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(From Georges Ifrah's The Universal History of Numbers)


 

FAQ in Ancient Indian Mathematics

If you have questions  on Vedic Mathematics or related topics, don't be bashful, send them to  Aryabhatta@indicethos.org.

No question will be deemed too simple or obvious.

 

Contents

 

 

 

 

 

 

 

 

 

 

Answers

  1. What were the strengths of the ancient Indian Mathematicians

The Indian Mathematicians of  the ancient era were primarily number theorists. Their interest in this field arose from a need to do astronomical calculations, which in turn were needed to advise the farmers on the proper timing to plant crops and for performing rituals . In particular they excelled in Diophantine Equations (probably erroneously credited to Diophantus , because the Vedics were adapt at  it long before Diophantus), algebraic equations in which only solutions in integers are permitted. Examples of  Diophantine equations that the ancients dealt with are

 

  • ax + by = 1:  this is a linear Diophantine.

  • xn + yn = zn: For n = 2 there are infinitely many solutions (x,y,z), the Pythagorean triples. For larger values of n, Fermat's last theorem states that no positive integer solutions x, y, z satisfying the above equation exist.

  • x2 - n y2 = 1: (Pell's equation) which is named, mistakenly, after the English mathematician John Pell. It was studied by Brahmagupta and much later by Fermat.

Another field or approach which the ancients favored was algebraic geometry, where geometry was studied predominantly using algebraic equations. Again the need was for designing ritual altars and their proper orientation with respect to the heavenly bodies and for following the rules of Vastu sastra (architecture) .

The use of concise mathematical symbolism (<,>,=,*,(), inf., sup., the integral sign, the derivative sign, the sigma sign etc is a relatively recent revolution in mathematics, When calculus was invented by Newton and Leibniz independently, they used different notations. Newton used dots on top of the quantity and called them fluxions, while Leibniz used the 'd' notation. Unfortunately when all the major advancements in science and mathematics were happening in Europe in the 17th and 18th century, India was stagnant, caught in interminable wars and conquests.

Ironically it was the Indian place value notation that triggered the advance in Europe. Prior to this it was common to express all problems in wordy sentences and verse.

As to the value of PI (cannot be expressed as a fraction) it is impossible to calculate it without the use of some variant of the 'method of exhaustion'. In this particular instance it was a matter of using increasingly larger number of isosceles triangles( forming an n sided polygon) within the circle. This is the germ of the method of series expansions. The ancient Indians were well aware of this technique and used it for several trigonometric calculations. The felicity with which Indians did series expansions extended to Srinivasa Ramanujam in recent times. He was the incomparable master and there may be none like him on the face of this earth again.

Again there is no claim that the Indians did everything. For example there is no evidence that the Indians were familiar with the representation of complex numbers and complex variables or advanced topics such as the calculus of variations.

The ancient vedic Indics were interested in very practical aspects of mathematics, namely the positions of the stars, developing a Panchanga (calendar), ordinary mathematics for everyday use, measurements, such as that of land and weights etc.. There is no evidence that they were familiar with the science of mechanics for instance, which developed in Europe in the 18th century after Newton.
 

Of course, the most important single concept that they developed was the decimal place value system. While other civilizations like the Maya and the Oaxaca of ancient Mexico developed place value systems independently, there is no evidence that they mastered the technique of using numbers with such facility  as the Vedics did. The Vedics also developed facility with areas such as Trigonometry and there is mention that they manipulated arrays of numbers in the same manner as we use matrices today.

Even with all these skills, it must be recognized that this forms only a small part of what constitutes the corpus of modern mathematics today. It must also be recognized that the Greeks and the Babylonians also had developed mathematical prowess, if one accepts that the axiomatic approach to geometry as primarily a Greek development.

Despite all these caveats, it must be admitted that for the era in which  they lived the Vedic contributions to the sum of human knowledge especially in mathematics was considerable and should therefore be a matter of great inspiration for those of us who consider themselves part of the Indic civilization. Further there is much study yet to be done and many more manuscripts which remain uninvestigated so the quest for what was the state of knowledge of the ancients in fields such as mathematics has just begun

  1. Why do you use the term Vedic mathematics

Strictly speaking the later contributors like Varahamihira did not live in the Vedic era, but the methods used show a continuity in development till the Modern age and hence the use of the term Vedic , signifying techniques which came into use in the distant past seems justified

  1. Why even bother with Vedic mathematics when we have progressed much farther in the intervening millennia ?

    I will quote the answer I gave in a discussion forum (Bharat-Rakshak)a few years ago. "

    First, do we understand the corpus of VM and Vedic Science (i certainly don't know enough about VM to make an authoritative statement). What part of what we know to be contributions, were purely Vedic and what part were contributions by mathematicians like Aryabhatta and Bhaskaracharya who came much later ?

