Mathematics through ages

In the pursuit of Mathematics India has a long tradition. Indian History shows that there was never a point of time when Mathematics was not being developed in India. Even a cursory study on the subject leaves one with a feeling of awe and appreciation at the depth and vastness of ancient Indian thought. A lot of effort has been put in locating manuscripts as there were many periods in Indian history which were regarded as dormant periods in Indian Mathematics but with the location and study of these manuscripts, it was found that they were actually very active periods. The term ‘Indian’ generally refers to the Indian Subcontinent which implies the diaspora and all those people wherever they may be geographically located but whose roots can be traced to the Indian subcontinent.

Mathematics in Pre Historic Times

Maths existed even during the Indus Valley Civilisation around 3000BCE. Its two famous cities of Harappa and Mohenjo-daro give evidence of this fact as it can be seen that for the purpose of construction of the buildings, a standardised measurement which was decimal in nature was followed. After the invention of the kiln, this civilisation had an advanced brick making technology as the bricks were needed in the construction of buildings and to construct embankments to control floods. They designed a ruler which was named the Mohenjo-daro ruler. It had a unit of length approximately 3.4 cms or 1.32 inches and it was divided into ten equal parts. They considered that for the stability of the brick structure it had to have dimensions of proportions 4:2:1. After designing the ruler, the dimensions of the bricks were integral multiples of this unit of length. It is said that several such scales were found during excavations and an appreciable fact is the accuracy with which they were marked. Examining the excavated ruins and measuring them depict the accuracy with which these units of length were used in construction in this civilisation. Then there was a decline in the Indus Valley Civilisation probably due to climatic disasters like floods, drought and epidemics or change in the climatic patterns or invasion of the Indo Aryan people from the North.

Vedic Times

Vedic theology and philosophy gave a stimulus for the development of certain aspects of Mathematics. The religious texts of the Vedic times like Yajurveda Samhitha shows evidence of the use of large numbers as high as 10 to the power 12. The Shatapata Brahmana which is a part of Shukla Yajur Veda describes the geometrical construction of altars which were needed for performing Yajnas. Thus they put the method of making bricks of the Indus Valley civilisation to a new use.

In the Shulba Sutras, the method used to construct the sacrificial fire altars with the help of Maths is written. The altars had to be constructed occupying the same area but with different shapes. Each shape depended on the reason for performing the Yajna, for eg one desiring heaven after death would have a fire altar constructed in the shape of a falcon, one desiring attainment of Brahman would have an altar in the shape of a tortoise etc. The altar construction depicted the earliest expression of the Pythagorean theorem. The four Shulba Sutras named after their authors were Baudhayana, Manava, Apastamba and Katyayana. These Sutras indicated the method to approximate the square root of a number by using rational numbers through a procedure which is now known as series expansion. They introduced the concept of irrational numbers which are numbers that cannot be written as a ratio of two whole numbers. This concept of series expansion indicates the branch of ‘Algebra’ being slowly developed just as the construction of altars brought out the concept of ‘Geometry’.

The works of Panini the famous Grammarian indicates the use of the null operator, context free grammars and some forms which are used nowadays in computer science. It also shows Boolean logic which are used these days in set theory, statistics and in the development of computer science and digital logic.

Pingala was a musical theorist and a scholar who authored the Chhandas Shastra which consists of Sanskrit classical poetry wherein he is said to have stumbled on Pascal triangle, binomial coefficients and combinatorial identity.

Katyayana wrote the Katyayana Sulba Sutra which consists of the Pythagorean theorem, computation of the square root of 2 upto five decimal places and geometry.

Vedic Maths

In recent times Vedic Maths was revived and rediscovered and has made Mathematics simple to follow and easy to solve. It lifts the veil of secrecy from Maths and demystifies numbers. According to Vedic Maths all mathematics is based on 16 Sutras or aphorisms or word formulae. These formulae take into account the way the mind naturally works and this helps the student in arriving at the solution in the shortest possible time and in the easiest possible way. The whole system is said to be beautifully unified and interrelated. It is said to be adaptable and flexible and is said to promote creativity, increase mental agility, is efficient, fast, easy and appeals to everyone.

Jain Mathematics

BHADRABAHU

Jain philosophy under the guidance of Mahavira spoke about concepts of the Infinite. This in turn led to infinity as a mathematical concept and development in other fields such as theory of indices, geometry, number theory, fractions and simple algebraic equations (Beejganita Samikaran). Thus Jain mathematicians were crucial links between the Mathematicians of the Vedic Period and the later Classical period. It is said that they were the first to use the word ’Shunya’ to denote zero. They were the first to recognise the fact that there was difference in the infinities and all were not the same or equal. Their philosophical theory of cosmological structures led to the development of sequences and progressions which is contained in their text Triloka Prajnapati. Important Jain Mathematicians like Bhadrabahu, Yativrisham Acharya and Umasvati also authored texts on Mathematics.

Brahmi Numerals, Concept of Zero and The Place Value System

The Brahmi Numerals which came from the Brahmi script later evolved into the Gupta numerals and subsequently into the Devnagari numerals. It is said that a place value system was known even in other cultures but the first decimal place value system was found in the Indian system. Also the other system did not have a symbol for zero. Many brilliant mathematicians struggled as they were finding it difficult to elevate zero to the same status as the other numbers. To formulate the rules of arithmetic to include zero proved to be the main problem.

