Indeterminate Equations of the First Degree - Revision history

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	== References ==		== References ==
	<references />		<references />
	~~[[Category:Mathematics]]~~
	[[Category:Algebra]]		[[Category:Algebra]]
			[[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]]~~

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alpha>Ramamurthy S: Category added

2022-11-15T07:40:19Z

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

alpha>Ramamurthy S at 15:42, 11 August 2022

2022-08-11T15:42:45Z

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	Bhaskara I observes: "The dividend and divisor will become prime to each other on being divided by the residue of their mutual division. The operation of the pulveriser should be considered in relation to them.".		Bhaskara I observes: "The dividend and divisor will become prime to each other on being divided by the residue of their mutual division. The operation of the pulveriser should be considered in relation to them.".
			[[File:Hindu astronomer, 19th-century illustration.jpg\|alt=Brahmagupta\|thumb\|Brahmagupta]]
	Brahmagupta says: "Divide the multiplier and the divisor mutually and find the last residue; those quantities being divided by the residue will be prime to each other."		Brahmagupta says: "Divide the multiplier and the divisor mutually and find the last residue; those quantities being divided by the residue will be prime to each other."

alpha>Ramamurthy S at 17:00, 10 May 2022

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			{{Infobox person
			\| name = Indeterminate Equations - 1st Degree
			}}

	In algebra, an '''indeterminate equation''' is an equation for which there is more than one solution<ref>{{Cite web\|title=Indeterminate equation\|url=https://en.wikipedia.org/wiki/Indeterminate_equation}}</ref>.		In algebra, an '''indeterminate equation''' is an equation for which there is more than one solution<ref>{{Cite web\|title=Indeterminate equation\|url=https://en.wikipedia.org/wiki/Indeterminate_equation}}</ref>.

alpha>Ramamurthy S: /* External Links */

2022-05-10T15:11:01Z

External Links

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	== Terminology ==		== Terminology ==
	Hindus called the subject of indeterminate analysis of the first degree as kuṭṭaka , kuṭṭākāra, kuṭṭīkāra or simply kuṭṭa. The names kuṭṭākāra and kuṭṭa appear as early as the Mahā-Bhāskarīya of Bhāskara I (522) . In the commentary of Āryabhaṭīya by Bhāskara I the terms kuṭṭaka and kuṭṭākāra can be found. The terms kuṭṭaka , kuṭṭākāra and kuṭṭa was used by [[Brahmagupta]]. [[Mahavira\|Mahāvīra]] had a preferential liking for the term kuṭṭīkāra.		Hindus called the subject of indeterminate analysis of the first degree as kuṭṭaka , kuṭṭākāra, kuṭṭīkāra or simply kuṭṭa. The names kuṭṭākāra and kuṭṭa appear as early as the Mahā-Bhāskarīya of Bhāskara I (522) . In the commentary of Āryabhaṭīya by Bhāskara I the terms kuṭṭaka<ref>{{Cite web\|title=Aryabhatiya\|url=https://en.wikipedia.org/wiki/Aryabhatiya}}</ref> and kuṭṭākāra can be found. The terms kuṭṭaka , kuṭṭākāra and kuṭṭa was used by [[Brahmagupta]]. [[Mahavira\|Mahāvīra]] had a preferential liking for the term kuṭṭīkāra.

	In the type 1 problem the quantities a , b are called "divisors", the sanskrit names are bhāgahāra, bhājak, cheda etc. and R<sub>1</sub> and R<sub>2</sub> "reminders", the sanskrit names are agra, śeṣa etc.		In the type 1 problem the quantities a , b are called "divisors", the sanskrit names are bhāgahāra, bhājak, cheda etc. and R<sub>1</sub> and R<sub>2</sub> "reminders", the sanskrit names are agra, śeṣa etc.
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	Hence <math>{\displaystyle x = {\frac{by+c}{a}}} \quad {\displaystyle y = {\frac{ax+c}{b}}} </math> according as <math>		Hence <math>{\displaystyle x = {\frac{by+c}{a}}} \quad {\displaystyle y = {\frac{ax+c}{b}}} </math> according as <math>
	R_1>R_2</math> or <math>R_2 > R_1</math> a positive integer.		R_1>R_2</math> or <math>R_2 > R_1</math> a positive integer.
			[[File:Description of Kuttaka in Aryabhatiya.jpg\|alt=Description of Kuttaka as given by Aryabhata in Aryabhatiya\|thumb\|Description of Kuttaka as given by Aryabhata in Aryabhatiya]]
	Āryabhaṭa says : "Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. The residue (and the divisor corresponding to the smaller remainder) being mutually divided, the last residue should be multiplied by such an optional		Āryabhaṭa says : "Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. The residue (and the divisor corresponding to the smaller remainder) being mutually divided, the last residue should be multiplied by such an optional

