Mathematical Tradition in

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South Asia

Acknowledgement

�The contents of the slides are from various internetsources, and NPTEL lectures.

�Instructor is not claiming that the contents belongs to

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�Instructor is not claiming that the contents belongs tohim

Mathematics3

“Yatha shikha mayuranam, naganam manayo yatha,

tadvedanga shastranam, ganitam nurdhni samsthita”

Development of Indian Mathematics

� Ancient and Early classical Period (till 500 CE)

Later Classical Period (500 -1250)

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� Later Classical Period (500 -1250)

� Medieval Period (1250 - 1850)

� Modern India

Ancient Period

� Broad classification of Knowledge – Mundaka-Upanisad

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Mathematics in the Sulbasutra texts

� One of the prime occupations of the vedic people seem to havebeen performing sacrifices, for which altars of prescribed shapesand sizes were needed.

� Recognizing that manuals would be greatly helpful in constructingsuch altars, the vedic priests have composed a class of texts calledSulba-sutras.

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Sulba-sutras.

� These texts (earliest of which is dated prior to 800 BCE), form a part of much larger corpus known as Kalpasutras that include:

The extant Sulbasutras

So far seven different Sulbasutra texts have been identified by scholars. They are:

1. Baudhayana Sulbasutra

2. Apastamba Sulbasutra

Katyayana Sulbasutra

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3. Katyayana Sulbasutra

4. Manava Sulbasutra,

5. Maitrayana Sulbasutra

6. Varaha Sulbasutra and

7. Vadhula Sulbasutra

� Of them, Bodhayana Sulbasutra is considered to be the most ancient one. (prior to 800 BCE).

Topics covered in the Baudhayana-sulbasutra

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The Sulva (Pythagorean) theorem

� A clear enunciation of the so-called ‘Pythagorean’ theorem called bhuja-koti-karna-nyaya in the later literature is described in Bodhayana Sulvasutra (1.12) as follows

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Manava version of the Sulva theorem

� The presentation of the theorem in Manava-sulvasutra differs from Bodhayana Sulvasutra both in form and in style.

� Here it is given in the form of a verse as follows:

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� Using modern notation the result may be expressed as:

Decimal Place Value System & Representation of numbers

� Generally most of us do not get to know or have opportunities to get to know answers to questions like:

� When did we start counting ?

� Were there other systems of counting?

� What are the different ways of representing numbers? etc.

As we keep using decimal system of numeration right from our childhood,

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� As we keep using decimal system of numeration right from our childhood, we are so familiar with that, that we tend to think that it has been there for ever.

� It is indeed pretty old. But how old?

� One of the most ancient literature Rg-veda presents the number 3339 using word numeration:

Ingenuity of the advent of Place value system & Zero

� Laplace while describing the contribution of Indians to mathematics observes:

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Origin of the present day numeration?

� In tracing the origin of the present day numeration, George Ifrah has done a splendid job in his book The Universal History of Numbers.

� While introducing the topic of representation of numbers by Indian astronomers, Georges Ifrah observes (pp. 107-108):

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Indian astronomers, Georges Ifrah observes (pp. 107-108):

Evolution of Numerals: Brahmi Modern14

� It has taken more than 18 centuries (3rd BCE – 15th CE) for the numerical notation to acquire the present form.

� The present form seems to have got adopted ‘permanently’ with the advent of printing press in Europe. However, there are as many as 15 different scripts used in India even today ( Nagri, Bengali, Tamil, Punjabi etc.).

Earliest explicit use of decimal place value system

� The earliest comprehensive astronomical/mathematical work that is available to us today is Āryabhaṭīya (499 CE).

� The degree of sophistication with which Āryabhaṭa has presented the number of revolutions made by the planets etc., clearly points to the fact that they had perfect knowledge of zero and the place value system.

Moreover, his algorithms for finding square-root, cube-root etc. are also

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� Moreover, his algorithms for finding square-root, cube-root etc. are also based on this.

� The system developed by Āryabhaṭa is indeed unique in the whole history of written numeration.

� Not only unique but also quite ingenious and sophisticated. Numbers of the order of 1016 can be represented by a single character

� However, it was not made use of by anybody other than Āryabhaṭa as it is too complicated to read!

Vowels employed in Devanagari script

� The chart below presents a summary of the vowels used in the Devanagarıscript:

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Consonants employed in Devanagari script

� The chart below presents a summary of the consonents used in the Devanagari script:

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Assigning numerical values to consonants

� Aryabhata’s scheme of assigning numerical values to the 33 consonants can be represented as:

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Varga

Avarga

Vocalizing the place value

� Aryabhata’s idea of making vowels decide the place value of the 33 consonants is quite novel and ingenious.

