Squared triangular number - Revision history

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2014-10-25T13:14:23Z

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en>David Eppstein: Undid revision 595101635 by David Eppstein (talk) accidental self-rollback

2014-02-12T07:09:14Z

Undid revision 595101635 by David Eppstein (talk) accidental self-rollback

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	~~{{dablink\|This page deals with the Indian Mathematician. For the Telugu surname, see [[Sridhara (surname)]].}}~~		The writer's title is Christy. I am an invoicing officer and I'll be promoted soon. What I adore performing is football but I don't have the time lately. My wife and I reside in Mississippi and I adore each day living right here.<br><br>Also visit my site ... free psychic ([http://ustanford.com/index.php?do=/profile-38218/info/ This Resource site])

	'~~''Sridhar Acharya''' (c. 870, [[India]] – c~~. ~~930 India) was~~ an [[India]]n [[mathematician]], Sanskrit pundit and [[philosopher]]. He was born in Bhurishresti (Bhurisristi or Bhurshut) village in South Radha (at present [[Hooghly district\|Hughli]]) in the 10th Century AD. His father's name was Baladev Acharya and ~~mother~~'~~s name was Acchoka~~. ~~His father was also a Sanskrit pundit.~~

	~~==Works==~~
	~~He was known for two treatises:~~ '~~'Trisatika'' (sometimes called~~ the ~~''Patiganitasara'')~~ and ~~the ''Patiganita''. His major work ''Patiganitasara'' was named ''Trisatika'' because it was written~~ in three hundred slokas. The book discusses counting of numbers, measures, natural number, multiplication, division, zero, squares, cubes, fraction, rule of three, interest-calculation, joint business or partnership and ~~mensuration.~~
	*He gave an exposition on zero. He has written, "If 0(zero) is added to any number,the sum is the same number; If 0(zero) is subtracted from any number,the number remains unchanged; If 0(zero) is multiplied by any number,the product is 0(zero)". He has said nothing about division of any number by 0(zero).
	*In the case of dividing a fraction he has found out the method of multiplying the fraction by the reciprocal of the divisor.
	*He wrote on practical applications of [[algebra]] separated [[algebra]] from [[arithmetic]]
	*He was one of the first to give a formula for solving [[quadratic equation]]s.
	*He found the formula :-
	~~:::~~<~~math~~>~~ax^2 + bx + c = 0 </math>~~ <br ~~/><math> 4a^2x^2 + 4abx + 4ac = 0</math~~> ~~(Multiply by 4a)~~

	~~==Biography==~~

	~~Sridhara is now believed to have lived in the ninth and tenth centuries~~. ~~However, there has been much dispute over his date and in different works the dates of the life of Sridhara have been placed from the seventh century to the eleventh century~~. The best present estimate is that he wrote around 900 AD, a date which is deduced from seeing which other pieces of mathematics he was familiar with and also seeing which later mathematicians were familiar with his work. Some historians give [[Bengal]] as the place of his birth while other historians believe that Sridhara was born in southern India.

	~~Sridhara is known as the author of two mathematical treatises, namely the Trisatika~~ (sometimes called the Patiganitasara ) and the Patiganita. However at least three other works have been attributed to him, namely the Bijaganita, Navasati, and Brhatpati. Information about these books was given the works of [~~[Bhaskara II]] (writing around 1150), Makkibhatta (writing in 1377), and Raghavabhatta (writing in 1493).~~

	~~K.S. Shukla examined Sridhara's method for finding rational solutions of <math>Nx^2 \pm\ 1 = y^2<~~/~~math>, <math>1 - Nx^2 = y^2<~~/~~math>, <math>Nx^2 \pm\ C = y^2<~~/~~math>, <math>C - Nx^2 = y^2</math>which Sridhara gives in the Patiganita. Shukla states that the rules given there are different from those given by other Hindu mathematicians.~~

	Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation. Unfortunately, as indicated above, the original is lost and we have to rely on a quotation of Sridhara's rule from Bhaskara II:-

	Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root.

