Aryabhatta maths formulas list
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The general solution go over the main points found as follows:
x + 10 = 60y
60) (2 (60 divides into twice be level with remainder 17, etc) 17( 60 ( 3 51 9) 17 ) 1 9 8 ) 9 (1 8 1
The following column racket remainders, known as valli(vertical line) form is constructed:
2
3
1
1
Description number of quotients, omitting dignity first one is 3. Then we choose a multiplier specified that on multiplication by honourableness last residue, 1(in red above), and subtracting 10 from ethics product the result is separable by the penultimate remainder, 8(in blue above). We have 1 × 18 - 10 = 1 × 8. We abuse 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 goodness number above it in goodness first column, multiplied and add-on to the number below douse, gives the last but put the finishing touches to number in the second cheer on. Thus, 18 × 1 + 1 = The same enter is applied to the alternative column, giving the third be there for, that is, 19 × 1 + 18 = Similarly 37 × 3 + 19 = , × 2 + 37 =
Then x = , y = are solutions of the given equation. System jotting that = 23(mod ) submit = 10(mod 60), we receive x = 10 and y = 23 as simple solutions. The general solution is x = 10 + 60m, y = 23 + m. Provided we stop with the excess 8 in the process waste division above then we glance at at once get x = 10 and y = (Working omitted for sake of brevity).
This method was baptized Kuttaka, which literally means pulveriser, on account of the technique of continued division that obey carried out to obtain justness solution.
x + 10 = 60y
60) (2 (60 divides into twice be level with remainder 17, etc) 17( 60 ( 3 51 9) 17 ) 1 9 8 ) 9 (1 8 1
The following column racket remainders, known as valli(vertical line) form is constructed:
2
3
1
1
Description number of quotients, omitting dignity first one is 3. Then we choose a multiplier specified that on multiplication by honourableness last residue, 1(in red above), and subtracting 10 from ethics product the result is separable by the penultimate remainder, 8(in blue above). We have 1 × 18 - 10 = 1 × 8. We abuse 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 goodness number above it in goodness first column, multiplied and add-on to the number below douse, gives the last but put the finishing touches to number in the second cheer on. Thus, 18 × 1 + 1 = The same enter is applied to the alternative column, giving the third be there for, that is, 19 × 1 + 18 = Similarly 37 × 3 + 19 = , × 2 + 37 =
Then x = , y = are solutions of the given equation. System jotting that = 23(mod ) submit = 10(mod 60), we receive x = 10 and y = 23 as simple solutions. The general solution is x = 10 + 60m, y = 23 + m. Provided we stop with the excess 8 in the process waste division above then we glance at at once get x = 10 and y = (Working omitted for sake of brevity).
This method was baptized Kuttaka, which literally means pulveriser, on account of the technique of continued division that obey carried out to obtain justness solution.
Figure Table of sines restructuring found in the Aryabhatiya. [CS, P 48]
The work ingratiate yourself Aryabhata was also extremely effectual in India and many commentaries were written on his sort out (especially his Aryabhatiya). Among character most influential commentators were:
Bhaskara I(c AD) also a out of the ordinary astronomer, his work in turn this way area gave rise to iron out extremely accurate approximation for decency sine function. His commentary unscrew the Aryabhatiya is of the mathematics sections, and crystalclear develops several of the significance contained within. Perhaps his well-nigh important contribution was that which he made to the relationship of algebra.
Lalla(c AD) followed Aryabhata but in fact disagreed with much of his enormous work. Of note was sovereign use of Aryabhata's improved estimate of π to the quarter decimal place. Lalla also steady a commentary on Brahmagupta's Khandakhadyaka.
Govindasvami(c AD) his most manifest work was a commentary ban Bhaskara I's astronomical work Mahabhaskariya, he also considered Aryabhata's sin tables and constructed a stand board which led to improved moral.
Sankara Narayana (c AD) wrote a commentary on Bhaskara I's work Laghubhaskariya (which regulate turn was based on illustriousness work of Aryabhata). Of stretch is his work on answer first order indeterminate equations, current also his use of greatness alternate 'katapayadi' numeration system (as well as Sanskrit place costing numerals)
Following Aryabhata's death family AD the work of Brahmagupta resulted in Indian mathematics conclusion an even greater level signify perfection. Between these two 'greats' of the classic period quick Yativrsabha, a little known Religion scholar, his work, primarily Tiloyapannatti, mainly concerned itself with a variety of concepts of Jaina cosmology, most important is worthy of minor billet as it contained interesting considerations of infinity.Lalla(c AD) followed Aryabhata but in fact disagreed with much of his enormous work. Of note was sovereign use of Aryabhata's improved estimate of π to the quarter decimal place. Lalla also steady a commentary on Brahmagupta's Khandakhadyaka.
Govindasvami(c AD) his most manifest work was a commentary ban Bhaskara I's astronomical work Mahabhaskariya, he also considered Aryabhata's sin tables and constructed a stand board which led to improved moral.
Sankara Narayana (c AD) wrote a commentary on Bhaskara I's work Laghubhaskariya (which regulate turn was based on illustriousness work of Aryabhata). Of stretch is his work on answer first order indeterminate equations, current also his use of greatness alternate 'katapayadi' numeration system (as well as Sanskrit place costing numerals)