    Second what part of it is already accepted in modern mathematics ( and I am not talking about arithmetical tricks to do multiplication and division). I remember in my Hall and Stevens text on geometry which i used in India, the proof of Pythagoras theorem took up a whole page. I would have loved to have the 4 alternate proofs offered by the VMB. I am interested for example in the development of Astronomy. Ptolemy used a concept called ecliptics (if i recall the use of epicycles - pl. don't harangue me on this, as this is of the top of my head) to get around the fact that the earth went around the sun and not the other way around. Did he borrow that from Vedic or was that a different stream of thought (Ptolemy of course predates al-Khwarismi and his 'Sindhind zij'). Given that Ptolemy post dates Vedic why did al-Khwarismi choose to rely more on Vedic ( a more ancient technology) when he presumably had access to both ?
     

    Third the point made by James is valid. The point is not to judge Vedic Science by the standards of 21st century and then trash it saying it is outmoded. Of course it is outmoded, in certain respects, you should expect that after 6 millennia. If it was not, that is tantamount to admitting that we have made no progress in the intervening millennia. The point is, does it give an alternate model at looking at nature (e.g. AyurVeda) that is equally valid, useful and perhaps more elegant.

    Fourth, these efforts at understanding our past, should not be restricted to Vedic systems but to other systems in India's past such as Yunani (etymology Ionian = Greek) which is attributed to Islamic savants. I am interested for example in tracing through the development of algebra (coined by al Khwarismi as al-gibr wal Maquaballah) from its ancient Vedic origins because of the efforts of Islamic savants in the middle ages, at a time when Europe was in the dark ages and was struggling with Roman numerals.

    My own view is that Vedic science and math is a forgotten science and for the most part does not contradict what has been discovered subsequently. First let us understand what it says before getting an anxiety neurosis that it is going to replace Western science in Indian schools.


     

  2. Where are these results to be found in the ancient texts

The earliest works on Mathematics by the Vedic savants are recorded in the Sulvasutras, the sacred books of altar construction in the Vedas, in particular the Apastambha Sulvasutra, the Baudhayana Sulvasutra and the Katyayana Sulvasutra (named after the mathematicians who developed the mathematics and wrote the sutras. The Sulvasutras are appended to a particular Veda (see FAQ on Hinduism for the typical contents of a Veda) can be translated as rope rules or "manuals of measurement", the modern term for which would be metrology. But in reality the scope of the investigations in the Sulvasutras is far broader and comprises among other fields, number theory, trigonometric, algebra, algebraic geometry, series expansions,  the concept of a rational number, etc. etc.. The dating of these Sulvasutras, while occurring after  the main corpus of the Veda was compiled, is of great antiquity, greater than that of Babylonian mathematics. It is only now that we are beginning to understand the extent of the antiquity of these ancient mathematicians. See for instance The origin of Mathematics by Lakshmikantham and Leela. in the list of references later in this page.

A word needs to be said about the use of Sutras (Aphorisms) as a means of communication and recording of results. The dictionary says that a sutra is Any of various aphoristic doctrinal summaries produced for memorization generally  during the millennia before the common era and later incorporated into Hindu literature. We must recall that writing materials during that period of history, were not plentiful and had to be laboriously produced probably by the author of the sutras himself . Whatever needed to be communicated had to be in as brief a form as possible. Hence the need for economy in language and the use of sutras. In fact there is evidence from which one can infer that the Vedics were the first to use symbols and mathematical equations and hence their prowess with Algebra .

Brevity has its downside however and the charge has been made by Europeans that the Vedics rarely provided proof. The reason was that the proof was generally terse and incomprehensible to the majority of the people. The notion that those who mastered these topics used brevity as a means to restricting this knowledge, overlooks the fact few in the populace would have the capability to pursue this  rigorously without the intellectual discipline that comes from years of study. Then as now Mathematics had the reputation of being a difficult subject to master.

Later texts were known as Siddhantas

  1. Who were the main contributors to Vedic Mathematics

The main contributors during the Vedic period were, as we mentioned earlier Baudhayana, Apastambha, and Katyayana. The following list of Mathematicians who were born in the geographical region corresponding to present day India was compiled by researchers at the University of  St. Andrews, Scotland, who have done an outstanding job in compiling a lot of information (this list includes some modern day  mathematicians born in the Indian sub continent also).