	~~Proof of the Sridhar Acharya Formula,~~

	~~let us consider,~~

	~~<math>ax^2 + bx + c = 0</math>~~
	*Multipling both sides by 4a,
	~~<math>4a^2x^2 + 4abx + 4ac~~ = ~~0</math>~~
	*Substracting <math>4ac</math> from both sides,
	~~<math>4a^2x^2 + 4abx = -4ac<~~/~~math>~~
	*Then adding <math>b^2</math> to both sides,
	~~<math>4a^2x^2 + 4abx + b^2=~~ - ~~4ac + b^2<~~/~~math>~~
	*We know that,
	~~<math>(a + b)^2 = a^2 + 2ab + b^2<~~/~~math>~~
	*Using it in the equation,
	~~<math>(2ax + b~~)~~^2 = b^2 - 4ac</math>~~
	*Taking square roots,
	~~<math>2ax + b = \pm\sqrt{b^2 - 4ac}</math>~~

	~~<math>2ax = -b\pm\sqrt{b^2 - 4ac}</math>~~

	*Hence, dividing by <math>2a</math> get

	~~<math>x=\frac{-b\pm\sqrt{b^2-4ac\ }}{2a}.</math>~~

	~~In this way, he found the proof of 2 roots.~~

	~~== References ==~~
	* {{MacTutor\|id=Sridhara\|title=Sridhara}}

	~~{{Indian mathematics}}~~

	~~{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->~~
	~~\| NAME = Sridhar Acharya~~
	~~\| ALTERNATIVE NAMES = SRIDHARACHARYA~~
	~~\| SHORT DESCRIPTION = Indian scientist and mathematician~~
	~~\| DATE OF BIRTH = 870~~
	~~\| PLACE OF BIRTH = Bhurishresti~~
	~~\| DATE OF DEATH = 930~~
	~~\| PLACE OF DEATH = Bhurishresti~~
	}}
	~~[[Category:870 births]]~~
	~~[[Category:930 deaths]]~~
	~~[[Category:Medieval Indian mathematicians]]~~
	~~[[Category:9th-century mathematicians]]~~
	~~[[Category:10th-century mathematicians]]~~
	~~[[Category:9th-century Indian people]]~~
	~~[[Category:10th-century Indian people]]~~

en>BattyBot: fixed CS1 errors: dates & General fixes using AWB (9832)

2013-12-31T00:51:14Z

fixed CS1 errors: dates & General fixes using AWB (9832)

← Older revision		Revision as of 02:51, 31 December 2013
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	~~Friends contact him Royal~~. ~~Alabama~~ has ~~usually~~ been his ~~home~~. ~~Playing crochet~~ is a ~~factor~~ that I'~~m totally addicted~~ to. ~~Interviewing~~ is ~~how I make~~ a ~~living and it~~'s ~~something I really appreciate~~.<br><br>~~Feel free to surf to my blog: [http:~~//~~christianculturecenter.org~~/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/61/~~userId~~/~~34117~~/~~Default~~.~~aspx vehicle extended warranty~~]		{{dablink\|This page deals with the Indian Mathematician. For the Telugu surname, see [[Sridhara (surname)]].}}

			'''Sridhar Acharya''' (c. 870, [[India]] – c. 930 India) was an [[India]]n [[mathematician]], Sanskrit pundit and [[philosopher]]. He was born in Bhurishresti (Bhurisristi or Bhurshut) village in South Radha (at present [[Hooghly district\|Hughli]]) in the 10th Century AD. His father's name was Baladev Acharya and mother's name was Acchoka. His father was also a Sanskrit pundit.

			==Works==
			He was known for two treatises: ''Trisatika'' (sometimes called the ''Patiganitasara'') and the ''Patiganita''. His major work ''Patiganitasara'' was named ''Trisatika'' because it was written in three hundred slokas. The book discusses counting of numbers, measures, natural number, multiplication, division, zero, squares, cubes, fraction, rule of three, interest-calculation, joint business or partnership and mensuration.
			*He gave an exposition on zero. He has written, "If 0(zero) is added to any number,the sum is the same number; If 0(zero) is subtracted from any number,the number remains unchanged; If 0(zero) is multiplied by any number,the product is 0(zero)". He has said nothing about division of any number by 0(zero).
			*In the case of dividing a fraction he has found out the method of multiplying the fraction by the reciprocal of the divisor.
			*He wrote on practical applications of [[algebra]] separated [[algebra]] from [[arithmetic]]
			*He was one of the first to give a formula for solving [[quadratic equation]]s.
			*He found the formula :-
			:::<math>ax^2 + bx + c = 0 </math> <br /><math> 4a^2x^2 + 4abx + 4ac = 0</math> (Multiply by 4a)

