Brahmagupta - Revision history

Jyoti: /* Quadratic Indeterminate Equations: The Cakravāla Method */

2018-01-11T10:20:14Z

‎Quadratic Indeterminate Equations: The Cakravāla Method

Jyoti: /* Quadratic Indeterminate Equations: The Cakravāla Method */

2018-01-11T10:18:38Z

‎Quadratic Indeterminate Equations: The Cakravāla Method

Jyoti: /* Quadratic Indeterminate Equations: The Cakravāla Method */

2018-01-11T10:16:57Z

‎Quadratic Indeterminate Equations: The Cakravāla Method

Jyoti: /* Quadratic Indeterminate Equations: The Cakravāla Method */

2018-01-11T10:15:51Z

‎Quadratic Indeterminate Equations: The Cakravāla Method

Jyoti at 10:12, 11 January 2018

2018-01-11T10:12:38Z

Jyoti at 09:56, 11 January 2018

2018-01-11T09:56:31Z

Jyoti: /* Brahmagupta II’s work */

2018-01-11T09:54:31Z

‎Brahmagupta II’s work

Jyoti at 09:49, 11 January 2018

2018-01-11T09:49:02Z

Jyoti at 09:46, 11 January 2018

2018-01-11T09:46:50Z

Jyoti at 09:36, 11 January 2018

2018-01-11T09:36:45Z

← Older revision		Revision as of 10:20, 11 January 2018
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	√2 ≈ 1+ 1/3 + 1/3.4 - 1/3.4.34 = 577/408 [(577)<sup>2</sup> - 2 (408)<sup>2</sup> = 1]		√2 ≈ 1+ 1/3 + 1/3.4 - 1/3.4.34 = 577/408 [(577)<sup>2</sup> - 2 (408)<sup>2</sup> = 1]
		+
		+

	मलू ं ि ध े वगाद ् गणु कगणु ािद यतु िवहीना ।		मलू ं ि ध े वगाद ् गणु कगणु ािद यतु िवहीना ।

@@ Line 324: / Line 324: @@
 √2 ≈ 1+ 1/3 + 1/3.4 - 1/3.4.34 = 577/408        [(577)<sup>2</sup> - 2 (408)<sup>2</sup> = 1]
-मलू ं ि ध े वगाद  ् गणु कगणु ािद यतु िवहीना । आ वधो गणु कगणु ः सहा  यघातने कृ तम  यम॥् व वध ै यं  थमं   पे ः पे वधत ु यः।
+मलू ं ि ध े वगाद  ् गणु कगणु ािद यतु िवहीना ।
-    पे श ो ध क   त े म लू े     पे क े   प ॥े

← Older revision		Revision as of 10:16, 11 January 2018
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	The Śulva-sūtra approximation (prior to 800 BCE):		The Śulva-sūtra approximation (prior to 800 BCE):

−	√2 ≈ 1+ 1/3 + 1/3.4 - 1/3.4.34 = 577/408 [(577)2 - 2 (408)2 = 1]	+	√2 ≈ 1+ 1/3 + 1/3.4 - 1/3.4.34 = 577/408 [(577)<sup>2</sup> - 2 (408)<sup>2</sup> = 1]
		+
		+	मलू ं ि ध े वगाद ् गणु कगणु ािद यतु िवहीना । आ वधो गणु कगणु ः सहा यघातने कृ तम यम॥् व वध ै यं थमं पे ः पे वधत ु यः।
		+	पे श ो ध क त े म लू े पे क े प ॥े

← Older revision		Revision as of 10:15, 11 January 2018
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	• Optimality of the cakravāla method.		• Optimality of the cakravāla method.
		+
		+	In Chapter XVIII of his Brāhmasphuña-siddhānta (c.628 CE), Brahmagupta considers the problem of solving for integral values of X, Y, the equation
		+
		+	X<sup>2</sup> – D Y<sup>2</sup> = K,
		+
		+	given a non-square integer D > 0, and an integer K.
		+
		+	X is called the larger root (jyeùñha-mūla), Y is called the smaller root (kaniùñha-mūla), D is the prakçti, K is the kùepa.
		+
		+	One motivation for this problem is that of finding rational approximations to square-root of D. If X, Y are integers such that X<sup>2</sup> - D Y<sup>2</sup> = 1, then,
		+
		+	\|√D-(x/y) \| ≤1/2xy,
		+
		+
		+	The Śulva-sūtra approximation (prior to 800 BCE):
		+
		+	√2 ≈ 1+ 1/3 + 1/3.4 - 1/3.4.34 = 577/408 [(577)2 - 2 (408)2 = 1]

