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Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century
Jain Jainism ( ), also known as Jain Dharma, is an Indian religion. Jainism traces its spiritual ideas and history through the succession of twenty-four tirthankaras (supreme preachers of ''Dharma''), with the first in the current time cycle being ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
possibly born in
Mysore Mysore (), officially Mysuru (), is a city in the southern part of the state of Karnataka, India. Mysore city is geographically located between 12° 18′ 26″ north latitude and 76° 38′ 59″ east longitude. It is located at an altitude of ...
, in
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
. He authored '' Gaṇitasārasan̄graha'' (''Ganita Sara Sangraha'') or the Compendium on the gist of Mathematics in 850 AD. He was patronised by the
Rashtrakuta Rashtrakuta (IAST: ') (r. 753-982 CE) was a royal Indian dynasty ruling large parts of the Indian subcontinent between the sixth and 10th centuries. The earliest known Rashtrakuta inscription is a 7th-century copper plate grant detailing their ...
king
Amoghavarsha Amoghavarsha I (also known as Amoghavarsha Nrupathunga I) (r.814–878 CE) was the greatest emperor of the Rashtrakuta dynasty, and one of the most notable rulers of Ancient India. His reign of 64 years is one of the longest precisely dated mo ...
. He separated
astrology Astrology is a range of Divination, divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positions of Celestial o ...
from mathematics. It is the earliest Indian text entirely devoted to mathematics. He expounded on the same subjects on which
Aryabhata Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
and
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
contended, but he expressed them more clearly. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems. He is highly respected among Indian mathematicians, because of his establishment of
terminology Terminology is a group of specialized words and respective meanings in a particular field, and also the study of such terms and their use; the latter meaning is also known as terminology science. A ''term'' is a word, compound word, or multi-wor ...
for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle. Mahāvīra's eminence spread throughout South India and his books proved inspirational to other mathematicians in
Southern India South India, also known as Dakshina Bharata or Peninsular India, consists of the peninsular southern part of India. It encompasses the States and union territories of India, Indian states of Andhra Pradesh, Karnataka, Kerala, Tamil Nadu, and T ...
. It was translated into the
Telugu language Telugu (; , ) is a Dravidian language spoken by Telugu people predominantly living in the Indian states of Andhra Pradesh and Telangana, where it is also the official language. It is the most widely spoken member of the Dravidian language fami ...
by
Pavuluri Mallana Pavuluri Mallana was an Indian mathematician of the 11th or early 12th century CE from present day Andhra Pradesh. Some historians consider him to be a contemporary of the Eastern Chalukya king Rajaraja Narendra (1022–1063 CE), while others plac ...
as ''Saara Sangraha Ganitamu''.Census of the Exact Sciences in Sanskrit by David Pingree: page 388 He discovered algebraic identities like ''a''3 = ''a'' (''a'' + ''b'') (''a'' − ''b'') + ''b''2 (''a'' − ''b'') + ''b''3. He also found out the formula for ''n''C''r'' as
'n'' (''n'' − 1) (''n'' − 2) ... (''n'' − ''r'' + 1)/ 'r'' (''r'' − 1) (''r'' − 2) ... 2 * 1 He devised a formula which approximated the area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number. He asserted that the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
of a
negative number In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
does not exist.


Rules for decomposing fractions

Mahāvīra's ''Gaṇita-sāra-saṅgraha'' gave systematic rules for expressing a fraction as the sum of unit fractions. This follows the use of unit fractions in
Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
in the Vedic period, and the Śulba Sūtras' giving an approximation of equivalent to 1 + \tfrac13 + \tfrac1 - \tfrac1. In the ''Gaṇita-sāra-saṅgraha'' (GSS), the second section of the chapter on arithmetic is named ''kalā-savarṇa-vyavahāra'' (lit. "the operation of the reduction of fractions"). In this, the ''bhāgajāti'' section (verses 55–98) gives rules for the following: * To express 1 as the sum of ''n'' unit fractions (GSS ''kalāsavarṇa'' 75, examples in 76): :: 1 = \frac1 + \frac1 + \frac1 + \dots + \frac1 + \frac1 * To express 1 as the sum of an odd number of unit fractions (GSS ''kalāsavarṇa'' 77): :: 1 = \frac1 + \frac1 + \dots + \frac1 + \frac1 * To express a unit fraction 1/q as the sum of ''n'' other fractions with given numerators a_1, a_2, \dots, a_n (GSS ''kalāsavarṇa'' 78, examples in 79): :: \frac1q = \frac + \frac + \dots + \frac + \frac * To express any fraction p/q as a sum of unit fractions (GSS ''kalāsavarṇa'' 80, examples in 81): : Choose an integer ''i'' such that \tfrac is an integer ''r'', then write :: \frac = \frac + \frac : and repeat the process for the second term, recursively. (Note that if ''i'' is always chosen to be the ''smallest'' such integer, this is identical to the
greedy algorithm for Egyptian fractions In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a sum ...
.) * To express a unit fraction as the sum of two other unit fractions (GSS ''kalāsavarṇa'' 85, example in 86): :: \frac1 = \frac1 + \frac1 where p is to be chosen such that \frac is an integer (for which p must be a multiple of n-1). :: \frac1 = \frac1 + \frac1 * To express a fraction p/q as the sum of two other fractions with given numerators a and b (GSS ''kalāsavarṇa'' 87, example in 88): :: \frac = \frac + \frac where i is to be chosen such that p divides ai + b Some further rules were given in the ''Gaṇita-kaumudi'' of
Nārāyaṇa Narayana (Sanskrit: नारायण, IAST: ''Nārāyaṇa'') is one of the forms and names of Vishnu, who is in yogic slumber under the celestial waters, referring to the masculine principle. He is also known as Purushottama, and is con ...
in the 14th century.


See also

*
List of Indian mathematicians chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India. Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians ...


Notes


References

*Bibhutibhusan Datta and Avadhesh Narayan Singh (1962). '' History of Hindu Mathematics: A Source Book''. * (Available, along with many other entries from other encyclopaedias for other Mahāvīra-s
online
) * * * * * * * {{DEFAULTSORT:Mahavira 9th-century Indian mathematicians 9th-century Indian Jains Scholars from Karnataka Acharyas