### "Vedic Mathematics" by Ajit Mishra's Online Classroom

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The religious texts of the Vedic Period provide evidence for the use of large numbers. By the time of the

*Yajurvedasaṃhitā-*(1200–900 BCE), numbers as high as 10

^{12}were being included in the texts.

^{}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:

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"Hail toThe Satapatha Brahmana (ca. 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.śata("hundred," 10^{2}), hail tosahasra("thousand," 10^{3}), hail toayuta("ten thousand," 10^{4}), hail toniyuta("hundred thousand," 10^{5}), hail toprayuta("million," 10^{6}), hail toarbuda("ten million," 10^{7}), hail tonyarbuda("hundred million," 10^{8}), hail tosamudra("billion," 10^{9}, literally "ocean"), hail tomadhya("ten billion," 10^{10}, literally "middle"), hail toanta("hundred billion," 10^{11},lit., "end"), hail toparārdha("one trillion," 10^{12}lit., "beyond parts"), hail to the dawn (us'as), 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 tosvarga(the heaven), hail tomartya(the world), hail to all."^{}

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The

*Śulba Sūtras*(literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. 700–400 BCE) list rules for the construction of sacrificial fire altars.

^{}Most mathematical problems considered in the

*Śulba Sūtras*spring from "a single theological requirement,"

^{}that of constructing fire altars which have different shapes but occupy the same area. 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.

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According to (Hayashi 2005, p. 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."

The diagonal rope (Since the statement is aakṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) <ropes> produce separately.^{}

*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.

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They contain lists of Pythagorean triples,

^{}which are particular cases of Diophantine equations.

^{}They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."

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Baudhayana (c. 8th century BCE) composed the

*Baudhayana Sulba Sutra*, the best-known

*Sulba Sutra*, which contains examples of simple Pythagorean triples, such as: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (12, 35, 37),

^{}as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square."

^{}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."

^{}Baudhayana gives a formula for the square root of two,

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^{[31]}This formula is similar in structure to the formula found on a Mesopotamian tablet

^{}from the Old Babylonian period (1900–1600 BCE):

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According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca. 1850 BCE

^{}"contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,

^{}indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "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.

^{}Dani goes on to say:

"As the main objective of theSulvasutraswas 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 theSulvasutras. The occurrence of the triples in theSulvasutrasis 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. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily."^{}

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