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Vedic Mathematics in the Ancient World

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ॐ पूर्णमदः पूर्णमिदम् पूर्णात् पूर्णमुदच्यते |

पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते ||

ॐ शान्तिः, शान्तिः, शान्तिः ||

Oum SaantiH, SaantiH, SaantiH.

That is Absolute, This is Absolute, Absolute arises out of Absolute, If Absolute is taken away from Absolute, Absolute remains OM Peace, Peace, Peace.

This section is dedicated to  the memory of Srinivasa Ramanujan (1887-1920), the man who knew infinity, arguably the greatest number theorist in all of human history

Introductory Remarks

The contributions of the ancient Indics are usually overlooked and rarely given sufficient credit in Western Texts (see for instance FAQ on Vedic Mathematics). The Wikipedia section on Indian Mathematics says the following;

Unfortunately, Indian contributions have not been given due acknowledgement in modern history, with many discoveries/inventions by Indian mathematicians now attributed to their western counterparts, due to Eurocentrism.

The historian Florian Cajori, one of the most celebrated historians of mathematics in the early 20th century, suggested that "Diophantus, the father of Greek algebra, got the first algebraic knowledge from India." This theory is supported by evidence of continuous contact between India and the Hellenistic world from the late 4th century BC, and earlier evidence that the eminent Greek mathematician Pythagoras visited India, which further 'throws open' the Eurocentric ideal.

More recently, evidence has been unearthed that reveals that the foundations of calculus were laid in India, at the Kerala School. Some allege that calculus and other mathematics of India were transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala was in continuous contact with China, Arabia, and from around 1500, Europe as well, thus transmission would have

Furthermore,  we cannot discuss Vedic mathematics without discussing Babylonian and Greek  Mathematics to give it the scaffolding and context. We will devote some attention to these developments to put the Indic contribution its proper context

However in recent years , there has been greater international recognition of the scope and breadth of  the Ancient Indic contribution to the sum of human knowledge especially in some fields of science and technology such as Mathematics and Medicine. Typical of this new stance is the following excerpt by researchers at St. Andrews in Scotland.

An overview of Indian mathematics

It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years. What is quite surprising is that there has been a reluctance to recognize this and one has to conclude that many famous historians of mathematics found what they expected to find, or perhaps even what they hoped to find, rather than to realize what was so clear in front of them.

We shall examine the contributions of Indian mathematics in this article, but before looking at this contribution in more detail we should say clearly that the "huge debt" is the beautiful number system invented by the Indians on which much of mathematical development has rested. Laplace put this with great clarity:-

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.

We shall look briefly at the Indian development of the place-value decimal system of numbers later in this article and in somewhat more detail in the separate article Indian numerals. First, however, we go back to the first evidence of mathematics developing in India.

Histories of Indian mathematics used to begin by describing the geometry contained in the Sulvasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita. Also it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times.

Equally exhaustive in its treatment is the Wiki encyclopedia, where in general the dates are still suspect. See for instance the   Wikipedia on Indian Mathematics

### Vedic number theory

Mathematicians in India were interested in finding integral solutions of Diophantine equations since the Vedic era. The earliest geometric use of Diophantine equations can be traced back to the Sulba Sutras, which were written between the 8th and 6th centuries BC. Baudhayana (c. 800 BC) found two sets of positive integral solutions to a set of simultaneous Diophantine equations, and also used simultaneous Diophantine equations with up to four unknowns. Apastamba (c. 600 BC) used simultaneous Diophantine equations with up to five unknowns.

### Jaina number theory

In India, Jaina mathematicians developed the earliest systematic theory of numbers from the 4th century BC to the 2nd century CE. The Jaina text Surya Prajinapti (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.

The Jains were the first to discard the idea that all infinites were the same or equal. They recognized five different types of infinity: infinite in one and two directions (one dimension), infinite in area (two dimensions), infinite everywhere (three dimensions), and infinite perpetually (infinite number of dimensions).

The highest enumerable number N of the Jains corresponds to the modern concept of aleph-null $\aleph_0$ (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of transfinite cardinal numbers, of which $\aleph_0$ is the smallest.

In the Jaina work on the theory of sets, two basic types of transfinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asmkhyata and ananata, between rigidly bounded and loosely bounded infinities.

