Indian mathematics

The chronology of Indian mathematics spans from the Indus valley civilization and the Vedas to modern times.

Indian mathematics has made outstanding contributions to the development of mathematics as we know it today. The Indian decimal notation of numbers, negative numbers and concept of zero have probably provided some of the biggest impetus' to advances in the field. Concepts from ancient and medieval India were carried to the Middle East, where they were studied extensively. From there they made their way to Europe.

Indian contributions to mathematics

• Decimal system — goes back as far as the Indus Valley civilization
• Hindu-Arabic numeral system, the modern positional notation numeral system used universally
• Zero — see Hindu-Arabic numeral system
• Negative numbers — see Brahmagupta
• Infinity — see Yajur Veda
• Transfinite numbers — see Ancient Jaina Mathematics: an Introduction, Jaina mathematics
• Irrational numbers — see Sulba Sutras
• Geometry — see Sulba Sutras
• Square roots and cube roots — see Baudhayana
• Metarules, transformations, recursions — see Panini
• Binary numbers — see Pingala
• Pascal's triangle — see Pingala
• The series popularly known as Fibonacci series
• One of the earliest statements of the Pythagorean theorem
• Pythagorean triples — see Sulba Sutras
• "James Gregory" series expansion — see Madhava
• Logarithms — see Jaina mathematics
• Quadratic equations and cubic equations — see Sulba_Sutras
• Turing Machine — see Panini
• The Panini Backus Normal Form — see Panini
• Algebra — see for example Aryabhata
• Algorithms — see Ancient India - Mathematics
• Differential calculus — see Bhaskara, Kerala School
• Mathematical analysis, including developments foundational to the development of calculus — see Madhava, Bhaskara, Jyeshtadeva

Harappan Mathematics (3300 BC - 1700 BC)

See: Indus Valley Civilization

The first appearance of evidence of the use of mathematics in the Indian subcontinent was in the Indus Valley Civilization, which dates back to around 3300 BC. Excavations at Harrapa, Mohenjo-daro and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The mathematics used by this early Harrapan civilisation was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced brick technology, which utilised ratios. The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding. [1]

The people of the Indus Valley Civilization achieved great accuracy in measuring length, mass, and time. They were among the first to develop a system of uniform weights and measures. Their measurements were extremely precise. Their smallest division, which is marked on an ivory scale found in Lothal, was approximately 1.704mm, the smallest division ever recorded on a scale of the Bronze Age. Harappan engineers followed the decimal division of measurement for all practical purposes, including the measurement of mass as revealed by their hexahedron weights.

Brick sizes were in a perfect ratio of 4:2:1, and the decimal system was used. Weights were based on units of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with each unit weighing approximately 28 grams, similar to the English ounce or Greek uncia, and smaller objects were weighed in similar ratios with the units of 0.871.

Also, many of the weights uncovered have been produced in definite geometrical shapes (cuboid, barrel, cone, and cylinder to name a few) which present knowledge of basic geometry, including the circle.

Unique Harappan inventions include an instrument which was used to measure whole sections of the horizon and the tidal dock. The engineering skill of the Harappans was remarkable, especially in building docks after a careful study of tides, waves, and currents.

This culture also produced artistic designs of a mathematical nature and there is evidence on carvings that these people could draw concentric and intersecting circles and triangles.

Further to the use of circles in 'decorative' design there is indication of the use of bullock carts, the wheels of which may have had a metallic band wrapped round the rim. This clearly points to the possession of knowledge of the ratio of the length of the circumference of the circle and its diameter, and thus values of pi.

Also of great interest is a remarkably accurate decimal ruler known as the Mohenjo-daro ruler. Subdivisions on the ruler have a maximum error of just 0.005 inches and, at a length of 1.32 inches, have been named the Indus inch. Furthermore, a correspondence has been noted between the Indus scale and brick size. Bricks (found in various locations) were found to have dimensions that were integral multiples of the graduations of their respective scales, which suggests advanced mathematical thinking.

Ancient Period (500 BC - 400 CE)

Sathanang Sutra, Bhagvati Sutra and Anoyogdwar Sutra are famous books of this time. Apart from these the book titled Tatvarthaadigyam Sutra Bhashya by Jaina philosopher Omaswati (135 BC) and the book titled Tiloyapannati of Aacharya (Guru) Yativrisham (176 BC) are famous writings of this time.

Indian mathematicians during this period used notations for squares, cube and other exponents of numbers. They gave shape to Beezganit Samikaran (Algebraic Equations).

Hindu Mathematics (500 BC - 400 CE)

Vaychali Ganit (300 BC - 200 BC)

This book discusses the following in detail - the basic calculations of mathematics, the numbers based on 10, fraction, square, cube, rule of false position, interest methods, questions on purchase and sale. The book has given the answers of the problems and also described testing methods.

Katyayana (200 BC)

His Katyayana Sulba Sutra presented much geometry, including a general version of the Pythagorean theorem and an accurate calculation of the square root of 2 (to 9 decimal places).

Surya Siddhanta (300 CE - 400 CE)

This text contains the roots of modern Trignometry. It mentions Zya (Sine), Otkram Zya (Inverse Sine) and Kotizya (Cosine).

Jaina Mathematics (400 BC - 400 CE)

Sathanang Sutra (300 BC - 200 CE)

The Sathanang Sutra mentioned five types of infinities.

Anoyogdwar Sutra (300 BC - 200 CE)

The Anoyogdwar Sutra mentioned four types of Pramaan (Measure). This Granth (book) also described permutations and combinations which were termed as Bhang and Vikalp.

