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Indian mathematics—which here is the mathematics that emerged in South Asia[1] from ancient times until the end of the 18th century—had its beginnings in the Bronze Age Indus Valley civilization (2600-1900 BCE) and the Iron Age Vedic culture (1500-500 BCE). In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhatta, Brahmagupta, and Bhaskara II. Indian mathematicians made early contributions to the study of the decimal number system,[2] zero,[3] negative numbers,[4] arithmetic, and algebra.[5] In addition, trigonometry, having evolved in the Hellenistic world and having been introduced into ancient India through the translation of Greek works,[6] was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there.[7] These mathematical concepts were transmitted to the Middle East, China, and Europe[5] and led to further developments that now form the foundations of many areas of mathematics. Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras in which a set of rules or problems 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 (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered as important as the ideas involved.[8][1] All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is the birch bark Bakhshali Manuscript, discovered in 1881 in the village of Bakhshali, near Peshawar (modern day Pakistan) and is likely from the seventh century CE.[9][10] A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) by mathematicians of the Kerala School in the fifteenth century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).[11] However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.[12][13][14][15] Some of the areas of mathematics studied in ancient and medieval India include the following
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