### Hellenistic number theory

Number theory was a favorite study among the Hellenistic mathematicians of Alexandria, Egypt from the 3rd century CE, who were aware of the Diophantine equation concept in numerous special cases. The first Hellenistic mathematician to study these equations was Diophantus.

Diophantus also looked for a method of finding integer solutions to linear indeterminate equations, equations that lack sufficient information to produce a single discrete set of answers. The equation x + y = 5 is such an equation. Diophantus discovered that many indeterminate equations can be reduced to a form where a certain category of answers is known even though a specific answer is not.

### Classical Indian number theory

Diophantine equations were extensively studied by mathematicians in medieval India, who were the first to systematically investigate methods for the determination of integral solutions of Diophantine equations. Aryabhata (499) gave the first explicit description of the general integral solution of the linear Diophantine equation ay + bx = c, which occurs in his text Aryabhatiya. This kuttaka algorithm is considered to be one of the most significant contributions of Aryabhata in pure mathematics, which found solutions to Diophantine equations by means of continued fractions. The technique was applied by Aryabhata to give integral solutions of simulataneous linear Diophantine equations, a problem with important applications in astronomy. He also found the general solution to the indeterminate linear equation using this method.

Brahmagupta in 628 handled more difficult Diophantine equations. He used the chakravala method to solve quadratic Diophantine equations, including forms of Pell's equation, such as 61x2 + 1 = y2. His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was subsequently translated into Latin in 1126. The equation 61x2 + 1 = y2 was later posed as a problem in 1657 by the French mathematician Pierre de Fermat. The general solution to this particular form of Pell's equation was found over 70 years later by Leonhard Euler, while the general solution to Pell's equation was found over 100 years later by Joseph Louis Lagrange in 1767. Meanwhile, many centuries ago, the general solution to Pell's equation was recorded by Bhaskara II in 1150, using a modified version of Brahmagupta's chakravala method, which he also used to find the general solution to other indeterminate quadratic equations and quadratic Diophantine equations. Bhaskara's chakravala method for finding the general solution to Pell's equation was much simpler than the method used by Lagrange over 600 years later. Bhaskara also found solutions to other indeterminate quadratic, cubic, quartic and higher-order polynomial equations. Narayana Pandit further improved on the chakravala method and found more general solutions to other indeterminate quadratic and higher-order polynomial equations.

EVIDENCE PROM EUROPE

INDIA: THE TRUE BIRTHPLACE OF
OUR NUMERALS

The views of savants and learned scholars from a non-Indian tradition about Indian mathematics

Severus Sebokt of Syria in 662 CE: (the following statement must  be understood in the context of  the alleged Greek claim that all mathematical knowledge emanated from them

"I shall not speak here of the science of the Hindus, who are not even Syrians, and not of their subtle discoveries in astronomy that are more inventive than those of the Greeks and of the Babylonians; not of their eloquent ways of counting nor of their art of calculation, which cannot be described in words - I only want to mention those calculations that are done with nine numerals. 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". (Nau, 1910)

Said al-Andalusi, probably the first historian of Science who in 1068 wrote Kitab Tabaqut al-Umam in Arabic  (Book of Categories of Nations) Translated into English by Alok Kumar in 1992

To their credit, the Indians have made great strides in the study of numbers (3) and of geometry. They have acquired immense information and reached the zenith in their knowledge of the movements of the stars (astronomy) and the secrets of the skies (astrology) as well as other mathematical studies. After all that, they have surpassed all the other peoples in their knowledge of medical science and the strengths of various drugs, the characteristics of compounds and the peculiarities of substances.

Albert Einstein in the 20th century also comments on the importance of Indian arithmetic: "We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made."

Quotes from Liberabaci (Book of the Abacus) by Fibonacci (1170-1250): The nine Indian numerals are ...with these nine and with the sign 0 which in Arabic is sifr, any desired number can be written. (Fibonacci learnt about Indian numerals from his Arab teachers in North Africa) .Fibonacci introduced Indian numerals into  Europe in 1202CE.

