earliest known use of negative numbers in mathematics was by Hindu mathematician Brahmagupta about ad 630. He presented rules for them in terms of fortunes (positive numbers) and debts (negative numbers).

ÖThe best-known Indian mathematician of the early period was Bhaskara, who lived in the 12th century. Bhaskara supplied the correct answer for division by zero as well as rules for operating with irrational numbers. Bhaskara wrote six books on mathematics, including Lilavati (The Beautiful), which summarized mathematical knowledge in India up to his time, and Karanakutuhala, translated as ìCalculation of Astronomical Wonders.î

The reality is that the so-called ìArabî contribution to mathematics was substantially built on prior knowledge of the Hindus and the Greeks and while the Greek influence and origins are frequently acknowledged, the Hindu contribution is very rarely mentioned.
We need to spread awareness about this and try and establish the facts whenever an opportunity arises ñ unless we do that, this ìhistoryî will be lost and become so little-known and distant as to become a myth.

Talking of forgotten Indian contribution to sciences and arts, here is another example of a glaring error in a recent news story in ìTIMEî Magazine and an email I sent in response

ìMay I point out two inaccuracies in your recent news story on an exhibition on Arab Science in Paris titled, ìAhead of Their Timeî (Time Magazine, Nov 21, í05; Pp48-49) by Ann Morrison?

In a paragraph about the Arabís interest in astronomy, Ann writes, ìÖThough the Arabs built many observatories during the Golden Age, not many survived. But viewers can see current images of two of these amazing outdoor structures in the Indian cities of Delhi and JaipurÖî

The observatories that Ann refers to in this paragraph were not built by Arabs but by the Hindu ruler Sawai Raja Jai Singh between 1724-1730 and were amongst the five that he built in Northern India (the other three were at Varanasi, Ujjain and Mathura) and are called Jantar Mantar (actually ìYantra Mantraî, yantra for instrument and mantra for formula).

The observatory in Delhi has also been depicted in a postage stamp and was the logo of the 1982 Asian Games, held in New Delhi, India.

To call them examples of Arab interest in the sciences is inaccurate and misleading.

In a later paragraph which details the interest of Arab scholars in astrology, Ann writes, ìÖAnother manuscript illustration from 17th century India, Astrologers working on a Nativityî, shows a procession of music makers and gift bearers wending their way through palace walls toward a newborn who would grow up to be the 14th century warrior Tamerlane…î

Again, this is an example of Indian art (and Indian interest in astrology) rather than having anything to do with Arabs or Arab art. Tamerlane himself was not an Arab king but from Central Asia (as were the Mughals).

As usual, I received neither an acknowledgement nor a response.

For those of you who would like to read more:

Hereís Alberuni on Pre-Islamic India’s Science, Math, and Architecture
And an interesting article on the origin of the decimal system.


The link to the article
 

johnee

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Indian Contribution to Mathematics

Does no one remember the Hindu contribution to Mathematics?


Whenever I read about the great ìArabicî contribution to Mathematics and Science (often in an apologetic tone of ìhow could these great people come to such a pass?î) the thing that really upsets me is the complete omission of any reference to the Hindu contribution to mathematics and numbers.

Slightly more than a year ago (Aug í04), in an article in the Sunday Times, Michael Portillo, eminent Conservative party leader in the UK and a one-time aspirant to the leadership of the Tory Party, wrote that, ìIslam brought back to the West knowledge of architecture, mathematics and astronomy that had been lost during the Dark Ages.î

In response, I wrote,

ìÖThe phrase ìbrought backî is at best, condescending and at worse, historically inaccurate.

For this knowledge, which Arab traders brought to Europe (typified in the Arabic numeral system – itself a misnomer, since the Arabs did not invent it but merely acted as the purveyors of this knowledge) was not Islamic or Arabic. In fact much of this knowledge was originally derived from ancient Vedic literature from India and passed through Arab traders and conquests to Middle East and eventually reaching Europe.

To quote from Carl B. Boyer in his “History of Mathematics”, ì…Mohammed ibn-Musa al-Khwarizmi, …, who died sometime before 850, wrote more than a half dozen astronomical and mathematical works, of which the earliest were probably based on the Sindhind derived from India. Besides … [he] wrote two books on arithmetic and algebra which played very important roles in the history of mathematics. … In this work, based presumably on an Arabic translation of Brahmagupta, al-Khwarizmi gave so full an account of the Hindu numerals that he probably is responsible for the widespread but false impression that our system of numeration is Arabic in origin. … [pages 227-228]…î.

In a translation of Alberuni ës ìIndicaî, a seminal work of this period (c.1030 AD), Edward Sachau, writes this in his introduction, ìMany Arab authors took up the subjects communicated to them by the Hindus and worked them out in original compositions , commentaries and extracts. A favourite subject of theirs was Indian mathematics…” etc.

Needless to say, the letter never got published.