Bakhshali Manuscript

It is the oldest mathematical manuscript and was discovered by a farmer while digging in the village of Bakhshali near Peshawar now in Pakistan. The leaflets of the Manuscript are now in fragments but preserved in the Oxford University and are written in verse with commentaries in prose consisting of rules, examples and solutions to the examples. The topics included are square roots, fractions, simple interest, profit and loss, algebra, arithmetic progressions and some geometry. Zero is represented by a dot referred to as ‘pujyam’. The equalisation problems led to the system of linear equations.

Classical Period

This period includes famous Indian mathematicians like AryabhataI Brahmagupta, Bhaskara, Sridhara, Mahavira, AryabhataII and BhaskaraII. It is said that during this period there were two centres emerging for mathematical research in North India one at Ujjain and the other at Kusumapura near Pataliputra.

Aryabhata I wrote his fundamental work called Aryabhatiya which contained important principles of Mathematics and which formed for many centuries the basis for all research carried out in Mathematics and Astronomy in India. Aryabhata I was the dominant figure at Kusumapura. He devised a general method for solving linear equations and called it the pulverizer or kuttaka method indicating that by pulverizing it into smaller numbers by a series of steps the solution could be arrived at. This was arrived at by Euclid later and referred to as Euclid’s Algorithm but actually when applied in reverse order was said to yield Aryabhata’s work. Aryabhata’s interest in astronomy led him to the study of the linear equations. Another important contribution of Aryabhata was his solution to the value of “Pi” to four decimal places and in the field of trigonometry.

The Ujjain centre of learning was famous for mathematicians like Brahmagupta, Varahamihira and Bhaskaracharya. Brahmagupta wrote the Brahma- Sphuta- Siddhanta which dealt with negative numbers and zero. He too was an astronomer like Aryabhata and the problems that arose in Astronomy led him to several mathematical concepts. He is also said to have given the solution for quadratic equations. He expanded and refined the kuttaka method and other works of Aryabhata and Brahmagupta. His famous book Siddhanta Shiromani is divided into four sections-Bijaganita on Algebra, Leelavati on Arithmetic, Goladhyaya on Celestial globe and Grahaganita on Mathematics of the planets.

It is said that the solution to Pell’s equation which is a topic in the field of algebraic number theory and arises in quadratic equations study was already discovered by Brahmagupta before being discovered and attributed to the English mathematician John Pell. Another mathematician Sridhara wrote Patiganita Sara a book on Algebra.

Astronomy and Mathematics

Astronomy was considered to be of divine origin so each family would delve into the subject and remain faithful to the revelations as presented by their gods. By making any big fundamental changes it would amount to changing religious beliefs of others so observations in Astronomy was generally kept low key. Mathematics was used as a tool to making mathematical calculations as, if one could prove ideas mathematically, the astronomical truths could be exhibited more easily. This led to the indepth study of Maths by all the scholars.

Mathematics in South India

In the ninth century Mahavira was a famous mathematician whose book Ganita-Sara –Sangraha echoes the work of Brahmagupta and was used as reference material in those days. Another renowned mathematician was Madhava from Kerala who belonged to the fourteenth century. He gave the value of “Pi’’ to eleven places far beyond that computed by Aryabhata. Madhava’s work laid the foundation for differential Calculus. He founded the Kerala School of Astronomy and Mathematics and among his noted followers were Neelakanta and Jyeshtadeva.

Mathematics in the Modern Ages

In recent times the most important Indian mathematicians has been Srinivasa Ramanujan. Ramanujan is the most famous modern mathematician and his most important discovery was the Arithmetic theory of modular forms which was later further developed by Hecke. He made extraordinary contributions to number theory, mathematical analysis, infinite series and continued fractions and was regarded by many as a genius and forerunner in the field of Mathematics.

Famous Quotes on India’s contribution to Mathematics

1) The ingenious method of expressing every possible number using a set of ten symbols with each symbol having a place value and an absolute value emerged in India, a profound and important idea which appears so simple to us that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions. We shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.-    19^th century French Mathematician Pierre-Simon Laplace

2) We owe a lot to the Indians who taught us how to count without which no worthwhile scientific discovery could have been made-  Albert Einstein

3) There has been no more revolutionary contribution than the one which the Indians made when they invented zero- Lancelot  Hogben the British zoologist and medical statistician.

4) It is true that even across the Himalayan barrier, India has sent to the West such gifts as grammar and logic, philosophy and fables, hypnotism and chess and above all numerals and the decimal system-Will Durant American Historian.

Shakuntala Devi

Shakuntala Devi was an Indian child prodigy in Maths, a legendary Maths wizard who was called the world’s fastest human computer. She gained world wide fame when she calculated the 23^rd root of a number which was more than hundred digits in 50 seconds and in the process was 12 seconds faster than the most advanced computer of that time. She was named in the Guiness Book Of World Records and she wrote a number of books on Maths, astrology and puzzles.

Conclusion

Undoubtedly Mathematics owes a huge debt to the great Indian Mathematicians who over many hundreds of years made outstanding contributions. Ancient history shows gaps in certain periods and hence the challenges in this field for Indian Mathematicians are in identifying and locating manuscripts of those Ancient periods and translating them into a language which can be understood and deciphered by modern scholars. In order to encourage the love of Maths, Olympiads and competitions are regularly held at national and international levels and people are reminded of our glorious past where Indian geniuses who contributed immensely to the world in the field of Mathematics have always reigned supreme from time immemorial.