...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]

The following quotes are from George Ifrah's book Universal History of Numbers
The real inventors of this fundamental discovery, which is no less important than such feats as the mastery of fire, the development of agriculture, or the invention of the wheel, writing or the steam engine, were the mathematicians and astronomers of Indian civilisation: scholars who, unlike the Greeks, were concerned with practical applications and who were motivated by a kind of passion for both numbers and numerical calculations.
There is a great deal of evidence to support this fact, and even the Arabo-Muslim scholars themselves have often voiced their agreement

The following is a succession of historical accounts in favor of this theory, given in chronological order, beginning with the most recent.
1. P. S. Laplace (1814): “The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.” [Dantzig. p. 26]
2. J. F. Montucla (179\$): “The ingenious number-system, which serves as the basis for modern arithmetic, was used by the Arabs long before it reached Europe. It would be a mistake, however, to believe that this invention is Arabic. There is a great deal of evidence, much of it provided by the Arabs themselves, that this arithmetic originated in India.” [Montucla, I, p. 375J
3. John Walls (1616-4703) referred to the nine numerals as Indian figures [Wallis (1695), p. 10]

4. Cataneo (1546) le noue figure de gli Indi, “the nine figures from India”. [Smith and Karpinski (1911), p.31
5. Willichius (1540) talks of Zyphrae !Indicate, “Indian figures”. [Smith and Karpinski (1911) p. 3]
6. The Crafte of Nombrynge (c. 1350), the oldest known English arithmetical tract: II fforthermore ye most vndirstonde that in this craft ben vsed teen figurys, as here bene writen for esampul 098 ^ 654321... in the quych we vse teen figwys of Inde. Questio II why Zen figurys of Inde? Soiucio. For as I have sayd afore thei werefondefrrst in Inde. [D. E. Smith (1909)1
7. Petrus of Dada (1291) wrote a commentary on a work entitled Algorismus by Sacrobosco (John of Halifax, c. 1240), in which he says the following (which contains a mathematical error): Non enim omnis numerus per quascumquefiguras Indorum repraesentatur “Not every number can be represented in Indian figures”. [Curtze (1.897), p. 251
8.Around the year 1252, Byzantine monk Maximus Planudes (1260—1310) composed a work entitled Logistike Indike (“Indian Arithmetic”) in Greek, or even Psephophoria kata Indos (“The Indian way of counting”), where he explains the following: “There are only nine figures. These are:
123456789
[figures given in their Eastern Arabic form].
A sign known as tziphra can be added to these, which, according to the Indians, means ‘nothing’. The nine figures themselves are Indian, and tziphra is written thus: 0”. [B. N., Pans. Ancien Fonds grec, Ms 2428, f” 186 r”]
9. Around 1240, Alexandre de Ville-Dieu composed a manual in verse on written calculation (algorism). Its title was Carmen de Algorismo, and it began with the following two lines:
Haec algorismus ars praesens dicitur, in qua Talibus Indorumfruimur bis quinquefiguris:
Algorism
is the art by which at present we use those Indian figures, which number two times five”. [Smith and Karpinski (1911), p. 11]
10. In 1202, Leonard of Pisa (known as Fibonacci), after voyages that took him to the Near East and Northern Africa, and in particular to Bejaia (now in Algeria), wrote a tract on arithmetic entitled Liber Abaci (“a tract about the abacus”), in which he explains the following:
Cum genitor meus a patria publicus scriba in duana bugee pro pisanis mercatoribus ad earn confluentibus preesset, me in pueritia mea ad se uenire faciens, inspecta utilitate el cornmoditate fiutura, ibi me studio abaci per aliquot dies stare uoluit et doceri. Vbi a mirabii magisterio in arte per nouem figuras Indorum introductus. . . Novem figurae Indorum hae sun!:
cum his itaque novemfiguris. et turn hoc signo o. Quod arabice zephirum appellatur, scribitur qui libel numerus: “My father was a public scribe of Bejaia, where he worked for his country in Customs, defending the interests of Pisan merchants who made their fortune there. He made me learn how to use the abacus when I was still a child because he saw how I would benefit from this in later life. In this way I learned the art of counting using the nine Indian figures... The nine Indian figures are as follows:

987654321

[figures given in contemporary European cursive form].

“That is why, with these nine numerals, and with this sign 0, called zephi rum in Arab, one writes all the numbers one wishes.”[Boncompagni (1857), vol.1]

11. C. U50, Rabbi Abraham Ben MeIr Ben Ezra (1092—1167), after a long voyage to the East and a period spent in Italy, wrote a work in Hebrew entitled: Sefer ha mispar (“Number Book”), where he explains the basic rules of written calculation.

He uses the first nine letters of the Hebrew alphabet to represent the nine units. He represents zero by a little circle and gives it the Hebrew name of galgal (“wheel”), or, more frequently, sfra (“void”) from the corresponding Arabic word.