Then, more recently, while reading the ìThe World is Flatî by Thomas L. Friedman , I came across this text in Chapter 11, “The Unflat World” (Pg 405), “As Nayan Chanda, the editor of YaleGlobal Online pointed out to me, it was the Arab-Muslim world that gave birth to algebra and algorithms, terms both derived form Arabic words. In other words, noted Chanda, “The entire modern information revolution, which is built to a large degree on algorithms, can trace its roots all the way back to Arab-Muslim civilization and the great learning centres of Baghdad and Alexandria,” which first introduced these concepts, then transferred them to Europe through Muslim Spain.

Dismayed, I wrote the following email to Nayan:

ìMay I respectfully point out that is not historically accurate and continuing research is providing evidence that the roots of the so-called Arab contribution to Mathematics and Science were further east in the lands of India and in the works of Indian mathematicians and scholars from several centuries ago.î

I then gave a couple of examples and concluded by saying:

ìI hope that you will re-consider your views in the light of these excerpts and a significant body of research that is now publicly available on this subject. I would be more than happy to provide more details if you wish.î

No acknowledgement was expected and none was received. I wanted to copy Thomas Friedman on it but could not find his contact details on his website ñ the only email address was that of his literary agent and PR agency.

This apparently widespread misunderstanding and ignorance – about the Hindu contribution to the number system and sciences – prompted me to dig deeper. Here is what I found:

From an online research piece on Al-Khwarizmi and his work (by Shawn Overbay, Jimmy Schorer, and Heather Conger)

ì Al-Khwarizmi wrote numerous books that played important roles in arithematic and algebra. In his work, De numero indorum (Concerning the Hindu Art of Reckoning), it was based presumably on an Arabic translation of Brahmagupta where he gave a full account of the Hindu numerals which was the first to expound the system with its digits 0,1,2,3,….,9 and decimal place value which was a fairly recent arrival from India. Because of this book with the Latin translations made a false inquiry that our system of numeration is arabic in origin. The new notation came to be known as that of al-Khwarizmi, or more carelessly, algorismi; ultimately the scheme of numeration making use of the Hindu numerals came to be called simply algorism or algorithm, a word that, originally derived from the name al-Khwarizmi, now means, more generally, any peculiar rule of procedure or operation.

Interestingly, as the article notes, ìThe Hindu numerals like much new mathematics were not welcomed by all. In 1299 there was a law in the commercial center of Florence forbidding their use; to this day this law is respected when we write the amount on a check in longhand (ernie.bgsu.edu).î From a very well-researched online article, ìNumbers: Their History and Meaningî

ìIt is now universally accepted that our decimal numbers derive from forms, which were invented in India and transmitted via Arab culture to Europe, undergoing a number of changes on the way. We also know that several different ways of writing numbers evolved in India before it became possible for existing decimal numerals to be marred with the place-value principle of the Babylonians to give birth to the system which eventually became the one which we use today.

Because of lack of authentic records, very little is known of the development of ancient Hindu mathematics. The earliest history is preserved in the 5000-year-old ruins of a city at Mohenjo Daro, located Northeast of present-day Karachi in Pakistan. Evidence of wide streets, brick dwellings an apartment houses with tiled bathrooms, covered city drains, and community swimming pools indicates a civilisation as advanced as that found anywhere else in the ancient Orient.

These early peoples had systems of writing, counting, weighing, and measuring, and they dug canals for irrigation. All this required basic mathematics and engineering.
And later in the article, ìThe special interest of the Indian system is that it is the earliest form of the one, which we use today. Two and three were represented by repetitions of the horizontal stroke for one. There were distinct symbols for four to nine and also for ten and multiples of ten up to ninety, and for hundred and thousand.î

and further ìÖKnowledge of the Hindu system spread through the Arab world, reaching the Arabs of the West in Spain before the end of the tenth century. The earliest European manuscript, which came from the Hindu numerals were modified in north-Spain from the year 976.î And finally an important point for those who maintain that the concept of zero was also evident in some other civilisations: ìOnly the Hindus within the context of Indo-European civilisations have consistently used zero.î

Fortunately, online encyclopaedias came across as less biased and more open in acknowledging the true source of the ìArabicî number system. For example, from MSN Encarta

ìThe system of numbers that we use today, with each number having an absolute value and a place value (units, tens, hundreds, and so forth) originated in India. Mathematicians in India also were the first to recognize zero as both an integer and a placeholder. When the Indian numeration system was developed is not known, but digits similar to the Arabic numerals used today have been found in a Hindu temple built about 250 bc.

In the 5th century Hindu mathematician and astronomer Aryabhata studied many of the same problems as Diophantus but went beyond the Greek mathematician in his use of fractions as opposed to whole numbers to solve indeterminate equations (equations that have no unique solutions). Aryabhata also figured the value of ìPî (pi) accurately to eight places, thus coming closer to its value than any other mathematician of ancient times. In astronomy, he proposed that Earth orbited the sun and correctly explained eclipses of the Sun and Moon.

