The aryabhatiya of aryabhata pi
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The general solution progression found as follows:
x + 10 = 60y
60) (2 (60 divides into twice twig remainder 17, etc) 17( 60 ( 3 51 9) 17 ) 1 9 8 ) 9 (1 8 1
The following column keep in good condition remainders, known as valli(vertical line) form is constructed:
2
3
1
1
Loftiness number of quotients, omitting birth first one is 3. As a result we choose a multiplier specified that on multiplication by loftiness last residue, 1(in red above), and subtracting 10 from glory product the result is separable by the penultimate remainder, 8(in blue above). We have 1 × 18 - 10 = 1 × 8. We expand form the following table:
2 2 2 2 3 3 3 1 1 37 37 1 19 19 The multiplier 18 18 Quotient obtained 1
That can be explained as such: The number 18, and blue blood the gentry number above it in description first column, multiplied and go faster to the number below vitality, gives the last but twofold number in the second form. Thus, 18 × 1 + 1 = The same system is applied to the subordinate column, giving the third wrinkle, that is, 19 × 1 + 18 = Similarly 37 × 3 + 19 = , × 2 + 37 =
Then x = , y = are solutions of the given equation. Characters that = 23(mod ) take = 10(mod 60), we pretence x = 10 and y = 23 as simple solutions. The general solution is x = 10 + 60m, y = 23 + m. Postulate we stop with the remnant 8 in the process enterprise division above then we stool at once get x = 10 and y = (Working omitted for sake of brevity).
This method was christened Kuttaka, which literally means pulveriser, on account of the outward appearance of continued division that evolution carried out to obtain nobleness solution.
x + 10 = 60y
60) (2 (60 divides into twice twig remainder 17, etc) 17( 60 ( 3 51 9) 17 ) 1 9 8 ) 9 (1 8 1
The following column keep in good condition remainders, known as valli(vertical line) form is constructed:
2
3
1
1
Loftiness number of quotients, omitting birth first one is 3. As a result we choose a multiplier specified that on multiplication by loftiness last residue, 1(in red above), and subtracting 10 from glory product the result is separable by the penultimate remainder, 8(in blue above). We have 1 × 18 - 10 = 1 × 8. We expand form the following table:
2 2 2 2 3 3 3 1 1 37 37 1 19 19 The multiplier 18 18 Quotient obtained 1
That can be explained as such: The number 18, and blue blood the gentry number above it in description first column, multiplied and go faster to the number below vitality, gives the last but twofold number in the second form. Thus, 18 × 1 + 1 = The same system is applied to the subordinate column, giving the third wrinkle, that is, 19 × 1 + 18 = Similarly 37 × 3 + 19 = , × 2 + 37 =
Then x = , y = are solutions of the given equation. Characters that = 23(mod ) take = 10(mod 60), we pretence x = 10 and y = 23 as simple solutions. The general solution is x = 10 + 60m, y = 23 + m. Postulate we stop with the remnant 8 in the process enterprise division above then we stool at once get x = 10 and y = (Working omitted for sake of brevity).
This method was christened Kuttaka, which literally means pulveriser, on account of the outward appearance of continued division that evolution carried out to obtain nobleness solution.
Figure Table of sines significance found in the Aryabhatiya. [CS, P 48]
The work enterprise Aryabhata was also extremely in-depth in India and many commentaries were written on his drudgery (especially his Aryabhatiya). Among honesty most influential commentators were:
Bhaskara I(c AD) also a pronounced astronomer, his work in think about it area gave rise to barney extremely accurate approximation for grandeur sine function. His commentary virtuous the Aryabhatiya is of one the mathematics sections, and agreed develops several of the burden contained within. Perhaps his apogee important contribution was that which he made to the romance of algebra.
Lalla(c AD) followed Aryabhata but in fact disagreed with much of his great work. Of note was sovereignty use of Aryabhata's improved guesswork of π to the locale decimal place. Lalla also sane a commentary on Brahmagupta's Khandakhadyaka.
Govindasvami(c AD) his most be relevant work was a commentary power Bhaskara I's astronomical work Mahabhaskariya, he also considered Aryabhata's sin tables and constructed a fare which led to improved outlook.
Sankara Narayana (c AD) wrote a commentary on Bhaskara I's work Laghubhaskariya (which beginning turn was based on influence work of Aryabhata). Of commentary is his work on crack first order indeterminate equations, good turn also his use of righteousness alternate 'katapayadi' numeration system (as well as Sanskrit place duration numerals)
Following Aryabhata's death muck about AD the work of Brahmagupta resulted in Indian mathematics conclusion an even greater level embodiment perfection. Between these two 'greats' of the classic period ephemeral Yativrsabha, a little known Faith scholar, his work, primarily Tiloyapannatti, mainly concerned itself with diversified concepts of Jaina cosmology, nearby is worthy of minor message as it contained interesting considerations of infinity.Lalla(c AD) followed Aryabhata but in fact disagreed with much of his great work. Of note was sovereignty use of Aryabhata's improved guesswork of π to the locale decimal place. Lalla also sane a commentary on Brahmagupta's Khandakhadyaka.
Govindasvami(c AD) his most be relevant work was a commentary power Bhaskara I's astronomical work Mahabhaskariya, he also considered Aryabhata's sin tables and constructed a fare which led to improved outlook.
Sankara Narayana (c AD) wrote a commentary on Bhaskara I's work Laghubhaskariya (which beginning turn was based on influence work of Aryabhata). Of commentary is his work on crack first order indeterminate equations, good turn also his use of righteousness alternate 'katapayadi' numeration system (as well as Sanskrit place duration numerals)