This is always free of charge. Sync your files with the cloud! 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 one indian girl pdf economy in verse in order to aid memorization by a student. Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of “practical mathematics”.

The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. Dholavira are demonstrated to have the ability to measure angles in a plane, as well as to determine the position of stars for navigation. The religious texts of the Vedic Period provide evidence for the use of large numbers.

With three-fourths Puruṣa went up: one-fourth of him again was here. 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. They contain lists of Pythagorean triples, which are particular cases of Diophantine equations.

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. The formula is accurate up to five decimal places, the true value being 1. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca. 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.

In all, three Sulba Sutras were composed. The marking should be started by putting 1 in the first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it.

Jain texts on mathematical topics were composed after the 6th century BCE. Jain mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the “classical period. A significant historical contribution of Jain mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints. In particular, their fascination with the enumeration of very large numbers and infinities led them to classify numbers into three classes: enumerable, innumerable and infinite. Prodigious energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity. For example, memorisation of the sacred Vedas included up to eleven forms of recitation of the same text. The texts were subsequently “proof-read” by comparing the different recited versions.

The knowers of the sūtra know it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable. Extreme brevity was achieved through multiple means, which included using ellipsis “beyond the tolerance of natural language,” using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables. The domestic fire-altar in the Vedic period was required by ritual to have a square base and be constituted of five layers of bricks with 21 bricks in each layer. After dividing the quadri-lateral in seven, one divides the transverse in three. In another layer one places the North-pointing. With the increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation.

India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world. The literate culture of Indian science goes back to at least the fifth century B. These became rare by the 13th century, derivations or proofs being favoured by then. It is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe. The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the “nine signs” of the Indians for expressing numbers. The earliest extant script used in India was the Kharoṣṭhī script used in the Gandhara culture of the north-west.

The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE. A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate. There are older textual sources, although the extant manuscript copies of these texts are from much later dates. Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE. It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE. The oldest extant mathematical manuscript in South Asia is the Bakhshali Manuscript, a birch bark manuscript written in “Buddhist hybrid Sanskrit” in the Śāradā script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE. The surviving manuscript has seventy leaves, some of which are in fragments.