The earliest known use of negative numbers in mathematics was by Hindu mathematician Brahmagupta about ad 630. He presented rules for them in terms of fortunes (positive numbers) and debts (negative numbers).

ÖThe best-known Indian mathematician of the early period was Bhaskara, who lived in the 12th century. Bhaskara supplied the correct answer for division by zero as well as rules for operating with irrational numbers. Bhaskara wrote six books on mathematics, including Lilavati (The Beautiful), which summarized mathematical knowledge in India up to his time, and Karanakutuhala, translated as ìCalculation of Astronomical Wonders.î

The reality is that the so-called ìArabî contribution to mathematics was substantially built on prior knowledge of the Hindus and the Greeks and while the Greek influence and origins are frequently acknowledged, the Hindu contribution is very rarely mentioned.
We need to spread awareness about this and try and establish the facts whenever an opportunity arises ñ unless we do that, this ìhistoryî will be lost and become so little-known and distant as to become a myth.

Talking of forgotten Indian contribution to sciences and arts, here is another example of a glaring error in a recent news story in ìTIMEî Magazine and an email I sent in response

ìMay I point out two inaccuracies in your recent news story on an exhibition on Arab Science in Paris titled, ìAhead of Their Timeî (Time Magazine, Nov 21, í05; Pp48-49) by Ann Morrison?

In a paragraph about the Arabís interest in astronomy, Ann writes, ìÖThough the Arabs built many observatories during the Golden Age, not many survived. But viewers can see current images of two of these amazing outdoor structures in the Indian cities of Delhi and JaipurÖî

The observatories that Ann refers to in this paragraph were not built by Arabs but by the Hindu ruler Sawai Raja Jai Singh between 1724-1730 and were amongst the five that he built in Northern India (the other three were at Varanasi, Ujjain and Mathura) and are called Jantar Mantar (actually ìYantra Mantraî, yantra for instrument and mantra for formula).

The observatory in Delhi has also been depicted in a postage stamp and was the logo of the 1982 Asian Games, held in New Delhi, India.

To call them examples of Arab interest in the sciences is inaccurate and misleading.

In a later paragraph which details the interest of Arab scholars in astrology, Ann writes, ìÖAnother manuscript illustration from 17th century India, Astrologers working on a Nativityî, shows a procession of music makers and gift bearers wending their way through palace walls toward a newborn who would grow up to be the 14th century warrior Tamerlane…î

Again, this is an example of Indian art (and Indian interest in astrology) rather than having anything to do with Arabs or Arab art. Tamerlane himself was not an Arab king but from Central Asia (as were the Mughals).

As usual, I received neither an acknowledgement nor a response.

For those of you who would like to read more:

Hereís Alberuni on Pre-Islamic India’s Science, Math, and Architecture
And an interesting article on the origin of the decimal system.


The link to the article
 

Yusuf

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Johnee disagree with the title of this thread and what you are trying to prove, its more of indic maths than hindu. Religion has got to do nothing with the knowledge, invention or anything. Its the region where it came from. Arabs didn't grow a brain after they became muslims. The mesopotamian civilization is one of the earliest one and it contributed a lot to human development. So did the sumerian. They gave us the wheel. Now where does religion come into the picture? by the title of the thread then since a lot of invention in modern era has happened in countries in the west which are majority christian, then do we term all of them as christian this and christian that?

In short I don't agree to the religious connotation to knowledge.
 

johnee

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Yusuf,
Its perogative to agree or disagree. Its not religious connotation to knowledge but simply a reminder that a certain religious group has made contributions to a field. Knowlege is universal, no denying that. But different groups contribute to it and their contributions need to be credited in history.
 

Yusuf

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I would go more as indic contribution than a religious based. What if I were a hindu all my life and then converted to islam and one fine day eureka, I invent something? So what who is to be credited? Me being from india or a follower of a faith?
 

johnee

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BTW, its not my title, its the original title given by the author. Indian mathematicians in the past were of the Dharma-based traditions. And it is right that this is made clear today, lest there be any confusion
 

Yusuf

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Well then I disagree with the author mate.

For some reason it may be to do with trying to create an identity. Or credit or what.

I am reminded of how every time an american of indian origin or an indian in any other country gets successful or wins an election or something is quickly caught by indians especially the media who go ga ga about a person of indian origin reaching the zenith. That person may be a third gen indian there who may have never visited india, but indians don't tire of trying to "own" them.
 

johnee

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I would go more as indic contribution than a religious based. What if I were a hindu all my life and then converted to islam and one fine day eureka, I invent something? So what who is to be credited? Me being from india or a follower of a faith?
A hypothetic situation can at best only generate clueless guesses. Anyway, the point that needs notice is that in west the early scientists in quest of knowledge had to defy existing traditions and were witch hunted in the process. While, in contrast, the ancient scientists in India were aided(and not obstructed) by the existing traditions of Hinduism. Science and religion gelled well in ancient India as compared to west(or even modern India). This pattern needs notice and appreciation given the violent persecution of the early scientists(like Galileo) in the west.
 

johnee

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Well then I disagree with the author mate.

