Rational Triangles - Revision history

Admin at 04:11, 16 December 2022

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 == References ==
 <references />
 [[Category:Mathematics]]
-[[Category:Algebra]]
+[[Category:Navigational boxes| ]]
 [[Category:Organic Articles English]]
-[[Category:Vigyan Ready]]
+[[Category:Pages with broken file links]]

Indicwiki: 19 revisions imported from :alpha:Rational_Triangles

2022-12-16T04:10:42Z

19 revisions imported from alpha:Rational_Triangles

← Older revision	Revision as of 09:40, 16 December 2022
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alpha>Ramamurthy S: Category added

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	[[Category:Mathematics]]		[[Category:Mathematics]]
	[[Category:Algebra]]		[[Category:Algebra]]
			[[Category:Organic Articles English]]
			[[Category:Vigyan Ready]]

alpha>Ramamurthy S: Internal inks added

2022-10-14T05:58:33Z

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	== Rational Right Triangles - Early Solutions ==		== Rational Right Triangles - Early Solutions ==
	In Śulba solution for the [[Equations\|equation]] <math>x^2+y^2=z^2</math>-------(1) is available<ref>{{Cite book\|last=Datta\|first=Bibhutibhusan\|title=History of Hindu Mathematics\|last2=Narayan Singh\|first2=Avadhesh\|publisher=Asia Publishing House\|year=1962\|location=Mumbai}}</ref>. Baudhāyana<ref>{{Cite web\|title=Baudhāyana\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Baudhayana/}}</ref> (c. 800 B.C.), Āpastamba<ref>{{Cite web\|title=Āpastamba\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Apastamba/}}</ref> and Kātyāyana<ref>{{Cite web\|title=Kātyāyana\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Katyayana/}}</ref> (c. 500 B.C.) gave a method for the transformation of a rectangle into a square, which is the equivalent of the algebraical identity.		In Śulba solution for the [[Equations\|equation]] <math>x^2+y^2=z^2</math>-------(1) is available<ref>{{Cite book\|last=Datta\|first=Bibhutibhusan\|title=History of Hindu Mathematics\|last2=Narayan Singh\|first2=Avadhesh\|publisher=Asia Publishing House\|year=1962\|location=Mumbai}}</ref>. [[Baudhāyana]]<ref>{{Cite web\|title=Baudhāyana\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Baudhayana/}}</ref> (c. 800 B.C.), [[Āpastamba]]<ref>{{Cite web\|title=Āpastamba\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Apastamba/}}</ref> and [[Kātyāyana]]<ref>{{Cite web\|title=Kātyāyana\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Katyayana/}}</ref> (c. 500 B.C.) gave a method for the transformation of a rectangle into a square, which is the equivalent of the algebraical identity.

	<math>{\displaystyle mn = \left (m- \frac{m-n}{2} \right)^2 - \left (\frac{m-n}{2} \right)^2 } </math>		<math>{\displaystyle mn = \left (m- \frac{m-n}{2} \right)^2 - \left (\frac{m-n}{2} \right)^2 } </math>

alpha>Ramamurthy S: Image added

2022-10-13T06:48:10Z

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	Kātyāyana says: "As many squares (of equal size) as you wish to combine into one, the transverse line will be (equal to) one less than that; twice a side will be (equal to) one more than that; (thus) form (an isosceles) triangle. Its arrow (i.e., altitude) will do that."		Kātyāyana says: "As many squares (of equal size) as you wish to combine into one, the transverse line will be (equal to) one less than that; twice a side will be (equal to) one more than that; (thus) form (an isosceles) triangle. Its arrow (i.e., altitude) will do that."
			[[File:Isosceles Triangle.jpg\|alt=Isosceles Triangle\|thumb\|Isosceles Triangle]]

	~~<Triangle picture to be added here>~~

	For combining ''n'' squares of sides ''a'' each we form the isosceles triangle ABC such that <math>AB=AC=\frac{(n+1)a}{2}</math> and <math>BC=(n-1)a</math>		For combining ''n'' squares of sides ''a'' each we form the isosceles triangle ABC such that <math>AB=AC=\frac{(n+1)a}{2}</math> and <math>BC=(n-1)a</math>

alpha>Ramamurthy S: /* External Links */

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External Links

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	* [http://www.irphouse.com/ijmr/ijmrv6n1_12.pdf A Study on the Properties of Rational Triangles]		* [http://www.irphouse.com/ijmr/ijmrv6n1_12.pdf A Study on the Properties of Rational Triangles]
			*[https://hrj.episciences.org/158/pdf Rational Triangles with the same perimeter and the same area]

	== References ==		== References ==

alpha>Ramamurthy S: External link updated

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	== External Links ==		== External Links ==

	* [http://~~grail~~.~~cba~~.~~csuohio.edu~~/~~~somos~~/~~rattri~~.~~html~~ Rational Triangles]		* [http://www.irphouse.com/ijmr/ijmrv6n1_12.pdf A Study on the Properties of Rational Triangles]