� The scheme proposed by him is as follows:

� First he classifies the consonants into two groups (varga and avarga);

� Assigns values to them, and then states that their place value is decided by the vowel that is tagged to them.

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that is tagged to them.

� Here the value of an isolated consonant (generally pronounced/vocalized with a), is taken as such.

� However if they are tagged with other vowels (i, u, r, etc.) we need to multiply them by successive powers 100.

Aryabhata’s representation: Illustrative examples

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Aryabhata and his period(1)

� Aryabhata’s (476–550 CE)works

include the Āryabhaṭīya (499 CE, when

he was 23 years old) and

the Arya-siddhanta.

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� Āryabhaṭīya is made

up of four parts namely:

1. Gitikapada (in 13 verses)

2. Ganitapada (in 33 verses)

3. Kalakriyapada (in 25 verses)

4. Golapada (in 50 verses)

Aryabhata and his period(2)

� The four parts of the text Āryabhaṭīya

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Aryabhata and his period(3)

� Aryabhata's work are known from the Aryabhatiya. Aryabhatahimself may not have given it a name.The name "Aryabhatiya"is due to later commentators.

� It is also occasionally referred to as Arya-shata-aShTa

(literally, Aryabhata's 108)

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(literally, Aryabhata's 108)

� The extreme brevity of the text was elaborated incommentaries by his disciple Bhaskara I (Bhashya, c. 600 CE)and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465CE). In this Bhasya, Nilakantha had discussed infiniteseries expansions of trigonometric functions and problems ofalgebra and spherical geometry.

Approximate value of

� The Sulba-sutra-s, give the value of close to 3.088.

� Aryabhata (499 AD) gives an approximation which is correct to four decimal places.

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π

π

� [The word] asanna means close to or nearby.

kuttaka Problem

� Suppose there is an integer N which when divided by two integers a, b leaves remainders r1, r2 i.e.

N = ax + r1 = by + r2

� This equation may be written as

by – c = ax, where c = r1 – r2

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by – c = ax, where c = r1 – r2

� kuttaka problem: Given integers a, b, c we need to find integers x, y that will satisfy the above equation

� Usually the equation given above is called Diophantine equation.

� Aryabhata presents the solution in two verses Ganitapada, (32–33)

� Aryabhata's method of solving such problems, elaborated by Bhaskara in 621 CE, is called the kuṭṭaka method.

More on Āryabhaṭīya

� Area of a triangle, Circle, trapezium etc.

� Construction of sine-table

� Results for the summation of series of squares and cubes

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cubes

� Dealing with arithmetic progressions.

� Problems related to gnomonic shadow

� Finding the height of a lamp-post.

Brahmagupta

� Born in CE 598. Composed Brahmasphutasiddhanta

(24 chapters and a total 1008 verses) in CE 628. Commentary by Prthudakasvamim in CE 860

� Brahmagupta described as Ganakacakracudamani

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Brahmagupta described as Ganakacakracudamani(Jewel among the circle of Mathematicians) by Bhaskara - II.

� Brahmagupta holds a remarkable place in the history of Eastern Civilization. It was from his works that the Arabs learnt astronomy before they became acquainted with Ptolemy.

Mathematics of positive, negative and zero

� Brahmasphutasiddhanta (c. 628 CE) is the first available text that discusses the mathematics of zero (sunya-parikarma) along with operations with positives and negatives.

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Operations with karani or surds29

Bhaskara II

� Perhaps the most well known name among the ancient Indian astronomer-mathematicians. Designated as Bhaskara-II to differentiate him from his earlier namesake, who lived in the seventh century CE (Bhaskara-I).

� His main work was Siddhanta Shiromani and divided into four parts Lilavati, Bijaganita, Grahaganita and Goladhyaya.

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Lilavati, Bijaganita, Grahaganita and Goladhyaya.

� Bhaskara's work on calculus predates Newton and Leibniz by over half a millennium.

� He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhaskara was a pioneer in some of the principles of differential calculus.

Lilavati(I)

� Lilavati is Indian mathematician Bhaskara II's treatise on

mathematics, written in 1150.

� It is the first volume of his main work Siddhanta Shiromani.

� The book contains thirteen chapters which mainly includes:

� Arithmetical terms

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� Arithmetical terms

� Interest computation

� Arithmetical and geometrical progressions

� Solid geometry

� Plane geometry

� The shadow of the gnomon

� The kuttaka - a method to solve indeterminate equations and

� Combinations.