For some reason it may be to do with trying to create an identity. Or credit or what.

I am reminded of how every time an american of indian origin or an indian in any other country gets successful or wins an election or something is quickly caught by indians especially the media who go ga ga about a person of indian origin reaching the zenith. That person may be a third gen indian there who may have never visited india, but indians don't tire of trying to "own" them.
You are right about the media and the general attitude. But I dont see any connection of that with the present topic. I dont understand why you are against the credit being given where its due. Are you against Hindus being given the credit of the work they have done in mathematical(and other scientific field)? If so why?

As for creating identities, no new identity is being created. The identity already exists. Their positive contribution is being appreciated which has been unfairly credited to others.

For eg: mostly arabs are being credited with the present decimal number system we use. But we know that its actually hindus who invented it.
 
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johnee

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Veda: The Origing of the Pure Mathematics

The First Concept of the Infinity


oḿ pūrṇam adaḥ pūrṇam idaḿ

pūrṇāt pūrṇam udacyate

pūrṇasya pūrṇam ādāya

pūrṇam evāvaśiṣyate

oḿ — the Complete Whole; pūrṇam — perfectly complete; adaḥ — that; pūrṇam — perfectly complete; idam — this phenomenal world; pūrṇāt — from the all-perfect; pūrṇam — complete unit; udacyate — is produced; pūrṇasya — of the Complete Whole; pūrṇam — completely, all; ādāya — having been taken away; pūrṇam — the complete ; eva — even; avaśiṣyate — is remaining.

The Isha Upanishad of the Yajurveda (400 BC) states :

The "Complete Whole", that is said here must contain everything both within and beyond our experience, otherwise He cannot be complete. When the "Complete Whole" is taken away from the "Complete", what remains is the "Complete Whole" itself.

The Indian mathematical text Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

* Enumerable: lowest, intermediate and highest
* Innumerable: nearly innumerable, truly innumerable and innumerably innumerable
* Infinite: nearly infinite, truly infinite, infinitely infinite

First Concept of the Void, the 'Shunya'


Shunya is the Sanskrit term translated in English as Void & also as Nothingness. The Madhyamika School of Buddhism that shunya is the transcendent & indefinable & immanent in all beings. Scholars speaking for shunya say that it is not nothingness since even the illusory structure can't be sustained in nothingness. Void is a metaphysical reality. Nagarjuna, the scholar, logician of the said school says Shunyata is a positive principle. Kumarajiva, commenting on Nagarjuna mentions that it is on account of Shunyata that everything becomes possible (Prajnaparamita). From the Madhyamika point the reality is Shunya (Shunyam Tattvam).

How 'Shunya' Became Zero


The concept of ‘emptiness' was alien to other cultures, so when this philosophical concept was applied in the mathematical context, it was not only revolutionary, but also mystifying. Interaction between Hindus and Arabs resulted in adopting the Indian numeration in the 10th Century. The Arabs however changed the Sanskrit word ‘SHUNYA' TO ‘SIFR' but when the 12th century, Italian mathematician Leonardo Pisano Fibonacci after studying Arabian algebra, introduced the Hindu-Arabic numerals in Italy, they however Latinized the Arabic word ‘SIFR' to ‘ZEPHIRUM'. This over time over time became zero. In Germany and England however the metamorphosis took a different turn. In Germany when Jordanus Nemaririus introduced the Arabic system of numerals, he retained the original Arabic word, but modified it to CIFRA' In England however the word CIFRA became CIPHER. In the early period the new numeration incorporating ZERO was looked upon as a secret sign by the common people. In fact the word ‘decipher' clearly reveals the enigma associated with it.

Vedic Technique of Computing Squares of Two Digit Integers

The Vedic technique allows to perform lightning fast calculations, unbelievably quick and easy way to master Multiplication in five minutes.



First Concept of Irrational Numbers


The Hindus had a very good system of approximating irrational square roots. Three of the Sulva Sutras, written by Baudhayana around 800 BC contain the approximation, much before others could get anywhere close:



[Refer: "Mathematical Thought from Ancient to Modern Times: Volume 1, Morris Kline, Oxford University Press US, 1990". pp. 200. ]