 

List of  Ancient and Modern  Mathematicians born or lived in the Indian Subcontinent

Apastambha
Aryabhatta I
Aryabhatta II
Baudhayana

Bhadrabahu
Bhaskara I
Bhaskara II
Bose
Brahmadeva
Brahmagupta
De Morgan
Govindasvami

Harish-Chandra
Hemchandra
Jagannatha
Jyesthadeva
Kamalakara
Katyayana

Lagadha
Lalla
Madhava
Mahavira
Mahendra Suri
Manava

 

Narayana
Nilakantha Somayaji
Panini
Paramesvara
Patodi

Pingala
Pillai
Prthudakasvami
Rajagopal
Ramanujam
Ramanujan
Sankara

Sommerville
Sridhara
Sripati
Varahamihira
Vijayanandi

Virasena  Acharya
Henry Whitehead

Yajnavalkya
Yativrsabha

Yatavrisham Acharya
Yavanesvara

                                                         

 

 

 

We will discuss the contributions of some of the main mathematicians of the ancient era.

  1. Why are the contributions of the Vedic savants not as widely known as those of Greek, Arab mathematicians

In the early years of colonial rule by the British (the attitude persists among Western Indologists even today, although less so among mathematicians in the West)  there was great reluctance to believe the sacred texts, after they were first recognized  by Sir William Jones in the 1770's. A typical reaction was that of  W. W Rouse Ball in History of Mathematics . I posted the following in the Bharat Rakshak forum in2000

"Typical of the racism exhibited by the Brits and other Europeans is W.W. Rouse Ball in 'A short account of the History of mathematics' Dover Publications,1960, (originally appeared in 1908), page 146 'The Arabs had considerable commerce with India, and a knowledge of one or both of the two great Hindoo works on algebra had been obtained in the Caliphate of Al-Mansur (754-775 AD)though it was not until fifty or seventy years later that they attracted much attention. The algebra and arithmetic of the Arabs were largely founded on these treatises, and I therefore devote this section to the consideration of Hindoo mathematics. The Hindoos like the Chinese have pretended that they are the most ancient people on the face of the earth, and that to them all sciences owe their creation. But it is probable that these pretensions have no foundation; and in fact no science or useful art (except a rather fantastic architecture and sculpture) can be definitely traced back to the inhabitants of the Indian peninsula prior to the Aryan invasion. This seems to have taken place at some time in the fifth century or in the sixth century when a tribe of Aryans entered India by the north west part of their country. Their descendants, wherever they have kept their blood pure, may still be recognized by their superiority over the races they originally conquered; but as is the case with the modern Europeans, they found the climate trying and gradually degenerated. Note the blatant racism in the second paragraph and the venom that this author exhibits. [This message has been edited by Kaushal (edited 15-06-2000).] "

There are a lot of facile and unsubstantiated assumptions that are made here, later to be  proved false,  but we will deal with them on the chapter  where we discuss AIT. Thus there was great reluctance to admit that the dark skinned natives of the Indian subcontinent could be capable of intellectual effort. Even after the advent of the legendary Srinivasa Ramanujam, the great number theorist  in the early years of the 20th century,  from what is now Chennai, Tamil Nadu , such attitudes among British and European scholars were hard to dispel. We will have a lot more to say about Srinivasa Ramanujam later in these pages.

With the coming of the internet, and the great proficiency of the Indics in matters related to Information Technology, this state of affairs has begun to change. Both the Indics and Western savants have begun to realize the profound importance of these early developments in mathematics to the advancement of human civilization. See for instance a recent column on Place Value systems.

  1. Did the Vedics use symbols when they expressed their algebraic equations  ?

The use of concise mathematical symbolism (<,>,=,*,(), inf., sup., the integral sign, the derivative sign, the sigma sign etc is a relatively recent revolution in mathematics, When calculus was invented by Newton and Leibniz independently, they used different notations. Newton used dots on top of the quantity and called them fluxions, while Leibniz used the 'd' notation. Unfortunately when all the major advancements in science and mathematics were happening in Europe in the 17th and 18th century, India was stagnant, caught in interminable wars and conquests and was finally subjugated by the British.

Ironically it was the Indian place value notation that triggered the advance in Europe. Prior to this it was common to express all problems in wordy sentences and verse.

As to the value of PI (cannot be expressed as a fraction) it is impossible to calculate it without the use of some variant of the 'method of exhaustion'. In this particular instance it was a matter of using increasingly larger number of isosceles triangles( forming an n sided polygon) within the circle. This is the germ of the method of series expansions. The ancient Indians were well aware of this technique and used it for several trigonometric calculations. The felicity with which Indians did series expansions extended to Ramanujam. He was the incomparable master and there may be none like him on the face of this earth again.