alpha>Ramamurthy S: /* References */

2022-05-10T15:00:51Z

References

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	The subject of indeterminate analysis of the first degree was considered so important by ancient Hindu Algebraists that the whole science of algebra was once named after it. Āryabhaṭa II , Bhāskara II and others mentions precisely along with the sciences of arithmetic, algebra and astronomy.		The subject of indeterminate analysis of the first degree was considered so important by ancient Hindu Algebraists that the whole science of algebra was once named after it. Āryabhaṭa II , Bhāskara II and others mentions precisely along with the sciences of arithmetic, algebra and astronomy.

	On account of its special importance exclusive work on this entitled kuṭṭākāra śiromaṇi by Devarāja a commentator of of Āryabhaṭa I.		On account of its special importance exclusive work on this entitled kuṭṭākāra śiromaṇi by Devarāja<ref>{{Cite web\|title=Kuṭṭākāra Śirōmaṇi\|url=https://en.wikipedia.org/wiki/Ku%E1%B9%AD%E1%B9%AD%C4%81k%C4%81ra_%C5%9Air%C5%8Dma%E1%B9%87i}}</ref> a commentator of of Āryabhaṭa I.

	== Types of Problems ==		== Types of Problems ==
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	Hindus called the subject of indeterminate analysis of the first degree as kuṭṭaka , kuṭṭākāra, kuṭṭīkāra or simply kuṭṭa. The names kuṭṭākāra and kuṭṭa appear as early as the Mahā-Bhāskarīya of Bhāskara I (522) . In the commentary of Āryabhaṭīya by Bhāskara I the terms kuṭṭaka and kuṭṭākāra can be found. The terms kuṭṭaka , kuṭṭākāra and kuṭṭa was used by [[Brahmagupta]]. [[Mahavira\|Mahāvīra]] had a preferential liking for the term kuṭṭīkāra.		Hindus called the subject of indeterminate analysis of the first degree as kuṭṭaka , kuṭṭākāra, kuṭṭīkāra or simply kuṭṭa. The names kuṭṭākāra and kuṭṭa appear as early as the Mahā-Bhāskarīya of Bhāskara I (522) . In the commentary of Āryabhaṭīya by Bhāskara I the terms kuṭṭaka and kuṭṭākāra can be found. The terms kuṭṭaka , kuṭṭākāra and kuṭṭa was used by [[Brahmagupta]]. [[Mahavira\|Mahāvīra]] had a preferential liking for the term kuṭṭīkāra.

	In the type 1 problem the quantities a , b are called "divisors", the sanskrit names are bhāgahāra, bhājak, cheda etc.) and R<sub>1</sub> and R<sub>2</sub> "reminders", the sanskrit names are agra, śeṣa etc.		In the type 1 problem the quantities a , b are called "divisors", the sanskrit names are bhāgahāra, bhājak, cheda etc. and R<sub>1</sub> and R<sub>2</sub> "reminders", the sanskrit names are agra, śeṣa etc.