The First Conception of the Binary Number System Pingala was an Ancient Indian musical theorist who authored the famous Chandas Shastra (chandaḥ-śāstra, also Chandas Sutra chandaḥ-sūtra), a Sanskrit treatise on prosody considered one of the Vedanga. He developed advanced mathematical concepts for describing the patterns of prosody in the 400 BC. The shastra is divided into eight chapters. It was edited by Weber (1863). It is at the transition between Vedic meter and the classical meter of the Sanskrit epics. The 10th century mathematician Halayudha commented and expanded it. Pingala presents the first known description of a binary numeral system. He described the binary numeral system in connection with the listing of Vedic meters with short and long syllables. His discussion of the combinatorics of meter, corresponds to the binomial theorem. Halayudha' s commentary includes a presentation of the Pascal's triangle (called meru-prastaara). Pingala's work also contains the basic ideas of Fibonacci number (called maatraameru ). Use of zero is sometimes mistakenly ascribed to Pingala due to his discussion of binary numbers, usually represented using 0 and 1 in modern discussion, while Pingala used short and long syllables. Four short syllables (binary "0000") in Pingala's system, however, represented the number one, not zero. Positional use of zero dates from later centuries and would have been known to Halayudha but not to Pingala.

[Further Reading:

1. B. van Nooten und G. Holland, Rig Veda, a metrically restored text, Department of Sanskrit and Indian Studies, Harvard University, Harvard University Press, Cambridge, Massachusetts and London, England, 1994
2. H. Oldenberg, Prolegomena on Metre and Textual History of the Ṛgveda, Berlin 1888. Tr. V.G. Paranjpe and M.A. Mehendale, Motilal Banarsidass 2005 ISBN 81-208-0986-6]

First Knowledge of Binary Operations
In Jyotish Shastra (Astrology) they calculated time, position and motion of stars. In the book of Vedanga Jyotish (At least 1000 BC) we find that astrologers knew about binary operations like addition, multiplication, subtraction. For example read below-



Meaning: Multiply the date by 11, then add to it the "Bhansh" of "Parv" and then divide it by "Nakshatra" number. In this way the "Nakshtra" of date should be told. Knowledge of Combinational Identity One of the sutras of Pingala it is found: "Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting 1 in the first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put 1 in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives the combination with one syllable, the third the combination with two syllables, ... The text also indicates that Pingala was aware of the combinatorial identity:



Baudhayan Sulv Sutra - Also known as "Paithogorus Theorem"

Baudhayan Sulv Sutra (1000 BC) is the formula given below, invented nearly 500 years before Paithogorus was born. It says, i a Deerghchatursh (Rectangle) the Chetra (Square) of Rajju (hypotenuse) is equal to sum of squares of Parshvamani (base) and Triyangmani (perpendicular). Amazing, isn't it ?



[Refer: Explorations in Mathematics pp:50, By A.A. Hattangadi, Published by Orient Blackswan, 2002] Roots of Modern Trigonometry Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry. This ancient text uses the following as trigonometric functions for the first time:

* Sine (Jya).
* Cosine (Kojya).
* Inverse sine (Otkram jya).

(Knowledge of Tangent and Secent were also know)


Cosmological Time-cycles


* The average length of the sidereal year as 365.2563627 days, which is only 1.4 seconds longer than the modern value of 365.2563627 days.
* The average length of the tropical year as 365.2421756 days, which is only 2 seconds shorter than the modern value of 365.2421988 days.

[Refer Vedic Evidence of the Sidereal Year - by Glen R. Smith]
]

Mathematics - Vedas and Vedangas (auxiliary)


The Vedic civilization originated in India bears the literary evidence of Indian culture, literature, astronomy and mathematics. Written in Vedic Sanskrit the Vedic works, Vedas and Vedangas (and later Sulbasutras) are primarily religious in content, but embody a large amount of astronomical knowledge and hence a significant knowledge of mathematics. Some chronological confusion exists with regards to the appearance of the Vedic civilization. S Kak states in a very recent work that the time period for the Vedic religion stretches back potentially as far as 8000BC and definitely 4000BC. It is also worthwhile briefly noting the astronomy of the Vedic period which, given very basic measuring devices (in many cases just the naked eye), gave surprisingly accurate values for various astronomical quantities. These include the relative size of the planets the distance of the earth from the sun, the length of the day, and the length of the year. Some of Vedic works are:

* All four arithmetical operators (addition, subtraction, multiplication and division).
* A definite system for denoting any number up to 1055 and existence of zero.
* Prime numbers.

The Arab scholar Al-Biruni (973-1084 AD) discovered that the Indians had a number system that was capable of going beyond the thousands in naming the orders in decimal counting. It is in Vedic works that we also first find the term "ganita" which literally means "the science of calculation". It is basically the Indian equivalent of the word mathematics and the term occurs throughout Vedic texts and in all later Indian literature with mathematical content.

Among the other works mentioned, mathematical material of considerable interest is found:

* Arithmetical sequences, the decreasing sequence 99, 88, ... , 11 is found in the Atharva-Veda.
* Pythagoras's theorem, geometric, constructional, algebraic and computational aspects known. A rule found in the Satapatha Brahmana gives a rule, which implies knowledge of the Pythagorean theorem, and similar implications are found in the Taittiriya Samhita.
* Fractions, found in one (or more) of the Samhitas.
* Equations, 972x2 = 972 + m for example, found in one of the Samhitas.