Again there is no claim that the Indians did everything. For example there is no evidence that the Indians were familiar with the representation of complex numbers and complex variables or advanced topics such as the calculus of variations (mathematics enthusiasts will recall that Queen Dido of Carthage, who had the correct intuition for the solution of the Isoperimetric problem, was also the first recorded instance of a woman mathematician).

The ancient vedic Indics were interested in very practical aspects of mathematics, namely the positions of the stars, developing a Panchanga (calendar), ordinary mathematics for everyday use, measurements, such as that of land and weights etc.. There is no evidence that they were familiar with the science of mechanics for instance, which developed in Europe in the 18th century after Newton. However, their use of distant stars as invariants and their familiarity with the precession of the equinoxes , hints of an understanding if not the inklings of an inertial frame of reference and this is an area in which I need to educate myself further

  1. Can we see examples of what you mean by Vedic Mathematics

 

First we will make a distinction between the corpus of Vedic Mathematics (VM) from the 16 extant sutras that are still available to us courtesy of the book on Vedic mathematics (VMB) by Jagadguru BhaarathiKrishnatirtha, the late Sankaracharaya of Puri. here we will restrict ourselves to VMB, which is largely a text at High school level, although i will wager very few school children even in India are familiar with these techniques. So what does  VMB comprise of ?

 

Here is a good description

if this site does not  work try this

An example of the proof of Pythagoras theorem (originally given by Baudhayana)

(from Pacific Institute of Mathematical Sciences, by Jeganathan Sriskandarajah

  1. It is now commonly acknowledged that the Vedics invented the place value system of numerals we use today including the symbol for zero. Do we know the name of the person who invented the zero

Civilization centers where the Zero was used in the ancient world

The place value system has been used by people in the ancient world in disparate locations such as Babylonia , India, China  and by the Maya in what is now Central America. The popular conventional view in the west goes thus

 

India: 458 A.D. (debated)

The final independent invention of the zero was in India. However, the time and the independence of this invention has been debated. Some say that Babylonian astronomy, with its zero, was passed on to Hindu astronomers but there is no absolute proof of this, so most scholars give the Hindus credit for coming up with zero on their own.

The reason the date of the Hindu zero is in question is because of how it came to be.

Most existing ancient Indian mathematical texts are really copies that are at most a few hundred years old. And these copies are copies of copies of copies passed through the ages. But the transcriptions are error free…can you imagine copying a math book without making any errors? Were the Hindus very good proofreaders? They had a trick.

Math problems were written in verse and could be easily memorized, chanted, or sung. Each word in the verse corresponded to a number. For example,

viya dambar akasasa sunya yama rama veda
sky (0) atmosphere (0) space (0) void (0) primordial couple (2) Rama (3) Veda (4)
0 0 0 0 2 3 4

Indian place notation moved from left to right with ones place coming first. So the phrase above translates to 4,230,000.

Using a vocabulary of symbolic words to note zero is known from the 458 AD cosmology text Lokavibhaga. But as a more traditional numeral—a dot or an open circle—there is no record until 628, though it is recorded as if well-understood at that time so it’s likely zero as a symbol was used before 628.

Which it probably was, considering that 30 years previously, an inscription of a date using a zero symbol in the Hindu manner was made in Cambodia.

A striking note about the Hindu zero is that, unlike the Babylonian and Mayan zero, the Hindu zero symbol came to be understood as meaning “nothing.” This is probably because of the use of number words that preceded the symbolic zero.

 

Part of the reason that unquestioned precedence is not accorded the Vedics is the confusion with the dating  of Baudhayana ,Aryabhatta and Panini, who in my opinion are the leading contenders for the privilege of having  invented the zero. It is only recently that it has been universally accepted that the Vedics were the first to use the place value system extensively.

"There is wide ranging debate as to when the decimal place value system was developed, but there is significant evidence that an early system was in use by the inhabitants of the Indus valley by 3000 BC. Excavations at both Harappa and Mohenjo Daro have supported this theory. At this time however a 'complete' place value system had not yet been developed and along with symbols for the numbers one through nine, there were also symbols for 10, 20, 100 and so on.

The formation of the numeral forms as we know them now has taken several thousand years, and for quite some time in India there were several different forms. These included Kharosthi and Brahmi numerals, the latter were refined into the Gwalior numerals, which are notably similar to those in use today (see Figure 7.1). Study of the Brahmi numerals has also lent weight to claims that decimal numeration was in use by the Indus civilisation as correlations have been noted between the Indus and Brahmi scripts.