	In the type 2 problem β is called the "divisor" and γ is called the "interpolator" the sanskrit names kṣepa, kṣepaka etc. α is called the 'dividend' (bhājya), the unknown quantity (x) the "multiplier" the sanskrit names guṇaka, guṇākāra etc.and y the "quotient" the sankrit name phala. The unknown (x) sometimes known by rāśi meaning "the unknown number" as per Mahāvīra.		In the type 2 problem β is called the "divisor" and γ is called the "interpolator" the sanskrit names kṣepa, kṣepaka etc. α is called the 'dividend' (bhājya), the unknown quantity (x) the "multiplier" the sanskrit names guṇaka, guṇākāra etc.and y the "quotient" the sankrit name phala. The unknown (x) sometimes known by rāśi meaning "the unknown number" as per Mahāvīra.
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	* [[पहली घात के अनिश्चित समीकरण]]		* [[पहली घात के अनिश्चित समीकरण]]

			== External Links ==

			* [https://mathshistory.st-andrews.ac.uk/Biographies/Brahmagupta/ Brahmagupta]
			* [[:en:Kuṭṭaka\|Kuttaka]]

	== References ==		== References ==

alpha>Ramamurthy S: Internal link added

2022-05-10T14:13:24Z

Internal link added

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	In algebra, an '''indeterminate equation''' is an equation for which there is more than one solution<ref>{{Cite web\|title=Indeterminate equation\|url=https://en.wikipedia.org/wiki/Indeterminate_equation}}</ref>.		In algebra, an '''indeterminate equation''' is an equation for which there is more than one solution<ref>{{Cite web\|title=Indeterminate equation\|url=https://en.wikipedia.org/wiki/Indeterminate_equation}}</ref>.

	Āryabhaṭa I (476) <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>was the earliest Hindu Algebraist worked on the Indeterminate Equations of the First Degree. He provided a method for solving the simple indeterminate equation		[[Aryabhata\|Āryabhaṭa I]] (476) <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>was the earliest Hindu Algebraist worked on the Indeterminate Equations of the First Degree. He provided a method for solving the simple indeterminate equation

	<math>by-ax=c</math>		<math>by-ax=c</math>
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	where a, b and c are integers.He also provided how to extend this to solve Simultaneous Indeterminate Equations of the first degree.		where a, b and c are integers.He also provided how to extend this to solve Simultaneous Indeterminate Equations of the first degree.

	Bhāskara I (522) disciple of Āryabhaṭa I has displayed that the same method might be applied to solve the equation		[[Bhaskara I\|Bhāskara I]] (522) disciple of Āryabhaṭa I has displayed that the same method might be applied to solve the equation

	<math>by-ax=-c</math>		<math>by-ax=-c</math>
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	== Terminology ==		== Terminology ==
	Hindus called the subject of indeterminate analysis of the first degree as kuṭṭaka , kuṭṭākāra, kuṭṭīkāra or simply kuṭṭa. The names kuṭṭākāra and kuṭṭa appear as early as the Mahā-Bhāskarīya of Bhāskara I (522) . In the commentary of Āryabhaṭīya by Bhāskara I the terms kuṭṭaka and kuṭṭākāra can be found. The terms kuṭṭaka , kuṭṭākāra and kuṭṭa was used by Brahmagupta. Mahāvīra had a preferential liking for the term kuṭṭīkāra.		Hindus called the subject of indeterminate analysis of the first degree as kuṭṭaka , kuṭṭākāra, kuṭṭīkāra or simply kuṭṭa. The names kuṭṭākāra and kuṭṭa appear as early as the Mahā-Bhāskarīya of Bhāskara I (522) . In the commentary of Āryabhaṭīya by Bhāskara I the terms kuṭṭaka and kuṭṭākāra can be found. The terms kuṭṭaka , kuṭṭākāra and kuṭṭa was used by [[Brahmagupta]]. [[Mahavira\|Mahāvīra]] had a preferential liking for the term kuṭṭīkāra.

	In the type 1 problem the quantities a , b are called "divisors", the sanskrit names are bhāgahāra, bhājak, cheda etc.) and R<sub>1</sub> and R<sub>2</sub> "reminders", the sanskrit names are agra, śeṣa etc.		In the type 1 problem the quantities a , b are called "divisors", the sanskrit names are bhāgahāra, bhājak, cheda etc.) and R<sub>1</sub> and R<sub>2</sub> "reminders", the sanskrit names are agra, śeṣa etc.
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	The Sanskrit words kuṭṭa, kuṭṭaka , kuṭṭākāra, kuṭṭīkāra are all derived from the root kuṭṭ means "to crush", "to grind," "to pulverise" . They all mean the act or process of "breaking", "grinding", "pulverising" as well as an instrument for that, that is, "grinder", "pulveriser".		The Sanskrit words kuṭṭa, kuṭṭaka , kuṭṭākāra, kuṭṭīkāra are all derived from the root kuṭṭ means "to crush", "to grind," "to pulverise" . They all mean the act or process of "breaking", "grinding", "pulverising" as well as an instrument for that, that is, "grinder", "pulveriser".