March 20, 2015SAU

Lilavati(II)

� Lilavati was daughter of Bhaskara.

� Bhaskara studied Lilavati's horoscope and predicted that she would remain both childless and unmarried.

� Bhaskara calculated a auspicious moment for his daughter's wedding and made a device to serve the purpose, warned

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wedding and made a device to serve the purpose, warned Lilavati not to go near the device.

� Lilavati, could not hold her curiosity and went to see what her father had done.

� Wedding took place, but not at the auspicious time and her died a few days after the marriage.

� That time he promised to his daughter to write a book in her name.

March 20, 2015SAU

Medieval Period33

Srinivasa Ramanujan (1887-1920)

� Born on December 22, 1887 at Erode, Madras Presidency (now Pallipalayam, Erode, Tamil Nadu).

� While at school, he got a copy of S.L. Loney’s Plane Trigonometry which he soon mastered, but also was

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Trigonometry which he soon mastered, but also was surprised to see there some of the results that he had obtained himself.

� Around 1903, Ramanujan went through G. S. Carr’s Synopsis of Pure and Applied Mathematics (1880), a compendium of about 5000 results, which is said to have influenced him considerably.

Srinivasa Ramanujan (II)

� Ramanujan met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical Society.Ramanujan, wishing for a job at the revenue department where Ramaswamy Aiyer worked, showed him his mathematics notebooks. As Ramaswamy Aiyer later

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his mathematics notebooks. As Ramaswamy Aiyer later recalled:

“I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.”

Hardy-Ramanujan number 1729

� The number 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see Ramanujan. In Hardy's words

“I remember once going to see him when he was ill at Putney. I

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“I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number;

it is the smallest number expressible as the sum of two cubes in two different ways.”

1729 = 13 + 123 = 93 + 103.

Ramanujan’s Work

� During his short life, Ramanujan independently compiled nearly 3900 results (mostly identities and equations).

� He stated results that were both original and highly unconventional, such as the Ramanujan prime (Ramanujan primes are the least integers Rn for which there are at least n primes between x and x/2 for all x ≥ R . E.g. 2, 11, 17, 29, and 41.) and the Ramanujan

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x/2 for all x ≥ Rn. E.g. 2, 11, 17, 29, and 41.) and the Ramanujantheta function.

� He discovered mock theta functions in the last year of his life. For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Mass forms.

for |ab| < 1.

Ramanujan’s Work on Partitions

� The number of partitions p(n) is the number of distinct ways of representing n as a sum of positive integers, without taking the order into account. p(0) is taken to be1.

� Partitions of 4 are: 1 + 1 + 1 + 1; 1 + 1 + 2; 1 + 3; 2 + 2; 4: Hence, p(4) = 5

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Hence, p(4) = 5

� Ramanujan discovered and proved the congruences:

p(5m + 4) = 0 (mod 5),

p(7m + 5) = 0 (mod 7),

p(11m + 6) = 0 (mod 11)

� Ramanujan also conjectured that 5, 7 and 11 are the only primes for which such congruences hold; Ahlgren and Boylan proved this in 2003.

Ramanujan’s magic square

22 12 18 87

88 17 9 25

This square looks like

any other normal magic

square. But this is

formed by great

mathematician –

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10 24 89 16

19 86 23 11

mathematician –

Srinivasa Ramanujan.

What is so great in it?

Ramanujan’s magic square

22 12 18 87

88 17 9 25

Sum of numbers of

any row is 139.

What is so great in it.?

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10 24 89 16

19 86 23 11

Ramanujan’s magic square

22 12 18 87

88 17 9 25

Sum of numbers of

any column is also 139.

Oh, this will be there in any magic square.

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10 24 89 16

19 86 23 11

any magic square.

What is so great in it..?

Ramanujan’s magic square

22 12 18 87

88 17 9 25

Sum of numbers of

any diagonal is also

139.

Oh, this also will be there

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10 24 89 16

19 86 23 11

Oh, this also will be there in any magic square.

What is so great in it…?

Ramanujan’s magic square

22 12 18 87

88 17 9 25

Sum of corner

numbers is also 139.

Interesting?

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10 24 89 16

19 86 23 11

Ramanujan’s magic square

22 12 18 87

88 17 9 25

Look at these

possibilities. Sum of

identical coloured

boxes is also 139.

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10 24 89 16

19 86 23 11

Interesting..?

Ramanujan’s magic square

22 12 18 87

88 17 9 25

Look at these

possibilities. Sum of

identical coloured

boxes is also 139.

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10 24 89 16

19 86 23 11

Interesting..?