	== References ==		== References ==

alpha>Manidh: Added Redirecting link to the Hindi page

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Added Redirecting link to the Hindi page

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	== Mahāvīra's Definitions ==		== Mahāvīra's Definitions ==
	Mahāvīra<ref>{{Cite web\|title=Mahāvīra\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Mahavira/}}</ref> says A triangle or a quadrilateral whose sides, altitudes and other dimensions can be expressed in terms of rational numbers is called ''janya'' which means generated, formed or that which is generated or formed. Numbers which are involved in forming a particular figure are called its bīja-saṃkhyā (element-numbers) or simply bīja (element or seed).		Mahāvīra<ref>{{Cite web\|title=Mahāvīra\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Mahavira/}}</ref> says A triangle or a quadrilateral whose sides, altitudes and other dimensions can be expressed in terms of rational numbers is called ''janya'' which means generated, formed or that which is generated or formed. Numbers which are involved in forming a particular figure are called its bīja-saṃkhyā (element-numbers) or simply bīja (element or seed).

			== See Also ==
			[[परिमेय त्रिभुज]]

	== External Links ==		== External Links ==

alpha>Ramamurthy S: Internal link added

2022-08-22T07:31:42Z

Internal link added

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	== Rational Right Triangles - Early Solutions ==		== Rational Right Triangles - Early Solutions ==
	In Śulba solution for the equation <math>x^2+y^2=z^2</math>-------(1) is available<ref>{{Cite book\|last=Datta\|first=Bibhutibhusan\|title=History of Hindu Mathematics\|last2=Narayan Singh\|first2=Avadhesh\|publisher=Asia Publishing House\|year=1962\|location=Mumbai}}</ref>. Baudhāyana<ref>{{Cite web\|title=Baudhāyana\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Baudhayana/}}</ref> (c. 800 B.C.), Āpastamba<ref>{{Cite web\|title=Āpastamba\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Apastamba/}}</ref> and Kātyāyana<ref>{{Cite web\|title=Kātyāyana\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Katyayana/}}</ref> (c. 500 B.C.) gave a method for the transformation of a rectangle into a square, which is the equivalent of the algebraical identity.		In Śulba solution for the [[Equations\|equation]] <math>x^2+y^2=z^2</math>-------(1) is available<ref>{{Cite book\|last=Datta\|first=Bibhutibhusan\|title=History of Hindu Mathematics\|last2=Narayan Singh\|first2=Avadhesh\|publisher=Asia Publishing House\|year=1962\|location=Mumbai}}</ref>. Baudhāyana<ref>{{Cite web\|title=Baudhāyana\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Baudhayana/}}</ref> (c. 800 B.C.), Āpastamba<ref>{{Cite web\|title=Āpastamba\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Apastamba/}}</ref> and Kātyāyana<ref>{{Cite web\|title=Kātyāyana\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Katyayana/}}</ref> (c. 500 B.C.) gave a method for the transformation of a rectangle into a square, which is the equivalent of the algebraical identity.

	<math>{\displaystyle mn = \left (m- \frac{m-n}{2} \right)^2 - \left (\frac{m-n}{2} \right)^2 } </math>		<math>{\displaystyle mn = \left (m- \frac{m-n}{2} \right)^2 - \left (\frac{m-n}{2} \right)^2 } </math>
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	== Later Rational Solutions ==		== Later Rational Solutions ==
	Brahmagupta (628) says : "The square of the optional (iṣṭa) side is divided and then diminished by an optional number; half the		[[Brahmagupta]] (628) says : "The square of the optional (iṣṭa) side is divided and then diminished by an optional number; half the

	result is the upright, and that increased by the optional number gives the hypotenuse of a rectangle."		result is the upright, and that increased by the optional number gives the hypotenuse of a rectangle."

alpha>Ramamurthy S: Citation updated

2022-04-22T06:07:14Z

Citation updated

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	== Mahāvīra's Definitions ==		== Mahāvīra's Definitions ==
	Mahāvīra says A triangle or a quadrilateral whose sides, altitudes and other dimensions can be expressed in terms of rational numbers is called ''janya'' which means generated, formed or that which is generated or formed. Numbers which are involved in forming a particular figure are called its bīja-saṃkhyā (element-numbers) or simply bīja (element or seed).		Mahāvīra<ref>{{Cite web\|title=Mahāvīra\|url=https://mathshistory.st-andrews.ac.uk/Biographies/Mahavira/}}</ref> says A triangle or a quadrilateral whose sides, altitudes and other dimensions can be expressed in terms of rational numbers is called ''janya'' which means generated, formed or that which is generated or formed. Numbers which are involved in forming a particular figure are called its bīja-saṃkhyā (element-numbers) or simply bīja (element or seed).

			== External Links ==

			* [http://grail.cba.csuohio.edu/~somos/rattri.html Rational Triangles]

	== References ==		== References ==