← Older revision		Revision as of 10:12, 11 January 2018
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	3. Use Brahmagupta's formula to develop equations for the length of the two diagonals of the quadrilateral.		3. Use Brahmagupta's formula to develop equations for the length of the two diagonals of the quadrilateral.
		+
		+	==Quadratic Indeterminate Equations: The Cakravāla Method==
		+	[https://www.imsc.res.in/~knr/acadimsc13/mds.pdf ''M.D.Srinivas, CENTRE FOR POLICY STUDIES'']
		+
		+
		+	'''OUTLINE'''
		+
		+	• The vargaprakçti equation X<sup>2</sup> – D Y<sup>2</sup> = K, and Brahmagupta’s bhāvanā process (c. 628 CE).
		+
		+	• The cakravāla method of solution of Jayadeva (c. 10th cent) and Bhāskara (c.1150).
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		+	• Bhāskara’s examples X<sup>2</sup> – 61Y<sup>2</sup> = 1, X<sup>2</sup> – 67Y<sup>2</sup> = 1.
		+
		+	• Analysis of the cakravāla method by Krishnaswami Ayyangar.
		+
		+	• History of the so called "Pell’s Equation" X<sup>2</sup> - D Y<sup>2</sup> = 1.
		+
		+	• Solution of "Pell’s equation" by expansion of √D into a simple continued fraction (c. 18th cent).
		+
		+	• Bhāskara semi-regular continued fraction expansion of √D
		+
		+	• Optimality of the cakravāla method.

← Older revision		Revision as of 09:56, 11 January 2018
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	With appropriate perseverance and and algebraic subsitiutions/simplifications, Brahmagupta's theorem can be derived.		With appropriate perseverance and and algebraic subsitiutions/simplifications, Brahmagupta's theorem can be derived.

		+	[[File: ABCD is a quadrilateral with sides of length a, b, c, and d, such that ABCD is both cyclic and has a circle inscribed in it.jpg\|ABCD is a quadrilateral with sides of length a, b, c, and d, such that ABCD is both cyclic and has a circle inscribed in it <br/> From: [http://jwilson.coe.uga.edu/emt725/brahmagupta/brahmagupta.html Jim Wilson, ''The University of Georgia'']\|frame\|500px]]

	Extensions:		Extensions:

@@ Line 224: / Line 224: @@
 ==Brahmagupta's Formula==
 [http://jwilson.coe.uga.edu/emt725/brahmagupta/brahmagupta.html  Jim Wilson, ''The University of Georgia'']
 ''Problem: Develop a proof for Brahmagupta's Formula.''
 Who was Brahmagupta?
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 If ABCD is a rectangle the formula is clear.
 Consider the chord AC.
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 Assume the quadrilateral is not a rectangle. WNLOG, extend AB and CD until they meet at P.
 Label the extensions outside the circle e and f.

@@ Line 255: / Line 255: @@
 Now the area of the quadrilateral ABCD is the area of the larger triangle PBC less the area of the smaller triangle PDA. If A is the area of the quadrilateral and T is the area of triangle PBC
-A= T-(d<sup>2</sup><sup>Superscript text</sup>/b<sup>2</sup>) T = 1-(d<sup>2</sup>/b<sup>2</sup>) T = (b<sup>2</sup>-d<sup>2</sup>/b<sup>2</sup>)T
+A= T-(d<sup>2</sup> / b<sup>2</sup>) T = 1-(d<sup>2</sup>/b<sup>2</sup>) T = (b<sup>2</sup>-d<sup>2</sup>/b<sup>2</sup>)T
 OR

@@ Line 30: / Line 30: @@
 == Biography==
-[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brahmagupta.html ''School of Mathematics and Statistics, University of St Andrews, Scotland'']
+[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brahmagupta.html '' MacTutor- School of Mathematics and Statistics, University of St Andrews, Scotland'']
 URL: [http://www-history.mcs.st-andrews.ac.uk/Biographies/Brahmagupta.html ''Stable URL'']
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 Of particular interest to mathematics in this second work by Brahmagupta is the interpolation formula he uses to compute values of sines. This is studied in detail in [13] where it is shown to be a particular case up to second order of the more general Newton-Stirling interpolation formula.

@@ Line 52: / Line 52: @@
 The remaining fifteen chapters seem to form a second work which is major addendum to the original treatise. The chapters are: examination of previous treatises on astronomy; on mathematics; additions to chapter 1; additions to chapter 2; additions to chapter 3; additions to chapter 4 and 5; additions to chapter 7; on algebra; on the gnomon; on meters; on the sphere; on instruments; summary of contents; versified tables.
-Brahmagupta's understanding of the number systems went far beyond that of others of the period. In the Brahmasphutasiddhanta Ⓣ he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-
+Brahmagupta's understanding of the number systems went far beyond that of others of the period. In the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-
 When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.