However, all he did was adapt the Indian system to the first nine Hebrew letters (which he naturally had used since his childhood).
In the introduction, he provides some graphic variations of the figures, making it clear that they are of Indian origin, after having explained the place-value system: “That is how the learned men of India were able to represent any number using nine shapes which they fashioned themselves specifically to symbolise the nine units.” (Silberberg (1895), p.2: Smith and Ginsburg (1918): Steinschneider (1893)1

12. Around the same time, John of Seville began his Liberalgoarismi de practica arismetrice (“Book of Algoarismi on practical arithmetic”) with the following:

Numerus est unitatum cot/echo, quae qua in infinitum progredilur (multitudo enim crescit in infinitum), ideo a peritissimis Indis sub quibusdam regulis et certis lirnitibus infinita numerositas coarcatur, Ut de infinitis dfinita disciplina traderetur etfuga subtilium rerum sub alicuius artis certissima Jege ten eretur:

“A number is a collection of units, and because the collection is infinite (for multiplication can continue indefinitely), the indians ingeniously enclosed this infinite multiplicity within certain rules and limits so that infinity could be scientifically defined: these strict rules enabled them to pin down this subtle concept.
[B. N., Paris, Ms. lat. 16 202, r 51: Boncompagni (1857), vol. I, p. 261

13. C. 1143, Robert of Chester wrote a work entitled: Algoritmi de numero Indorum (“Algoritmi: Indian figures”), which is simply a trans lation of an Arabic work about Indian arithmetic. [Karpinski (1915); Wallis (1685). p. 121

14. C. 1140, Bishop Raimundo of Toledo gave his patronage to a work written by the converted jew Juan de 1.una and archdeacon Domingo Gondisalvo: the Liber Algorismi de numero Indorum (“Book of Algorismi of Indian figures) which is simply a translation into a Spanish and Latin version of an Arabic tract on Indian arithmetic. [Boncompagni (1857), vol. 11

15. C. 1130, Adelard of Bath wrote a work entitled: Algoritmi de numero Indorum (“Algoritmi: of Indian figures”), which is simply a translation of an Arabic tract about Indian calculation. [Boncompagni (1857), vol. Ii

16. C. 1125, The Benedictine chronicler William of Malmesbury wrote De gestis regum Anglorum, in which he related that the Arabs adopted the Indian figures and transported them to the countries they conquered, particularly Spain. He goes on to explain that the monk Gerbert of Aurillac, who was to become Pope Sylvester II (who died in 1003) and who was immortalized for restoring sciences in Europe, studied in either Seville or Cordoba, where he learned about Indian figures and their uses and later contributed to their circulation in the Christian countries of the West. L. Malmesbury (1596), f” 36 r’; Kopeck (1857), p. 35J

17. Written in 976 in the convent of Albelda (near the town of Logroño, in the north of Spain) by a monk named Vigila, the Coda Vigilanus contains the nine numerals in question, but not zero. The scribe clearly indicates in the text that the figures are of Indian origin:

Item de figuels aritmetice. Scire debemus Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et geometrica et ceteris liberalibu.c disciplinis concedere. Et hoc manifèstum at in novem figuris, quibus quibus designant unum quenque gradum cuiu.slibetgradus. Quatrum hec sunt forma:
9 8 7 6 5 4 3 2 1.

“The same applies to arithmetical figures. It should be noted that the Indians have an extremely subtle intelligence, and when it comes to arithmetic, geometry and other such advanced disciplines, other ideas must make way for theirs. The best proof of this is the nine figures with which they represent each number no matter how high. This is how the figures look:
9 8 7 6 5 4 3 2 1.”

'

Book Reviews

What India should Know

### What India Should Know - A book review

The distortion of India’s past by western historians

V. Lakshmikantham & J. Vasundhara Devi; What India Should Know, Bharatiya Vidya Bhavan, pp 308, Rs 250.00
By Manju Gupta

The deep-rooted prejudices about the qualities, traditions and religions of the East have been a pervasive and marked characteristic of Western thought of centuries. It was a thought reinforced in the 19th century by industrialisation and imperialism, and which resulted in identification of the East with backwardness and ungovernability.

We also agree that today scholarship means being at home with what is written by Western scholars, who have more than often discredited the ancient past of Indian culture and distorted the history and chronology of events.