It is uncertain how much longer it took for zero to be invented but there is little doubt that such a symbol was in existence by 500 BC, if not in widespread use. Evidence can be found in the work of the famous Indian grammarian Panini (5th or 6th century BCE ; ed. note - we believe Panini must predate any text written in Classical Sanskrit and at a minimum must have lived prior to 1700 BCE) and later the work of Pingala a scholar who wrote a work, Chhandas-Sutra (c. 200 BCE). The first documented evidence of the use of zero for mathematical purposes is not until around 2nd century AD (in the Bakhshali manuscript). The first recorded 'non-mathematical' use of zero dates even later, around 680 AD, the number 605 was found on a Khmer inscription in Cambodia. Despite this it seems certain that a symbol was in use prior to that time. B Datta and A Singh discuss the likelihood that the decimal place value system, including zero had been 'perfected' by 100 BC or earlier. Although there is no concrete evidence to support their claims, they are established on the very solid basis that new number systems take 800 to 1000 years to become 'commonly' used, which the Indian system had done by the 9th century AD.


My investigations to date lead me to conclude that  the honor of inventing  the zero and the place value system belongs to one of three individuals - Apastambha. Baudhayana, and/or Panini. Of course Aryabhatta was and is a perennial favorite for being the main suspect However, we are closing in on the target and should have the answer soon in short order. We will examine the evidence related to each and reach a conclusion. See also the reasons adduced by Ifrah  cited by me in the section on  Linguistics
 

In the Proposed skeleton of  Indian Chronology Baudhayana is dated at  3200BCE, which would make him contemporaneous or earlier than the Babylonians.

  1. What do we know about Panini

References

Books:
 

  1. G Cardona, Panini : a survey of research (Paris, 1976).
  2. G G Joseph, The crest of the peacock (London, 1991).

Articles:
 

  1. P Z Ingerman, 'Panini-Backus form' suggested, Communications of the ACM 10 (3)(1967), 137.

 

  1. What do we know about the life and mathematical contributions of Bhaskaracharya (or Bhaskara I)

  2. Life and contributions of Aryabhatta the Elder
    (under construction)

  3. What did the Brahmi script look like

An example of Brahmi script - Ashoka's first rock inscription at Girnar.

An example of Brahmi script - Ashoka's first rock inscription at Girnar.

Indian-standard silver drachm of the Greco-Bactrian king Agathocles (190 BC-180 BC), the obverse showing the Greek legend BASILEOS AGATOKLEOUS "King Agathocles", the reverse RAJANE AGATHUKLAYASA "King Agathocles" in Brahmi.

Indian-standard silver drachm of the Greco-Bactrian king Agathocles (190 BC-180 BC), the obverse showing the Greek legend BASILEOS AGATOKLEOUS "King Agathocles", the reverse RAJANE AGATHUKLAYASA "King Agathocles" in Brahmi.

To compare this with Babylonian see for instance  the following Babylonian tablets

  1. If the current chronology is wrong , what do  you suggest that the correct chronology be

Proposed Hypothesis for a consistent chronology

Event

Date

Birth of Dhirgatmas

Birth of Yajnavalkya

Shatapatha Brahmana

Brihadaranyaka Upanishad

Birth of Panini

 

4000 BCE

?

?

3500 BCE ?

Birth of Pingala

Birth of Veda Vyasa

Birth of Baudhayana

Birth   of Apastambha

Beginning of  Kali Yuga

Birth of Aryabhatta

Aryabhatteeyum written by Aryabhatta  

Gautama Buddha

Coronation of Chandragupta Maurya

Coronation of Asoka Maurya (may not be the same

Asoka who built the edicts)

Kanishka’s time

Panchatantra composed by Vishnusharman

Birth of Adi Sankara

Coronation of Chandragupta of  Imperial Gupta Dynasty

Varahamihira wrote Pancha Siddhanta

Vikramaditya  

Bhartrihari

Vikram era calendar

Birth of Brahmagupta.

Salivahana Era Calendar Saka

Bhaskara II wrote Siddhanta Siromani

?

3374 BCE

3200 BCE

 

3102 BCE

2765 BCE

2742BCE

1888-1807 BCE

1554 BCE

1472 BCE

 

1294 – 1234 BCE

? 1470 -510 BCE

509 BCE

 327 BCE

123 BCE

102 BCE to 78 CE

100 BCE

30 BCE

78 CE

486 CE

 

 

Partial Source : The Origin of Mathematics, Lakshmikantham and Leela, Page30, University Press of America Inc.,2000

 

  1. References on Vedic mathematics

  2. Links on Vedic mathematics

Excellent Link on Vedic Mathematics





 

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