Indian mathematics emerged in the Indian subcontinent from 1200 BCE. This further continued until the end of the 18th century. In the classical period of Indian mathematics (400 AD to 1200 AD), important contributions were made by scholars like Aryabhatta, Brahmagupta, and Bhaskara II.
The decimal number system in use today, were first recorded in Indian mathematics. Indian mathematicians made many early contributions. These contributions pertained to the concept of zero as a number, negative numbers, arithmetic, and algebra.
In addition to these, trigonometry was further advanced in India, particularly the modern definitions of sine and cosine which were developed there. These mathematical concepts were transmitted to the Middle East, China, along with Europe and led to further developments that now form the foundations of many areas of mathematics.
Ancient and medieval Indian mathematical works were all composed in Sanskrit. It usually consisted of a section of sutras in which a set of rules or problems were stated. These sutras were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary which explained the problem in more detail and provided justification for the solution.
In the prose section, the form was not considered as important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE. Thereafter they were transmitted both orally and in manuscript form.
The oldest extant of mathematical document produced on the Indian subcontinent is the birch bark Bakhshali Manuscript. It was discovered in 1881 in the village of Bakhshali, near Peshawar in the modern day Pakistan. This document was believed to be from the 7th century CE.
According to the archeological excavations conducted at Harappa, Mohenjodaro and other sites of the Indus Valley Civilization have uncovered evidence of the use of "practical mathematics." The people of the IVC manufactured bricks whose dimensions were in the proportion 4:2:1. This proportion was considered favorable for the stability of a brick structure.
They used a standardized system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, as also 500, with the unit weight equaling approximately 28 grams. They produced weights in regular geometric shapes. These shapes included hexahedra, barrels, cones, and cylinders, which demonstrated knowledge of basic geometry.
The inhabitants of Indus civilization, further, also tried to standardize measurement of length to a high degree of accuracy. They designed a ruler which was the Mohenjodaro ruler, whose unit of length was approximately 1.32 inches or 3.4 centimeters and was divided into ten equal parts.
It is a popular fact that the decimal place-value system which was first recorded in India, was then carried over to the Islamic world, thereby eventually reaching Europe. The earliest extant script used in India was the Kharoshti script used in the Gandhara culture of the north-west.
Almost contemporaneously, another script, the Brahmi script, also appeared on much of the sub-continent. Both scripts had numeral symbols and numeral systems, which were initially not based on a place-value system. The earliest surviving evidence of decimal place value numerals in India and south East Asia is from the middle of the first millennium CE.
A copper plate from Gujarat, India mentioned the date 595 CE, written in a decimal place value notation. There is, however, doubt as to the authenticity of the plate. Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia. At both these places, the Indian cultural influence was substantial.
Mathematicians of ancient and early medieval India were mostly all Sanskrit pundits. These were scholars in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar (vyakarna), exegesis (mimamsa) and logic (nyaya)." Memorization of "what is heard" through recitation played a major role in the transmission of sacred texts in ancient India. Scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia."
The commencement of mathematical activity in ancient India began as a part of a "methodological reflection" on the sacred Vedas. This took the form of works called Vedangas or, "Ancillaries of the Veda." Those who knew the sutras knew it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable.
The classical from 400-1200BCE period have often been regarded as the golden age of Indian Mathematics. This was because this period witnessed mathematicians such as Aryabhata Varahmihira Brahmagupta, Bhaskara I, Mahavira, and Bhaskara II. These scholars gave a broader and clearer shape to many branches of mathematics. Their contributions then spread to Asia, the Middle East, and eventually to Europe.
As opposed to Vedic mathematics, their works included both astronomical and mathematical contributions. As a matter of fact, mathematics of that period was included in the 'astral science.' It consisted of three sub-disciplines: mathematical sciences, horoscope astrology and divination. This tripartite division was seen in Varahamihira's compilation called 'Pancasiddhantika.'
Around the fifth and sixth century BCE, the Surya Siddhanta came into existence. The Surya Siddhants contained the roots of modern trigonometry. The author of this work is, however, not known. Moreover, because it contained many words of foreign origin, some scholars considered that the work was produced under the influence of Mesopotamia and Greece. This text has been referred by some of the subsequent mathematicians like Aryabhata and was the first to have used the following trigonometric functions:
1. Sine (Jya).
2. Cosine (Kojya).
3. Inverse sine (Otkram jya).
Apart from this, the text also contained the earliest uses of:
Around the 7th and 8th century, two separate fields namely arithmetic and algebra, began to emerge in Indian mathematics. Brahmagupta, in his astronomical work Brahma Sphuta Siddhanta of 628 CE, also dedicated two chapters to these fields. This was done in addition to propounding his famous theorem on the diagonals of acyclic quadrilateral.
Between the 9th and the 12th there emerged some noteworthy mathematicians who contributed to a great extent in the field of mathematics. For example, Virasena was the first to have used logarithms to base 2,3, and 4 while Mahavira in his book on numerical mathematics called, 'Ganit Saar Sangraha' included Zero, Squares, Cubes, Square roots, cube roots, Plane geometry, solid geometry etc.
Shridhara was another such mathematician. He propounded 'a good rule for finding the volume of a sphere' along with the formula for solving quadratic equations.
Aryabhata II wrote a commentary on Shridhara. Additionally, he also authored an astronomical treatise Maha-Siddhanta which discussed algebra, numerical mathematics and solutions of indeterminate equations.
Lastly came, Bhaskara II who was a mathematician-astronomer. He wrote a number of important treatises, namely the Siddhanta Shiromani, Lilavati, Bijaganita, Gola Addhaya, Griha Ganitam and Karan Kautoohal.
Some of his landmark contributions included: interest computation, plane as well as solid geometry, surds, identification of a positive number having two square roots, solutions to quadratic and cubic equations, proof of Pythagoras theorem etc to name a few.
Finally, from 1300 to 1600, there existed the Kerala School of mathematics. This school was founded by Madhava of Sangamagrama. This school consisted of members like Parameshvara, Neelkanta Somayji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar.
This school flourished between the 14th and 16th centuries. While finding solutions to astronomical problems, the Kerala school astronomers also simultaneously created a number of important mathematics concepts. Some of the most important results like the series expansion for trigonometric functions were given in Sanskrit verse in a book by Neelakanta called Tantrasangraha. There was also a commentary on this work called Tantrasangraha-vakhya whose authorship is not known.
The works of this Kerala School were first written up for the Western world by Englishman C.M. Whish in 1835. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries."
Hence to sum up, some of the areas of mathematics which was studied in ancient as well as medieval India include the following:
1. Arithmetic: Decimal system, Negative numbers, Zero, Binary numeral system, the modern positional notation numeral system, Floating point numbers, Number theory, Infinity, Transfinite numbers, Irrational numbers
2. Geometry: Square roots, Cube roots, Pythagorean triples Transformation, Pascal's triangle triples.
3. Algebra: Quadratic equations, Cubic equations and Quadratic equations (bi quadratic equations)
4. Mathematical logic: Formal grammars, formal language theory, the Panini-Backus form, Recursion
5. General mathematics: Fibonacci numbers, Earliest forms of Morse code, infinite series, Logarithms, indices, Algorithms, Algorism
6. Trigonometry: Trigonometric functions, Trigonometric series