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Ganita Kaumudi (Sanskrit: गणितकौमदी) is a treatise on mathematics written by Indian mathematician Narayana Pandita in 1356. It was an arithmetical treatise alongside the other algebraic treatise called "Bijganita Vatamsa" by Narayana Pandit.
Contents
Gaṇita Kaumudī contains about 475 verses of sūtra (rules) and 395 verses of udāharaṇa (examples). It is divided into 14 chapters (vyavahāra):[1]
1. Prakīrṇaka-vyavahāra
Weights and measures, length, area, volume, etc. It describes addition, subtraction, multiplication, division, square, square root, cube and cube root. The problems of linear and quadratic equations described here are more complex than in earlier works.[2] 63 rules and 82 examples[1]
2. Miśraka-vyavahāra
Mathematics pertaining to daily life: “mixture of materials, interest on a principal, payment in instalments, mixing gold objects with different purities and other problems pertaining to linear indeterminate equations for many unknowns”[2] 42 rules and 49 examples[1]
3. Śreḍhī-vyavahāra
Arithmetic and geometric progressions, sequences and series. The generalization here was crucial for finding the infinite series for sine and cosine.[2] 28 rules and 19 examples.[1]
4. Kṣetra-vyavahāra
Geometry. 149 rules and 94 examples.[1] Includes special material on cyclic quadratilerals, such as the “third diagonal”.[2]
5. Khāta-vyavahāra
Excavations. 7 rules and 9 examples.[1]
6. Citi-vyavahāra
Stacks. 2 rules and 2 examples.[1]
7. Rāśi-vyavahāra
Mounds of grain. 2 rules and 3 examples.[1]
8. Chāyā-vyavahāra
Shadow problems. 7 rules and 6 examples.[1]
9. Kuṭṭaka
Linear integer equations. 69 rules and 36 examples.[1]
10. Vargaprakṛti
Quadratic. 17 rules and 10 examples.[1] Includes a variant of the Chakravala method.[2] Ganita Kaumudi contains many results from continued fractions. In the text Narayana Pandita used the knowledge of simple recurring continued fraction in the solutions of indeterminate equations of the type .
11. Bhāgādāna
Contains factorization method,[1] 11 rules and 7 examples.[1]
12. Rūpādyaṃśāvatāra
Contains rules for writing a fraction as a sum of unit fractions. 22 rules and 14 examples.[1]
Unit fractions were known in Indian mathematics in the Vedic period:[3] the Śulba Sūtras give an approximation of √2 equivalent to . Systematic rules for expressing a fraction as the sum of unit fractions had previously been given in the Gaṇita-sāra-saṅgraha of Mahāvīra (c. 850).[3] Nārāyaṇa's Gaṇita-kaumudi gave a few more rules: the section bhāgajāti in the twelfth chapter named aṃśāvatāra-vyavahāra contains eight rules.[3] The first few are:[3]
- Rule 1. To express 1 as a sum of n unit fractions:[3]
- Rule 2. To express 1 as a sum of n unit fractions:[3]
- Rule 3. To express a fraction as a sum of unit fractions:[3]
- Pick an arbitrary number i such that is an integer r, write
- and find successive denominators in the same way by operating on the new fraction. If i is always chosen to be the smallest such integer, this is equivalent to the greedy algorithm for Egyptian fractions, but the Gaṇita-Kaumudī's rule does not give a unique procedure, and instead states evam iṣṭavaśād bahudhā ("Thus there are many ways, according to one's choices.")[3]
- Rule 4. Given arbitrary numbers ,[3]
- Rule 5. To express 1 as the sum of fractions with given numerators :[3]
- Calculate as , ,
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