	Gaṇeśa(1545) says: "Kuṭṭaka is a term for the multiplier, for multiplication is admittedly called by words importing 'injuring,' 'killing.' A certain given number being multiplied by another (unknown quantity), added or subtracted by a given interpolator and then divided by a given divisor leaves no remainder; that multiplier is the Kuṭṭaka. so it has been said by the ancients. This is a special technical term."		[[Development of Mathematics\|Gaṇeśa(1545)]] says: "Kuṭṭaka is a term for the multiplier, for multiplication is admittedly called by words importing 'injuring,' 'killing.' A certain given number being multiplied by another (unknown quantity), added or subtracted by a given interpolator and then divided by a given divisor leaves no remainder; that multiplier is the Kuṭṭaka. so it has been said by the ancients. This is a special technical term."

	Hence the subject of the indeterminate analysis of the first degree came to be designated by the term Kuṭṭaka .		Hence the subject of the indeterminate analysis of the first degree came to be designated by the term Kuṭṭaka .
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	R_1>R_2</math> or <math>R_2 > R_1</math> a positive integer.		R_1>R_2</math> or <math>R_2 > R_1</math> a positive integer.

	Āryabhaṭa says : "Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. The residue (and the divisor corresponding to the smaller remainder) being mutually divided, the last residue should be ~~mul6plicd~~ by such an optional		Āryabhaṭa says : "Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. The residue (and the divisor corresponding to the smaller remainder) being mutually divided, the last residue should be multiplied by such an optional

	Here a = divisor corresponding to greater remainder ; b= divisor corresponding to lesser remainder ; R<sub>1</sub> = greater remainder ; R<sub>2</sub> = lesser remainder		Here a = divisor corresponding to greater remainder ; b= divisor corresponding to lesser remainder ; R<sub>1</sub> = greater remainder ; R<sub>2</sub> = lesser remainder

alpha>Ramamurthy S: /* References */

2022-05-10T12:31:12Z

References

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	"Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. The residue (and the divisor corresponding to the smaller remainder) being mutually divided (until the remainder becomes zero), the last quotient should be multiplied by an optional integer and then added (in case the number of quotients of the mutual division is even) or subtracted (in case the number of quotients is odd) by the difference of the remainders. (Place the other quotients of the mutual division successively one below the other in a column; below them the result just obtained and underneath it the optional integer). Any number below (i.e., the penultimate) is multiplied by the one just above it and then added by that just below it. Divide the last number (obtained by doing so repeatedly) by the divisor corresponding to the smaller remainder; then multiply the residue by the divisor corresponding to the greater remainder and add the greater remainder. (The result will be) the number corresponding to the two divisors."		"Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. The residue (and the divisor corresponding to the smaller remainder) being mutually divided (until the remainder becomes zero), the last quotient should be multiplied by an optional integer and then added (in case the number of quotients of the mutual division is even) or subtracted (in case the number of quotients is odd) by the difference of the remainders. (Place the other quotients of the mutual division successively one below the other in a column; below them the result just obtained and underneath it the optional integer). Any number below (i.e., the penultimate) is multiplied by the one just above it and then added by that just below it. Divide the last number (obtained by doing so repeatedly) by the divisor corresponding to the smaller remainder; then multiply the residue by the divisor corresponding to the greater remainder and add the greater remainder. (The result will be) the number corresponding to the two divisors."

			== See Also ==

			* [[पहली घात के अनिश्चित समीकरण]]

	== References ==		== References ==

alpha>Ramamurthy S: Category added

2022-04-21T08:13:56Z

Category added

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