Sulba Sutras


The Sulba-sutras, dated from around 800-200 BC, contain the first 'use' of irrational numbers, quadratic equations of the form a x2 = c and ax2 + bx = c. Indeed an early method for calculating square roots can be found in some Sutras, the method involves repeated application of the formula:
A =
(a2 + r) = a + r/2a,
r being small. The Bakhshali Manuscript The Bakhshali manuscript was written on leaves of birch, in combination of Sanskrit and Prakrit. This may go some way to explaining the number of inaccurate translations. Many of the historians who have been involved in translating ancient Indian works have done poorly, due to the obscure script, or alternatively because they did not understand the mathematics full or to play down the importance of ancient Indian works, because they challenge the Eurocentric ideal. The Bakhshali manuscript highlights developments in Arithmetic and Algebra. The arithmetic contained within the work is of such a high quality that it has been suggested:

...In fact [the] Greeks [are] indebted to India for much of the developments in Arithmetic. [LG, P 53]

There are eight principal topics 'discussed' in the Bakhshali manuscript:

* Examples of the rule of three (and profit and loss and interest).
* Solution of linear equations with as many as five unknowns.
* The solution of the quadratic equation (development of remarkable quality).
* Arithmetic (and geometric) progressions.
* Compound Series (some evidence that work begun by Jainas continued).
* Quadratic indeterminate equations (origin of type ax/c = y).
* Simultaneous equations.
* Fractions and other advances in notation including use of zero and negative sign.
* Improved method for calculating square root (and hence approximations for irrational numbers). The improved method (shown below) allowed extremely accurate approximations to be calculated:
A =
t(a2 + r) = a + r/2a - {(r/2a)2 / 2(a + r/2a)}


Decimal Numeration and the Place-value System


The development of a decimal place value system of numeration and there is now very little doubt among historians that this invention originated from the Indian. That said, it was considered, until recently, that Arabic scholars were responsible for the system, as C Srinivasiengar writes:

...During the earlier decades of this century (20th) attempts were made to credit this invention wholly or in part to the Arabs. [CS, P 2]

The last significant case of an attempt to abolish the Indian decimal place value system was in Sweden in the early 18th century. This is clearly a very brief overview of the phenomenal development of the decimal place value system, without which it is accepted 'higher mathematics' would not be possible. A quote from G Halstead who commented: ...The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habituation and a name, a picture, a symbol but helpful power, is the characteristic of the Hindu race from whence it sprang. No single mathematical creation has been more potent for the general on go of intelligence and power. [CS, P 5]
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johnee

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Yusuf,
the above article shows the hindu background(or vedic background to be more precise) was not just an accident but infact a delibrate contributor to the achievements.
 

xebex

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Johnee I don think attaching mathematical inventions with religion is appropriate. Lets, say can we say the modern science is 'Christian science'?? the credit for those inventions should be given to the civilization, to be precise, to the guy who invented it despite his religion.
 

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Johnee I don think attaching mathematical inventions with religion is appropriate. Lets, say can we say the modern science is 'Christian science'?? the credit for those inventions should be given to the civilization, to be precise, to the guy who invented it despite his religion.
Xebex, read the last posted article. That article shows how the ancient indian contributions were directly related to the tradition. And thats the reason for this thread. It needs noted and appreciated that the ancient Indian traditions were not obstructive to the development of science but in contrast directly encouraging. You know that the development of modern science has started in west but it was not encouraged by the existing traditions, instead they did everything within their power to stifle it. So, while the ancient indian science and its achievements can be credited to the existing religions and the traditions the same is not the case with the western(or modern) science. Hope, you understand the difference and dont use a broad brush for all things. Religion and science are not anti to each other, they have never been in India, this syndrome has been mostly a hallmark of west or western born traditions or religions.
 

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Varahamihira


Varahamihira (505BC-587BC) produced the Pancha Siddhanta (The Five Astronomical Canons). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal



places of accuracy and the following formulas relating sine and cosine functions:


Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by:


Brahmagupta's Theorem on rational triangles: A triangle with rational sides a,b,c and rational area is of the form:


[For more refer to this Link ]
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Aryabhata


Aryabhata stands as a pioneer of the revival of Indian mathematics, and the so called 'classical period', or 'Golden era' of Indian mathematics. Arguably the Classical period continued until the 12th century, although in some respects it was over before Aryabhata's death following a costly, if ultimately successful, war with invading Huns which resulted in the eroding of the Gupta
culture. Some of his contributions are:

* Tables of sine values
* The Aryabhatiya is the first historical work of the dated type, which definitely uses some of these (trigonometric) functions and contains a table of sines.
* Of the mathematics contained within the Aryabhatiya the most remarkable is an approximation for π (pi), which is surprisingly accurate. The value given is: π = 3.1416
* In the field of 'pure' mathematics his most significant contribution was his solution to the indeterminate equation: ax - by = c