The book under review, written by mathematicians Dr V. Lakshmikantham and Dr J. Vasundhara Devi, begins by throwing light on the confusion till today between Gupta Chandragupta and Maurya Chandragupta. They point out that actually Gupta Chandragupta flourished in 327 BC and was the contemporary of Alexander, while Maurya Chandragupta lived in 1534 BC. “But the Western historians wrongly identified Alexander’s contemporary with Maurya Chandragupta, thus affecting more than 1,200 years in the history of ancient India. This colossal blunder upset the whole scheme and brought terrible chaos into the Puranic dates of India.” They point out that it was Sir William Jones, “the first historian of India”, who changed this date to effect a sort of similitude between the Biblical and Indian conceptions of time and they add, “twelve centuries of time after the Mahabharata war (3138 BC) and 10 centuries before that are struck off like this and the history the Indians got to know is put upon this wrong base. The Western scholars have not only bungled facts and tampered with texts, but even gone to the extent to hurling abuse at ancient Indian historians and sages.”

The authors feel that colonisation had affected the Indian mind in certain aspects. Through Macaulay’s education policies, the British ensured that they left behind an inferiority complex among the Indians by constantly denigrating Indian culture. “This is why the intellectuals of India today repeat what their masters said before and ape them after having hated them,” say the authors.

They add that another masterstroke of the British was the propagation of the “absurd” theory of Aryan invasion according to which India was invaded by a tribe called Aryans who originated in western Russia and imposed upon the Dravidians of India, the hateful caste system. They continue, “To the Aryans are attributed Sanskrit, the Vedic religion, as well as India’s greatest spiritual texts, the Vedas and a host of writings like the Upanishads. The Aryan invasion myth has shown that the Indian civilisation was not that ancient and that it was secondary to the cultures that influenced the Western world. Also, whatever good thing India had developed has been a consequence of the influence of the West.”

The book deals with the general prejudice about the East, the distortion of Indian history and the superficial translation of the Vedas by Western scholars. The authors comment ironically that the “supposedly enlightened writers” such as Edward Gibbon who never set foot east of Switzerland, in his History of the Roman Empire, loved to make play of the “despicable people of the East”, and Voltaire, who never travelled beyond Berlin, “fantasised about the misery and bigotry of the Eastern nation”. They add, “The most conspicuous example was Lord Macaulay, who carried his all-consuming racist hatred of the East to ridiculous depths by asserting that the entire corpus of knowledge that the Orient possessed could be contained in half a thimble.” They add that the world is but one and the East and West bifurcation is a mythical boundary.

The catastrophic event of the formation of a Mediterranean Sea resulted in the loss of culture and civilisation existing in Europe. The history of the Greeks, Roman and the British are traced briefly and so is the awakening of Europe from the “dark ages”.

The book ridicules the theory of Aryan invasion and gives in points the reasons for its dismissal. It says that the Aryans spread from the Bharatavarsha in different directions to spread the Aryan culture. “There was never any Aryan invasion of India or any Aryan-Dravidian war. The cradle of civilisation was not Sumeria in Mesopotamia, but the Sapta Sindhu, the land of seven rivers in north-west India.”

Then it expounds on the misrepresentation of the two Chandraguptas and tries to set right the chronology of events in India.

It points out that the Aryan invasion theory was aimed at dividing India into factions. It explains that the Aryans were extremely sensitive to the high walks of life, righteousness and nobility, both in thought and action. That is, the Aryans followed the Vedic Dharma, also called the Sanatana Dharma. Dharma is “that nature which makes a thing what it is.” Thus Manava Dharma implies that human beings “should be true to their own essential nature, which is divine; therefore, all efforts in life should be directed towards maintaining the dignity of the atma (the self) and not plodding through life like helpless animals. Thus Dharma is the ‘law of being’.”

The book exposes the deliberate distortions wrought by Orientalists in their efforts to write the history of India.

The book traces the great traditions laid down by Sanatana Dharma throughout the world that endured in Bharatakhand in the 12th century.

And the authors try to synthesise India with its glorious heritage and the present technological advances ready for taking India into the twenty-first century. The chapter ends on a positive note that this entry “will have a new awakening and the humanity will be much more spiritual than it has been.”

