Trairāśika is the Sanskrit term used by Indian astronomers and mathematicians of the pre-modern era to denote what is known as the "rule of three" in elementary mathematics and algebra. In the contemporary mathematical literature, the term "rule of three" refers to the principle of cross-multiplication which states that if a b = c d {\displaystyle {\tfrac {a}{b}}={\tfrac {c}{d}}} then a d = b c {\displaystyle ad=bc} or a = b c d {\displaystyle a={\tfrac {bc}{d}}} . The antiquity of the term trairāśika is attested by its presence in the Bakhshali manuscript, a document believed to have been composed in the early centuries of the Common Era.
The trairāśika rule Basically trairāśika is a rule which helps to solve the following problem:
"If p {\displaystyle p} produces h {\displaystyle h} what would i {\displaystyle i} produce?" Here p {\displaystyle p} is referred to as pramāṇa ("argument"), h {\displaystyle h} as phala ("fruit") and i {\displaystyle i} as ichcā ("requisition"). The pramāṇa and icchā must be of the same denomination, that is, of the same kind or type like weights, money, time, or numbers of the same objects. Phala can be a of a different denomination. It is also assumed that phala increases in proportion to pramāṇa. The unknown quantity is called icchā-phala, that is, the phala corresponding to the icchā. Āryabhaṭa gives the following solution to the problem:
"In trairāśika, the phala is multiplied by ichcā and then divided by pramāṇa. The result is icchā-phala." In modern mathematical notations, icchā-phala = phala × icchā pramāṇa . {\displaystyle {\text{icchā-phala }}={\tfrac {{\text{phala}}\times {\text{icchā}}}{\text{pramāṇa}}}.}
The four quantities can be presented in a row like this:
pramāṇa | phala | ichcā | icchā-phala (unknown) Then the rule to get icchā-phala can be stated thus: "Multiply the middle two and divide by the first."
Illustrative examples 1. This example is taken from Bījagaṇita, a treatise on algebra by the Indian mathematician Bhāskara II (c. 1114–1185).
Problem: "If two and a half pala-s (a unit of weight) of saffron be obtained for three-sevenths of a nishca (a unit of money); say instantly, best of merchants, how much is got for nine nishca-s?" Solution: pramāṇa = 3 7 {\displaystyle {\tfrac {3}{7}}} nishca, phala = 2 1 2 {\displaystyle 2{\tfrac {1}{2}}} pala-s of saffron, icchā = 9 {\displaystyle 9} nishca-s and we have to find the icchā-phala. icchā-phala = phala × icchā pramāṇa = ( 2 1 2 ) × 9 3 7 = 52 1 2 {\displaystyle {\text{icchā-phala }}={\tfrac {{\text{phala}}\times {\text{icchā}}}{\text{pramāṇa}}}={\tfrac {(2{\tfrac {1}{2}})\times 9}{\tfrac {3}{7}}}=52{\tfrac {1}{2}}} pala-s of safron. 2. This example is taken from Yuktibhāṣā, a work on mathematics and astronomy, composed by Jyesthadeva of the Kerala school of astronomy and mathematics around 1530.
Problem: "When 5 measures of paddy is known to yield 2 measures of rice how many measures of rice will be obtained from 12 measures of paddy?" Solution: pramāṇa = 5 measures of paddy, phala = 2 measures of rice, icchā = 12 measures of rice and we have to find the icchā-phala. icchā-phala = phala × icchā pramāṇa = 2 × 12 5 = 24 5 {\displaystyle {\text{icchā-phala }}={\tfrac {{\text{phala}}\times {\text{icchā}}}{\text{pramāṇa}}}={\tfrac {2\times 12}{5}}={\tfrac {24}{5}}} measures of rice.
Vyasta-trairāśika: Inverse rule of three The four quantities associated with trairāśika are presented in a row as follows:
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