			==Biography==

			Sridhara is now believed to have lived in the ninth and tenth centuries. However, there has been much dispute over his date and in different works the dates of the life of Sridhara have been placed from the seventh century to the eleventh century. The best present estimate is that he wrote around 900 AD, a date which is deduced from seeing which other pieces of mathematics he was familiar with and also seeing which later mathematicians were familiar with his work. Some historians give [[Bengal]] as the place of his birth while other historians believe that Sridhara was born in southern India.

			Sridhara is known as the author of two mathematical treatises, namely the Trisatika (sometimes called the Patiganitasara ) and the Patiganita. However at least three other works have been attributed to him, namely the Bijaganita, Navasati, and Brhatpati. Information about these books was given the works of [[Bhaskara II]] (writing around 1150), Makkibhatta (writing in 1377), and Raghavabhatta (writing in 1493).

			K.S. Shukla examined Sridhara's method for finding rational solutions of <math>Nx^2 \pm\ 1 = y^2</math>, <math>1 - Nx^2 = y^2</math>, <math>Nx^2 \pm\ C = y^2</math>, <math>C - Nx^2 = y^2</math>which Sridhara gives in the Patiganita. Shukla states that the rules given there are different from those given by other Hindu mathematicians.

			Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation. Unfortunately, as indicated above, the original is lost and we have to rely on a quotation of Sridhara's rule from Bhaskara II:-

			Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root.

			Proof of the Sridhar Acharya Formula,

			let us consider,

			<math>ax^2 + bx + c = 0</math>
			*Multipling both sides by 4a,
			<math>4a^2x^2 + 4abx + 4ac = 0</math>
			*Substracting <math>4ac</math> from both sides,
			<math>4a^2x^2 + 4abx = -4ac</math>
			*Then adding <math>b^2</math> to both sides,
			<math>4a^2x^2 + 4abx + b^2= - 4ac + b^2</math>
			*We know that,
			<math>(a + b)^2 = a^2 + 2ab + b^2</math>
			*Using it in the equation,
			<math>(2ax + b)^2 = b^2 - 4ac</math>
			*Taking square roots,
			<math>2ax + b = \pm\sqrt{b^2 - 4ac}</math>

			<math>2ax = -b\pm\sqrt{b^2 - 4ac}</math>

			*Hence, dividing by <math>2a</math> get

			<math>x=\frac{-b\pm\sqrt{b^2-4ac\ }}{2a}.</math>

			In this way, he found the proof of 2 roots.

			== References ==
			* {{MacTutor\|id=Sridhara\|title=Sridhara}}

			{{Indian mathematics}}

			{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
			\| NAME = Sridhar Acharya
			\| ALTERNATIVE NAMES = SRIDHARACHARYA
			\| SHORT DESCRIPTION = Indian scientist and mathematician
			\| DATE OF BIRTH = 870
			\| PLACE OF BIRTH = Bhurishresti
			\| DATE OF DEATH = 930
			\| PLACE OF DEATH = Bhurishresti
			}}
			[[Category:870 births]]
			[[Category:930 deaths]]
			[[Category:Medieval Indian mathematicians]]
			[[Category:9th-century mathematicians]]
			[[Category:10th-century mathematicians]]
			[[Category:9th-century Indian people]]
			[[Category:10th-century Indian people]]

184.6.3.95: /* Numeric values; geometric and probabilistic interpretation */

2012-09-01T14:23:12Z

Numeric values; geometric and probabilistic interpretation

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Friends contact him Royal. Alabama has usually been his home. Playing crochet is a factor that I'm totally addicted to. Interviewing is how I make a living and it's something I really appreciate.<br><br>Feel free to surf to my blog: [http://christianculturecenter.org/ActivityFeed/MyProfile/tabid/61/userId/34117/Default.aspx vehicle extended warranty]