The book concludes by saying that the Sanatana Dharma “is much more open than any other religion to new ideas, scientific thought and social experimentation. Many principles basic to Sanatana Dharma initially appeared strange to the West, such as yoga, meditation, reincarnation and methods of interiorisation, but these principles have now found worldwide acceptance. Sanatana Dharma is, of course, a world religion…”

(Bharatiya Vidya Bhavan, Kulpati Munshi Marg, Mumbai - 400 007.)

 http://www.archaeologyonline.net/artifacts/mankind-invention.html MANKIND’S GREATEST INVENTION The story of the search for the perfect number system By N.S. Rajaram The Universal History of Numbers, 3 volumes by Georges Ifrah (2005 PB). Penguin, India. “Finally it all came to pass as though across the ages and the civilizations, the human mind had tried all the possible solutions to the problem of writing numbers, before universally adopting the one which seemed the most abstract, the most perfected and the most effective of all.” In these memorable words, the French-Moroccan scholar Georges Ifrah, the author of the monumental but somewhat flawed The Universal History of Numbers, sums up the many false starts by many civilizations until the Indians hit upon a method of doing arithmetic which surpassed and supplanted all others— one without which science, technology and everything else that we take for granted would be impossible. This was the positional or the place value number system. It is without a doubt the greatest mathematical discovery ever made, and arguably India’s greatest contribution to civilization. The three-volume Indian edition is the English version of the 1994 French edition. It tells the story of humanity’s 3000-year struggle to solve the most basic and yet the most important mathematical problem of all— counting. The first two volumes recount the tortuous history of the long search that culminated in the discovery in India of the ‘modern’ system and its westward diffusion through the Arabs. The third volume, on the evolution of modern computers, is not on the same level as the first two. Better accounts exist. The term ‘Arabic numerals’ is a misnomer; the Arabs always called them ‘Hindi’ numerals. What is remarkable is the relatively unimportant role played by the Greeks. They were poor at arithmetic and came nowhere near matching the Indians. Babylonians a thousand years before them were more creative, and the Maya of pre-Colombian America far surpassed them in both computation and astronomy. The Greek Miracle is a modern European fantasy. The discovery of the positional number system is a defining event in history, like man’s discovery of fire. It changed the terms of human existence. While the invention of writing by several civilizations was also of momentous consequence, no writing system ever attained the universality and the perfection of the positional number system. Today, in the age of computers and the information revolution, computer code has all but replaced writing and even pictures. This would be impossible without the Indian number system, which is now virtually the universal alphabet as well. What makes the positional system perfect is the synthesis of three simple yet profound ideas: zero as a numerical symbol; zero having ‘nothing’ as its value; and the zero as a position in a number string. Other civilizations, including the Babylonian and the Maya, discovered one or other feature but failed to achieve the grand synthesis that gave us the modern system. Of the world’s civilizations, the Mayas came closest. They, like the Babylonians, had an idea of the zero, but never learnt how to operate with it. In Ifrah’s words: “The measure of genius of the Indian civilization, to which we owe our modern, system, is all the greater in that it was the only one in all history to have achieved this triumph.” Modern civilization rests on the modern number system. The decimal system is just a special case of it. The synthesis was possible due to the Indians’ capacity for abstract thought: they saw numbers not as visual aids to counting, but as abstract symbols. While other number systems, like the Roman numerals for example, expressed numbers visually, Indians early broke free of this shackle and saw numbers as pure symbols with values. We see it in other fields also. The great grammarian Panini describes the Indian alphabet in purely phonetic terms, without reference to symbols. It is the same in music. While the Western notation depends on both the form and the location of notes written across staves, the Indian notation can use any seven symbols. The economy and precision of the positional system has made all others obsolete. Some systems could be marvels of ingenuity, but led to incredible complexities. The Egyptian hieroglyphic system needed 27 symbols to write a number like 7659. Another indispensable feature of the Indian system is its uniqueness. Once written, it has a single value no matter who reads it. This was not always the case with other systems. In one Maya example, the same signs can be read as either 4399 or 4879. It was even worse in the Babylonian system, where a particular number string can have a value ranging from 1538 to a fraction less than one! So a team of scribes had be on hand to cross check numbers for accuracy as well as interpretation. The Universal History of Numbers is an impressive achievement but not a definitive work. It has several drawbacks— errors of omission and commission that are perhaps unavoidable when one tries to cover a vast area spanning space, time and civilizations. The author’s discussion of palaeography sometimes goes awry due to his reliance on secondary sources, some of which go back to the nineteenth century. He accepts as proven conclusions that are contentious and even demonstrably false. (Like his acceptance of the non-existent Aramaeo-Brahmi as the source of the Brahmi alphabet.) These, however, do not seriously detract from a marvelous work. The books may be read by anyone with an interest in mathematics. In summary, Georges Ifrah has opened the gates for what promises to be a major new pathway for research. It is now for others to rise to the challenge. _________ N.S. Rajaram is a mathematician who has written on ancient history.

