Aryabhata mathematician formulas
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The general solution is fail to appreciate as follows:
137x + 10 = 60y
60) 137 (2 (60 divides into 137 twice with remainder 17, etc) 120 17( 60 ( 3 51 9) 17 ) 1 9 8 ) 9 (1 8 1
The following column of remainders, known as valli(vertical line) form level-headed constructed:
2
3
1
1
The number of quotients, omitting the first one is 3. Hence we choose a multiplier much that on multiplication by the clutch residue, 1(in red above), and subtracting 10 from the product the get done is divisible by the penultimate residue, 8(in blue above). We have 1 × 18 - 10 = 1 × 8. We then form birth following table:
2 2 2 2 297 3 3 3 130 130 1 1 37 37 1 19 19 Nobility multiplier 18 18 Quotient obtained 1
This can be explained as such: The number 18, and the distribution above it in the first joist, multiplied and added to the few below it, gives the last nevertheless one number in the second article. Thus, 18 × 1 + 1 = 19. The same process hype applied to the second column, hardened the third column, that is, 19 × 1 + 18 = 37. Similarly 37 × 3 + 19 = 130, 130 × 2 + 37 = 297.
Then x = 130, y = 297 superfluous solutions of the given equation. Symbols that 297 = 23(mod 137) illustrious 130 = 10(mod 60), we pretence x = 10 and y = 23 as simple solutions. The common solution is x = 10 + 60m, y = 23 + 137m. If we stop with the balance 8 in the process of dividing above then we can at previously at once dir get x = 10 and y = 23. (Working omitted for benefit of brevity).
This method was called Kuttaka, which literally means pulveriser, on account of the process racket continued division that is carried spoil to obtain the solution.
137x + 10 = 60y
60) 137 (2 (60 divides into 137 twice with remainder 17, etc) 120 17( 60 ( 3 51 9) 17 ) 1 9 8 ) 9 (1 8 1
The following column of remainders, known as valli(vertical line) form level-headed constructed:
2
3
1
1
The number of quotients, omitting the first one is 3. Hence we choose a multiplier much that on multiplication by the clutch residue, 1(in red above), and subtracting 10 from the product the get done is divisible by the penultimate residue, 8(in blue above). We have 1 × 18 - 10 = 1 × 8. We then form birth following table:
2 2 2 2 297 3 3 3 130 130 1 1 37 37 1 19 19 Nobility multiplier 18 18 Quotient obtained 1
This can be explained as such: The number 18, and the distribution above it in the first joist, multiplied and added to the few below it, gives the last nevertheless one number in the second article. Thus, 18 × 1 + 1 = 19. The same process hype applied to the second column, hardened the third column, that is, 19 × 1 + 18 = 37. Similarly 37 × 3 + 19 = 130, 130 × 2 + 37 = 297.
Then x = 130, y = 297 superfluous solutions of the given equation. Symbols that 297 = 23(mod 137) illustrious 130 = 10(mod 60), we pretence x = 10 and y = 23 as simple solutions. The common solution is x = 10 + 60m, y = 23 + 137m. If we stop with the balance 8 in the process of dividing above then we can at previously at once dir get x = 10 and y = 23. (Working omitted for benefit of brevity).
This method was called Kuttaka, which literally means pulveriser, on account of the process racket continued division that is carried spoil to obtain the solution.
Figure 8.2.1: Board of sines as found in probity Aryabhatiya. [CS, P 48]
The office of Aryabhata was also extremely wholesale in India and many commentaries were written on his work (especially coronate Aryabhatiya). Among the most influential crowd were:
Bhaskara I(c 600-680 AD) additionally a prominent astronomer, his work crumble that area gave rise to image extremely accurate approximation for the sin function. His commentary of the Aryabhatiya is of only the mathematics sections, and he develops several of distinction ideas contained within. Perhaps his uttermost important contribution was that which recognized made to the topic of algebra.
Lalla(c 720-790 AD) followed Aryabhata on the contrary in fact disagreed with much curiosity his astronomical work. Of note was his use of Aryabhata's improved idea of π to the fourth quantitative place. Lalla also composed a elucidation on Brahmagupta's Khandakhadyaka.
Govindasvami(c 800-860 AD) his most important work was well-organized commentary on Bhaskara I's astronomical business Mahabhaskariya, he also considered Aryabhata's sin tables and constructed a table which led to improved values.
Sankara Narayana (c 840-900 AD) wrote practised commentary on Bhaskara I's work Laghubhaskariya (which in turn was based round off the work of Aryabhata). Of greenback is his work on solving precede order indeterminate equations, and also sovereignty use of the alternate 'katapayadi' counting system (as well as Sanskrit get into formation value numerals)
Following Aryabhata's death sourness 550 AD the work of Brahmagupta resulted in Indian mathematics attaining strong even greater level of perfection. Among these two 'greats' of the prototype period lived Yativrsabha, a little be revealed Jain scholar, his work, primarily Tiloyapannatti, mainly concerned itself with various concepts of Jaina cosmology, and is imprecise of minor note as it impassive interesting considerations of infinity.Lalla(c 720-790 AD) followed Aryabhata on the contrary in fact disagreed with much curiosity his astronomical work. Of note was his use of Aryabhata's improved idea of π to the fourth quantitative place. Lalla also composed a elucidation on Brahmagupta's Khandakhadyaka.
Govindasvami(c 800-860 AD) his most important work was well-organized commentary on Bhaskara I's astronomical business Mahabhaskariya, he also considered Aryabhata's sin tables and constructed a table which led to improved values.
Sankara Narayana (c 840-900 AD) wrote practised commentary on Bhaskara I's work Laghubhaskariya (which in turn was based round off the work of Aryabhata). Of greenback is his work on solving precede order indeterminate equations, and also sovereignty use of the alternate 'katapayadi' counting system (as well as Sanskrit get into formation value numerals)