Bakhshali Manuscript (200 BC - 400 CE)

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 the use of zero and negative sign. An improved method for calculating square roots allowed extremely accurate approximations for irrational numbers to be calculated. [2]

Classical Period (400 CE - 1200 CE)

This period is known as often known as the golden age of Indian Mathematics. Although earlier Indian mathematics was also very significant, this period saw great mathematicians such as Aryabhatta, Brahmagupta, Mahavira Acharya and Bhaskara Acharya give a broad and clear shape to almost all the branches of mathematics. Their contributions to mathematics would eventually spread to the Middle East, Europe, China and other parts of the world.

Aryabhata I (476-550)

He was a resident of Patna in India. He described the important fundamental principles of Mathematics in 332 Shlokas.He calculated the value of pi correct upto four decimal places.

Bhaskara I (600-680)

He worked on Indeterminate equations. He expanded the work of Aryabhatt in his books titled Mahabhaskariya, Aryabhattiya Bhashya and Laghu Bhaskariya .

Brahmagupta (598-668)

His famous work is his book titled Brahm-sfut. Brahmgupt gave a method of calculating the volume of Prism and Cone, described how to sum a Geometric progression.

Shridhara Acharya (650-850)

He wrote books titled "Nav Shatika", "Tri Shatika", "Pati Ganit".

Mahavira Acharya (850)

He wrote the book titled "Ganit Saar Sangraha" on Numerical Mathematics. He derived formulae to calculate the area of ellipse and quadrilateral inside a circle.

Aryabhata II (920-1000)

He wrote a book titled Maha Siddhanta. This book discusses Numerical Mathematics (Ank Ganit) and Algebra.

Shripati Mishra (1019-1066)

He wrote the books titled Siddhanta Shekhar and Ganit Tilak. He worked mainly on permutations and combinations.

Nemichandra Siddhanta Chakravati (1100)

His book is titled Gome-mat Saar.

Bhaskara Acharya (Bhaskara II) (1114-1185)

He wrote excellent books, namely Siddhanta Shiromani, Leelavati Beezganitam, Gola Addhaya, Griha Ganitam and Karan Kautoohal.

Keralese Mathematics (1300-1600)

See: Kerala School

The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th centuries and has its intellectual roots with Aryabhatta who lived in the 5th century. The lineage continues down to modern times but the original research seems to have ended with Narayana Bhattathiri (1559-1632). These astronomers, in attempting to solve problems, invented revolutionary ideas of calculus. They discovered the theory of infinite series, tests of convergence (often attributed to Cauchy), differentiation, term by term integration, iterative methods for solution of non-linear equations, and the theory that the area under a curve is its integral. They achieved most of these results upto several centuries before European mathematicians.

Jyeshtadeva consolidated the Kerala School's discoveries in the Yuktibhasa, the world's first calculus text.

Narayana Pandit (1340-1400)

Narayana Pandit, the earliest of the notable Keralese mathematicians, had written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati).

Although the Karmapradipika contains very little original work, seven different methods for squaring numbers are found within it, a contribution that is wholly original to the author. Narayana's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, using the second order indeterminate equation nq² + 1 = p² (Pell's equation). Mathematical operations with zero, several geometrical rules and discussion of magic squares and similar figures are other contributions of note. Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II's work. Narayana has also made contributions to the topic of cyclic quadrilaterals.

Madhava of Sangamagramma (1350-1425)

Madhava is the founder of the Kerala School and considered to be one of the greatest mathematician-astronomers of the Middle Ages. It is vaguely possible that he may have written Karana Paddhati a work written sometime between 1375 and 1475 but all we really know of Madhava comes from works of later scholars.

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, and thus he is considered the father of mathematical analysis. Madhava was responsible for a number of discoveries, including the Madhava-Gregory series, Madhava-Newton power series, Euler's series, and the power series for pi (usually attributed to Leibniz). Madhava is responsible for laying the foundations for the development of calculus, which was then further developed by his successors at the Kerala School.

Parameshvara (1370-1460)

Parameshvara wrote commentaries on the work of Bhaskara I, Aryabhata and Bhaskara II, and his contributions to mathematics include an outstanding version of the mean value theorem. Furthermore Paramesvara gave a mean value type formula for inverse interpolation of sine, and is thought to have been the first mathematician to give the radius of circle with inscribed cyclic quadrilateral, an expression that is normally attributed to Lhuilier (1782).

Nilakantha Somayaji (1444-1544)

In his most notable work Tantra Samgraha (which 'spawned' a later anonymous commentary Tantrasangraha-vyakhya and a further commentary by the name Yuktidipaika, written in 1501) he elaborates and extends the contributions of Madhava. Sadly none of his mathematical works are extant, however it can be determined that he was a mathematician of some note. Nilakantha was also the author of Aryabhatiya-bhasa a commentary of the Aryabhatiya. Of great significance is the presence of mathematical proof (inductive) in Nilakantha's work.

Jyesthadeva (1500-1575)

Jyesthadeva was another member of the Kerala School. His key work was the Yukti-bhasa (written in Malayalam, a regional language of Kerala). Similarly to the work of Nilakantha it is almost unique in the history of Indian mathematics, in that it contains both proofs of theorems and derivations of rules. He also studied various topics found in many previous Indian works, including integer solutions of systems of first degree equations solved using kuttaka.

Charges of Eurocentrism

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. 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 was 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 aswell, thus transmission would have been possible. There is no evidence by way of relevant manuscripts but the evidence of methodological similarities, communication routes and a suitable chronology for transmission is hard to dismiss.

See also

• Indian mathematicians
• History of mathematics

External links

• History of Indian Mathematics
• An overview of Indian mathematics
• Indian Mathematics: Redressing the balance
• Bakhshali Manuscript
• Eurocentrism in Mathematics; Transmission of Calculus to Europe
• The Kerala School, European Mathematics and Navigation
• [3]
• History of Mathematics -Indian Contribution