Chronological list of Mathematicians

Indic mathematicians through the ages

FAQ On the Mathematics of the Vedics

Mathematics Historians

Mathematics in ancient India

A sample of vedic mathematics

Ancient Indian Mathematics

Indian Mathematics

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The first mathematics which we shall describe in this article developed in the Indus valley. The earliest known urban Indian culture was first identified in 1921 at Harappa in the Punjab and then, one year later, at Mohenjo-Daro, near the Indus River in the Sindh. Both these sites are now in Pakistan but this is still covered by our term "Indian mathematics" which, in this article, refers to mathematics developed in the Indian subcontinent. The Indus civilisation (or Harappan civilisation as it is sometimes known) was based in these two cities and also in over a hundred small towns and villages. It was a civilisation which began around 2500 BC and survived until 1700 BC or later. The people were literate and used a written script containing around 500 characters which some have claimed to have deciphered but, being far from clear that this is the case, much research remains to be done before a full appreciation of the mathematical achievements of this ancient civilisation can be fully assessed. "

The above statement must be revised based on new archaeological discoveries. More than 400 sites have been found along the banks of the dried up river bed of the ancient river Saraswathi. These sites include the submerged city of Bet Dwaraka, the city ruled by Sri Krishna during the episodes of the Mahabharata and the great Bharata war that is described in detail in that epic. The important point to note is that a prerequisite to do numerical work is a script. So, there must have been a script by the time the Saraswathi Sindhu civilization was flourishing not just centered in the two cities of Mohenjo Daro and Harappa but along dozens of urban towns and cities like Dholavira, Lothal, Dwaraka and others. European historians often wonder what happened to the  denizens of the Indus Valley civilization. Ockhams razor suggests the right answer . Nothing catastrophic happened to these people and we the modern Indics are the descendants of this civilization which was spread over a huge area stretching from Haryana in the north to the present day province of Maharashtra to places like Prathishtan (later Pathan) which eventually became the capital of   the Satavahana Kingdoms are in fact a successor to the Urban civilizations that existed prior to them. This makes eminent sense because the word Brahmi signifies the goddess Saraswathi (consort of Brahma) and is therefore also considered to be the Guardian deity of Knowledge and the one who is credited with blessing us with the gift of a script. There are a group of  Brahmanas in the Konkan area of present day state of Karnataka  who call themselves Saraswath Brahmanas and legend has it that they migrated from the banks of the Saraswath river when it eventually dried out. In fact  the Gowda Saraswath Brahmanas   have done extremely well over the succeeding centuries and have prospered far in excess of their proportion in the population. In fact our family records show that about 15 generations ago my ancestor by  the name of Hanuman Bhat migrated to the Andhra country , to escape the  turmoil caused by the interminable wars and the tyranny of Aurangazeb , from the area which is considered present day Konkan

We often think of Egyptians and Babylonians as being the height of civilization and of mathematical skills around the period of the Indus civilisation, yet V G Childe in New Light on the Most Ancient East (1952) wrote:-

India confronts Egypt and Babylonia by the 3rd millennium with a thoroughly individual and independent civilisation of her own, technically the peer of the rest. And plainly it is deeply rooted in Indian soil. The Indus civilisation represents a very perfect adjustment of human life to a specific environment. And it has endured; it is already specifically Indian and forms the basis of modern Indian culture.

The Sutra Era of Vedic Mathematics

Decimal vs Binary The advent of the computer has given rise to number systems other than Decimal, for example Binary ( base2 ) and hexadecimal systems (base 16). It can be seen that as the value of the  'base' decreases the more the number of digits that are needed to represent the number. This is of no consequence for a computer but for the ancients who had to indulge in a lot of mental arithmetic the base number had to be as large as possible while remembering the properties of the   first (base -1) numbers. Obviously the choice of a decimal had something to do  also with the fact that we have 10 fingers and   10 toes.