Aryabhata's Table of Sign vales shown in "Hindu Sine" Column



The Aryabhatiya was translated into Arabic by Abu'l Hassan al-Ahwazi (before 1000 AD) as Zij al-Arjabhar and it is partly through this translation that Indian computational and mathematical methods were introduced to the Arabs, which will have had a significant effect on the forward progress made by mathematics. The historian A Cajori even goes as far as to suggest that:

...Diophantus, the father of Greek algebra, got the first algebraic knowledge from India. [RG4, P 12]

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Bhaskaracharya II

Bhaskara II, is regarded almost without question as the greatest Hindu mathematician of all time and his contribution to not just Indian, but world mathematics is undeniable. As L Gurjar states:

...Because of his work India gave a definite 'quota' to the forward world march of the science. [LG, P 104]

He wrote Siddhanta Siromani in 1150 AD, which contained four sections:

* Lilavati (arithmetic)
* Bijaganita (algebra)
* Goladhyaya (sphere/celestial globe)
* Grahaganita (mathematics of the planets)

Bhaskara is to be the first to show that:

deltasin x = cos x deltax

Bhaskara was fully acquainted with the principle of differential calculus, and that his researches were in no way inferior to Newton's, asides the fact that it seems he did not understand the utility of his researches, and thus historians of mathematics generally neglect his outstanding achievement, which is extremely regrettable. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'. He also gives the (now) well known results for sin(a + b) and sin(a - b). There is also evidence of an early form of Rolle's theorem if f(a) = f(b) = 0 then f '(x) = 0 for some x with a <>, in Bhaskara's work.

Lilavati: Definitions


In Lilavati, the beautiful definitions of different units demonostrates the advanced arithmetics of the Vedic-era.

1. Having bowed to [Ganesa] who causes the joy of those who worship him, who, when thought of, removes obstacles, the elephant-headed one whose feet are honored by multitudes of gods, I state the arithmetical rules of true computation, the beautiful Lilavati, clear and providing enjoyment to the wise by its concise, charming and pure quarter-verses.
2. Two times ten varatakas [cowrie] are a kakini [shell], and four of those are a pana [copper coin]. Sixteen of those are considered here [to be] a dramma [coin, "drachma"], and so sixteen drammas are a niska [gold coin].
3. Two yavas [barley grain (a weight measure)] are here considered equal to a gunja [berry]; three gunjas are a valla [wheat grain] and eight of those are a dharana [rice grain]. Two of those are a gadyanaka, so a ghataka is defined [to be] equal to fourteen vallas.
4. Those who understand weights call half of ten gunjas a masa [bean], and sixteen of [the weights] called masa a karsa, and four karsas a pala. A karsa of gold is known as a suvarna [lit. "gold"].
5. An angula [digit] is eight yavodaras [thick part of a barley grain]; a hasta [hand] is four times six angulas. Here, a danda [rod] is four hastas, and a krosa [cry] is two thousand of those.
6. A yojana is four krosas. Likewise, ten karas [hand, hasta] are a vamsa [bamboo]; a nivartana is a field bounded by four sides of twenty vamsas [each].
7. A twelve-edged [solid] with width, length, and height measured by one hasta is called a cubic hasta. In the case of grain and so forth, a measure [equal to] a cubic hasta is called in treatises a "Magadha kharika".
8. And a drona [bucket] is a sixteenth part of a khari; an adhaka is a fourth part of a drona. Here, a prastha is a fourth part of an adhaka; by earlier [authorities], a kudava is defined [as] one-fourth of a prastha.


[For more Dept of Mathematics, Brown Univercity]


Ancient Scripts in Lilavati


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Madhava of Sangamagramma

Born in Cochin on the Kerala coast Madhava of Sangamagramma (c. 1340 - 1425) is one of the greatest mathematician-astronomer of medieval India. Sadly most of his mathematical works are currently lost, although it is possible extant work may yet be 'unearthed'. His most significant contribution was in moving on from the finite procedures of ancient mathematics to 'treat their limit passage to infinity', which is considered to be the essence of modern classical analysis. Although there is not complete certainty it is thought Madhava was responsible for the discovery of all of the following results:

1. theta = tan theta - (tan3 theta)/3 + (tan5theta)/5 - ... ,
2. rtheta= {r(rsintheta)/1(rcostheta)}-{r(rsintheta)3/3(rcostheta)3}+{r(rsintheta)5/5(rcostheta)5}- ...
3. sintheta = theta - theta3/3! + theta5/5! - .. Madhava-Newton power series.
4. costheta = 1 - theta2/2! + theta4/4! - ..., Madhava-Newton power series.
5. p/4 approx 1 - 1/3 + 1/5 - ... plusminus 1/n plusminus (-fi(n+1)), i = 1,2,3, and where f1 = n/2, f2 = (n/2)/(n2 + 1) and f3n/2)2 + 1)/((n/2)(n2 + 4 + 1))2
6. pd approx 2d + 4d/(22 - 1) - 4d/(42 - 1) + ... plusminus 4d/(n2 + 1), etc, this resulted in improved approximations of p, a further term was added to the above expression, allowing Madhava to calculate p to 13 decimal places. The value p = 3.14159265359 is unique to Kerala and is not found in any other mathematical literature. A value correct to 17 decimal places (3.14155265358979324) is found in the work Sadratnamala. R Gupta attributes calculation of this value to Madhava, (so perhaps he wrote this work, although this is pure conjecture).
7. tan -1x = x - x3/3 + x5/5 - ..., Madhava-Gregory series, power series for inverse tangent,
8. sin(x + h) approx sin x + (h/r)cos x - (h2/2r2)sin x
9. It is interesting that a four-term approximation formula for the sine function so close to the Taylor series approximation was known in India more than two centuries before the Taylor series expansion was discovered by Gregory about 1668.

G Joseph states:

...We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition. [GJ, P 293]

With regards to Keralese contributions as a whole, M Baron writes (in D Almeida, J John and A Zadorozhnyy):

...Some of the results achieved in connection with numerical integration by means of infinite series anticipate developments in Western Europe by several centuries. [DA/JJ/AZ1, P 79]


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Possible Transmission of Indian Mathematics to Europe

There is a very recent paper (written by D Almeida, J John and A Zadorozhnyy) of great interest, which goes as far as to suggest Indian mathematics may have been transmitted to Europe. It is true that India was in continuous contact with China, Arabia, and at the turn of the 16th century, Europe, thus transmission might well have been possible. However the current theory is that Indian calculus remained localised until its discovery by Charles Whish in the late 19th century. There is no evidence of direct transmission by way of relevant manuscripts but there is evidence of methodological similarities, communication routes and a suitable chronology for transmission. A key development of pre-calculus Europe, that of generalisation on the basis of induction, has deep methodological similarities with the corresponding Kerala development (200 years before). There is further evidence that John Wallis (1665) gave a recurrence relation and proof of Pythagoras theorem exactly as Bhaskara II did. The only way European scholars at this time could have been aware of the work of Bhaskara would have been through Indian 'routes'. The need for greater calendar accuracy and inadequacies in sea navigation techniques are thought to have led Europeans to seek knowledge from their colonies throughout the 16th and 17th centuries. The requirements of calendar reform were imperative with the dating of Easter proving extremely problematic, by the 16th century the European 'Julian' calendar was becoming so inaccurate that without correction Easter would eventually take place in summer! There were significant financial rewards for 'anyone' who could 'assist' in the improvement of navigation techniques. It is thought 'information' was sought from India in particular due to the influence of 11th century Arabic translations of earlier Indian navigational methods. vents also suggest it is quite possible that Jesuits (Christian missionaries) in Kerala were 'encouraged' to acquire mathematical knowledge while there. It is feasible that these observations are mere coincidence but if indeed it is true that transmission of ideas and results between Europe and Kerala occurred, then the 'role' of later Indian mathematics is even more important than previously thought.

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Examples of Vedic-age Algebra

1. Use Bhaskara’s method to find two integers such that the square of their sum plus the cube of their sum equals twice the sum of their cubes. (This is a problem from Chapter 7 of the Vija Ganita.)


2. Show that the formula given by Brahmagupta for the area of a quadrilateral is correct if and only if the quadrilateral can be inscribed in a circle.


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Examples of Vedic-age Trigonometry
1. Show that Aryabhata’s list of sine differences can be interpreted in our language as the table whose nth entry is


Use a computer to generate this table for n = 1,2,...24 and compare the result with Aryabhata’s table.


2. If the recursive procedure described by Aryabhata is followed faithfully (as a computer can do), the result is the following sequence.
225, 224, 222, 219, 215, 210, 204, 197, 189, 181, 172, 162, 151, 140, 128, 115, 102, 88, 74, 60, 45, 30, 15, 0
Compare this list with Aryabhata’s list, and note the systematic divergence. These differences should be approximately 225 times the cosine of the appropriate angle. That is What does that fact



suggest about the source of the systematic errors in the recursive procedure described by Aryabhata? Answer. The differences form the sequence 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 7, 7, 7. What we have here is an algebraic function attempting to approximate a transcendental function and losing track of it. If Aryabhata did use this formula, he must have started over with a new “seed” several times in order to avoid the error accumulation shown here.

3. Use Aryabhata’s procedure to compute the altitude of the Sun above the horizon in London (latitude 51 degree 32min) at 10:00AM on the vernal equinox. Assume that the sun rises at 6:00 AM on that day and sets at 6:00 PM.



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