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 = (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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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]

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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

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:

sin x = cos x x

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 visit Dept of Mathematics, Brown Univercity]

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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. = tan - (tan³ )/3 + (tan⁵)/5 - ... ,
2. r= {r(rsin)/1(rcos)}-{r(rsin)³/3(rcos)³}+{r(rsin)⁵/5(rcos)⁵}- ...
3. sin = - ³/3! + ⁵/5! - .. Madhava-Newton power series.
4. cos = 1 - ²/2! + ⁴/4! - ..., Madhava-Newton power series.
5. p/4 1 - 1/3 + 1/5 - ... 1/n (-f_i(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 2d + 4d/(2² - 1) - 4d/(4² - 1) + ... 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) 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. 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.