Equations - Revision history

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	== References ==		== References ==
	~~[[Category:Mathematics]]~~

			<references />

	[[Category:Algebra]]		[[Category:Algebra]]
	~~<references />~~		[[Category:Articles using infobox templates with no data rows]]
			[[Category:Articles with hCards]]
	[[Category:Equations]]		[[Category:Equations]]
			[[Category:Mathematics]]
	[[Category:Organic Articles English]]		[[Category:Organic Articles English]]
	[[Category:~~Vigyan Ready~~]]		[[Category:Pages using duplicate arguments in template calls]]

Indicwiki: 36 revisions imported from :alpha:Equations

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alpha>Ramamurthy S: Category added

2022-11-15T07:28:39Z

Category added

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alpha>Ramamurthy S: Category added

2022-11-15T07:27:54Z

Category added

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alpha>Ramamurthy S: /* Linear Equations */

2022-05-10T13:53:17Z

Linear Equations

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	== Algebraic Expressions and Algebraic Equations ==		== Algebraic Expressions and Algebraic Equations ==
	[[File:Equation illustration colour.svg\|alt=Equation\|thumb]]		[[File:Equation illustration colour.svg\|alt=Algebraic Equation\|thumb\|Algebraic Equation]]
	Algebraic expression can be understood with the following example.<ref>{{Cite book\|title=A Primer to Bhāratīya Gaṇitam , Bhāratīya-Gaṇita-Praveśa- Part-1\|publisher=Samskrit Promotion Foundation\|year=2021\|isbn=978-81-951757-2-7}}</ref>		Algebraic expression can be understood with the following example.<ref>{{Cite book\|title=A Primer to Bhāratīya Gaṇitam , Bhāratīya-Gaṇita-Praveśa- Part-1\|publisher=Samskrit Promotion Foundation\|year=2021\|isbn=978-81-951757-2-7}}</ref>

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	=== Linear Equations ===		=== Linear Equations ===
	[[File:Linear equation for y=3x+1.png\|alt=Linear Equation\|thumb]]		[[File:Linear equation for y=3x+1.png\|alt=Linear Equation\|thumb\|Linear Equation]]
	Mahāvīra gives the following examples leading to simultaneous linear equations together with rules for the solution of each.		Mahāvīra gives the following examples leading to simultaneous linear equations together with rules for the solution of each.

alpha>Ramamurthy S: Images added

2022-05-10T13:44:44Z

Images added

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	== Algebraic Expressions and Algebraic Equations ==		== Algebraic Expressions and Algebraic Equations ==
			[[File:Equation illustration colour.svg\|alt=Equation\|thumb]]
	Algebraic expression can be understood with the following example.<ref>{{Cite book\|title=A Primer to Bhāratīya Gaṇitam , Bhāratīya-Gaṇita-Praveśa- Part-1\|publisher=Samskrit Promotion Foundation\|year=2021\|isbn=978-81-951757-2-7}}</ref>		Algebraic expression can be understood with the following example.<ref>{{Cite book\|title=A Primer to Bhāratīya Gaṇitam , Bhāratīya-Gaṇita-Praveśa- Part-1\|publisher=Samskrit Promotion Foundation\|year=2021\|isbn=978-81-951757-2-7}}</ref>

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	=== Linear Equations ===		=== Linear Equations ===
			[[File:Linear equation for y=3x+1.png\|alt=Linear Equation\|thumb]]
	Mahāvīra gives the following examples leading to simultaneous linear equations together with rules for the solution of each.		Mahāvīra gives the following examples leading to simultaneous linear equations together with rules for the solution of each.

alpha>Ramamurthy S: /* Forming Equations */

2022-05-10T13:29:22Z

Forming Equations

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	We need to form the equation ( ''samī-karaṇa, samī-kāra'' or ''samī-kriyā; from sama, equal and kṛ ,'' to do; hence literally , making equal) from the given conditions of the proposed problem. This may require the application of one or more fundamental operations of algebra or arithmetic.		We need to form the equation ( ''samī-karaṇa, samī-kāra'' or ''samī-kriyā; from sama, equal and kṛ ,'' to do; hence literally , making equal) from the given conditions of the proposed problem. This may require the application of one or more fundamental operations of algebra or arithmetic.

	Bhāskara II says: "Let ''yāvat-tāvat'' be assumed as the value of the unknown quantity. Then doing precisely as has been specifically told-by subtracting, adding, multiplying or dividing the two equal sides of an equation should be very carefully built.		[[Bhaskara II\|Bhāskara II]] says: "Let ''yāvat-tāvat'' be assumed as the value of the unknown quantity. Then doing precisely as has been specifically told-by subtracting, adding, multiplying or dividing the two equal sides of an equation should be very carefully built.

	== Algebraic Expressions and Algebraic Equations ==		== Algebraic Expressions and Algebraic Equations ==
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	By looking at the positions of x<sup>2</sup>, x<sup>1</sup>, x<sup>0</sup> (constant term), there is some pattern ? The standard way of writing an equation starts from the highest power of x. Then the powers of x were written in the descending order up to its lowest power. This format of writing equation was followed by mathematicians from ancient times.		By looking at the positions of x<sup>2</sup>, x<sup>1</sup>, x<sup>0</sup> (constant term), there is some pattern ? The standard way of writing an equation starts from the highest power of x. Then the powers of x were written in the descending order up to its lowest power. This format of writing equation was followed by mathematicians from ancient times.

