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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 7th century CE, 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 15th 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).
In the middle ones put the sum of the digits in the two squares above each. Of these lines, the second gives the combinations with one syllable, the third the combinations with two syllables, ... 3rd century BCE) is notable for being the last of the Vedic mathematicians."Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India." As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras.The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time.There is evidence that in his work on the enumeration of syllabic combinations, Pingala stumbled upon both Pascal's triangle and binomial coefficients, although he did not have knowledge of the binomial theorem itself.Pingala's work also contains the basic ideas of Fibonacci numbers (called maatraameru).In the prose section, the form (and therefore its memorization) was not considered so 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 altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians." Since the statement is a sūtra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca.
It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together." which expresses √2 in the sexagesimal system, and which is also accurate up to 5 decimal places (after rounding). 1850 BCE indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE.
600 BCE), contained results similar to the Baudhayana Sulba Sutra.
An important landmark of the Vedic period was the work of Sanskrit grammarian, Pāṇini (c. His grammar includes early use of Boolean logic, of the null operator, and of context free grammars, and includes a precursor of the Backus–Naur form (used in the description programming languages).
For example, the mantra (sacrificial formula) at the end of the annahoma ("food-oblation rite") performed during the aśvamedha, and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion: lit., "beyond parts"), hail to the dawn (uṣas), hail to the twilight (vyuṣṭi), hail to the one which is going to rise (udeṣyat), hail to the one which is rising (udyat), hail to the one which has just risen (udita), hail to svarga (the heaven), hail to martya (the world), hail to all.