Ramanujan’s magic square

22 12 18 87

88 17 9 25

Look at these central

squares.

Interesting…?

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10 24 89 16

19 86 23 11

Ramanujan’s magic square

22 12 18 87

88 17 9 25

Can you try these

combinations?

Interesting…..?

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10 24 89 16

19 86 23 11

Ramanujan’s magic square

22 12 18 87

88 17 9 25

Try these combinations

also?

Interesting.…..?

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10 24 89 16

19 86 23 11

Ramanujan’s magic square

22 12 18 87

88 17 9 25

NOWNOWNOWNOW

LETS FACE LETS FACE LETS FACE LETS FACE

THE THE THE THE

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10 24 89 16

19 86 23 11

THE THE THE THE

CLIMAXCLIMAXCLIMAXCLIMAX

Ramanujan’s magic square

22 12 18 87

88 17 9 25

Recall !!

birth of Srinivasa

Ramanujan?

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10 24 89 16

19 86 23 11

Ramanujan’s magic square

22 12 18 87

88 17 9 25

It is 22nd Dec 1887.

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10 24 89 16

19 86 23 11

Ramanujan’s magic square

22 12 18 87

88 17 9 25

It is 22nd Dec 1887.

Yes. It is 22.12.1887

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10 24 89 16

19 86 23 11

Qaiser Mushtaq

� Born 28 February 1954, , is prominent Pakistani mathematician

� Has made numerous contributions in the field of Group theory and Semigroup.

� His research contributions in the fields of group theory and LA-semigroup Theory.

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semigroup Theory.

� Mushtaq has over a hundred research papers to his credit.

� Honours and awards

� Young Scientist of the South Award (1993) from Third World Academy of Sciences, Italy

� Gold Medal of Honour (1987) from United States

� Salam Prize in Mathematics (1987) and so many prizes.

Qaiser Mushtaq

� Born 28 February 1954, , is prominent Pakistani mathematician

� Has made numerous contributions in the field of Group theory and Semigroup.

� His research contributions in the fields of group theory and LA-semigroup Theory.

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semigroup Theory.

� Mushtaq has over a hundred research papers to his credit.

� Honours and awards

� Young Scientist of the South Award (1993) from Third World Academy of Sciences, Italy

� Gold Medal of Honour (1987) from United States

� Salam Prize in Mathematics (1987) and so many prizes.

Asghar Qadir

� Born July 1946 is a renowned Pakistani mathematician .

� Considered as one of the top mathematicians in Pakistan.

� Qadir is author of the book "Relativity: An Introduction to the

Special Theory" which has been translated in several different languages and is widely read by science students in colleges

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languages and is widely read by science students in colleges throughout Asia.

� Qadir has made important and significant contributions to the fields of differential equations, theoretical cosmology and the mathematical physics.

� He has published more than 140 research papers. He is the author of 12 books, 22 research level articles, 7 teaching journal papers, 32 popular articles, and 48 research preprints.

Mathematics awards

� Abel Prize

� Fields Medal

� Leelavati Award

� ICTP Ramanujan Prize

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� ICTP Ramanujan Prize

� SASTRA Ramanujan Prize

� Infosys Prize

� Srinivasa Ramanujan Medal

� And many more…

Prize for young mathematicians

� A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the International Centre for Theoretical Physics (ICTP), which nominate members of the prize committee.

� The Shanmugha Arts, Science, Technology & Research

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� The Shanmugha Arts, Science, Technology & Research Academy (SASTRA), based in the state of Tamil Nadu in South India, has instituted the SASTRA Ramanujan Prize of $10,000 to be given annuallyto a mathematician not exceeding the age of 32 for outstanding contributions in an area of mathematics influenced by Ramanujan.

Abel Prize

� an international prize presented by the King of Norway to one or more outstanding mathematicians.

� Named after Norwegian mathematician Niels HenrikAbel (1802–1829)

� Established in 2001 by the Government of Norway.

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� Established in 2001 by the Government of Norway.

� The Norwegian Government gave the prize an initial funding of NOK 200 million (about US$23 million) in 2001.

� S. R. Srinivasa Varadhan "for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviation” in 2007.

Fields Medal

� The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years.

� The Fields Medal and the Abel Prize have often been

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The Fields Medal and the Abel Prize have often been described as the "mathematician's Nobel Prize" (but different at least for the age restriction).

� The prize comes with a monetary award, which since 2006 has been C$15,000 (in Canadian dollars).

� Manjul Bhargava of Indian origin “for developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves” in 2014

Further Study

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http://nptel.ac.in/courses/111101080/

Thanks

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