	Brahmagupta called an equation as samakaraṇa or samīkaraṇa. It means 'making equal. The two sides of an equation (LHS and RHS) were written one below the other. The symbol '=' was not used. Both the sides of an equation were made same by finding the appropriate value(s) for the unknown(s).		[[Brahmagupta]] called an equation as samakaraṇa or samīkaraṇa. It means 'making equal. The two sides of an equation (LHS and RHS) were written one below the other. The symbol '=' was not used. Both the sides of an equation were made same by finding the appropriate value(s) for the unknown(s).

	Pṛthūdakasvāmin (864 CE) in his commentary on the Brāhma-sphuṭa-siddhānta writes the equation 40x - 48 = x<sup>2</sup> + 51 as below		Pṛthūdakasvāmin (864 CE) in his commentary on the Brāhma-sphuṭa-siddhānta writes the equation 40x - 48 = x<sup>2</sup> + 51 as below
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	=== Solution of Linear Equations ===		=== Solution of Linear Equations ===
	~~āryabhaṭaI~~(499) says:		[[Aryabhata\|āryabhaṭa I]](499) says:

	"The difference of the known "amounts" relating to the two persons should be divided by the difference of the coefficients of the unknown. The quotient will be the value of the unknown, if their possessions be equal."		"The difference of the known "amounts" relating to the two persons should be divided by the difference of the coefficients of the unknown. The quotient will be the value of the unknown, if their possessions be equal."
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	~~Naraya~~ writes:
			Narayana writes:

	"From one side clear off the' unknown and from the other the known quantities; then divide the residual known by the residual coefficient of the unknown. Thus will certainly become known the value of the unknown. "		"From one side clear off the' unknown and from the other the known quantities; then divide the residual known by the residual coefficient of the unknown. Thus will certainly become known the value of the unknown. "
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	=== Rule of Concurrence ===		=== Rule of Concurrence ===
	One topic commonly discussed by almost all Hindu writers goes by the special name of ''sankramana'' (concurrence). According to Nārāyana(1350), it is also called ''sankrama'' and ''sankraama''. Brahmagupta (628) includes it in algebra while others consider it as falling within the scope of arithmetic. As explained by the commentator ~~Ganga-,dhara~~ (1420), the subject of discussion here is "the investigation of two quantities concurrent or grown together in the form of their sum and difference."		One topic commonly discussed by almost all Hindu writers goes by the special name of ''sankramana'' (concurrence). According to [[Development of Mathematics\|Nārāyana(1350)]], it is also called ''sankrama'' and ''sankraama''. Brahmagupta (628) includes it in algebra while others consider it as falling within the scope of arithmetic. As explained by the commentator Gangadhara (1420), the subject of discussion here is "the investigation of two quantities concurrent or grown together in the form of their sum and difference."

	In other words ''sankramana'' is the solution of the simultaneous equations		In other words ''sankramana'' is the solution of the simultaneous equations
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	"The square of the difference of the quantities together with twice their product is equal to the sum of their squares. The square-root of this result plus twice the product is the sum."		"The square of the difference of the quantities together with twice their product is equal to the sum of their squares. The square-root of this result plus twice the product is the sum."

	For the solution of (2) the following rule is given by Mahāvīra(850) : "Subtract four times the area (of a rectangle) from the square of the semi-perimeter; then by saṅkramaṇa between the square-root of that (remainder) and the semi-perimeter, the base and the upright are obtained."		For the solution of (2) the following rule is given by [[Mahavira\|Mahāvīra(850)]] : "Subtract four times the area (of a rectangle) from the square of the semi-perimeter; then by saṅkramaṇa between the square-root of that (remainder) and the semi-perimeter, the base and the upright are obtained."

	<math>{\displaystyle x = {\frac{1}{2}}}(a + {\sqrt{a^2-4b}})</math>		<math>{\displaystyle x = {\frac{1}{2}}}(a + {\sqrt{a^2-4b}})</math>

alpha>Ramamurthy S: /* External Links */

2022-05-10T12:25:19Z

External Links

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	Now we get the values of x and y.		Now we get the values of x and y.

			== See Also ==

			* [[समीकरण]]

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

alpha>Ramamurthy S: Category updated

2022-04-21T08:17:09Z

Category updated

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	[[Category:Mathematics]]		[[Category:Mathematics]]
	[[Category:Algebra]]		[[Category:Algebra]]
			<references />
			[[Category:Equations]]

alpha>Ramamurthy S: External Links updated

2022-04-21T07:58:45Z

External Links updated

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	Now we get the values of x and y.		Now we get the values of x and y.

			== External Links ==

			* [https://ia902604.us.archive.org/3/items/indianmathematic00kayerich/indianmathematic00kayerich.pdf Indian Mathematics]
			* [http://www.ms.uky.edu/~sohum/ma330/files/chennai_talks/Emch_Sridharan_Srinivas%20-%20Contributions%20ot%20the%20History%20of%20Indian%20Mathematics%20(2005).pdf Contributions to the History of Indian Mathematics]

	== References ==		== References ==
	[[Category:Mathematics]]		[[Category:Mathematics]]
	[[Category:Algebra]]		[[Category:Algebra]]