Parikarmāṣṭaka - Fundamental Operations - Revision history

Ramamurthy S: Internal links updated

2022-11-30T08:40:12Z

Internal links updated

← Older revision		Revision as of 14:10, 30 November 2022
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	Āryabhaṭa II (950) defines subtraction as " The taking out (of some number) from the ''sarvadhana'' (total) is subtraction. What remains is called ''śeṣa'' (remainder) ".		Āryabhaṭa II (950) defines subtraction as " The taking out (of some number) from the ''sarvadhana'' (total) is subtraction. What remains is called ''śeṣa'' (remainder) ".

	[[~~Bhaskara II\|~~Bhāskara II]] has mentioned about these operations in his work on Līlavatī.		[[Bhāskara II]] has mentioned about these operations in his work on Līlavatī.

	कार्यः क्रमादुत्क्रमतोऽथवाऽङ्कयोगो यथास्थानकमन्तरं वा ॥ <small>(Līlavatī , vs.12, p.12)</small>		कार्यः क्रमादुत्क्रमतोऽथवाऽङ्कयोगो यथास्थानकमन्तरं वा ॥ <small>(Līlavatī , vs.12, p.12)</small>
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	==== Tatstha-guṇana ====		==== Tatstha-guṇana ====
	Ancient Indian Mathematicians enhanced the several methods for multiplication to perform multiplication more efficiently and easily. Tatstha-guṇana is one of those methods making multiplication involving three or more digits faster. Indian Mathematicians like Śrīdhara, [[~~Mahavira\|~~Mahāvīra]], Śripati have mentioned this method. Tatstha-guṇana is also known as vajrābhyāsa.		Ancient Indian Mathematicians enhanced the several methods for multiplication to perform multiplication more efficiently and easily. Tatstha-guṇana is one of those methods making multiplication involving three or more digits faster. Indian Mathematicians like Śrīdhara, [[Mahāvīra]], Śripati have mentioned this method. Tatstha-guṇana is also known as vajrābhyāsa.

	Gaṇeśa (c.1545) explains Tatstha-guṇana as "That method of multiplication in which the numbers stand in the same place is called Tatstha-guṇana. It is as follows: after setting the multiplier under the multiplicand multiply unit by unit and the note the result underneath. Then as in vajrābhyāsa multiply unit by ten and ten by unit, add together and set down the result in the line. Next multiply unit by hundred, hundred by unit and ten by ten, add together and set down the result as before; and so on with the rest of the digits. This being done, the line of results is the product.".		Gaṇeśa (c.1545) explains Tatstha-guṇana as "That method of multiplication in which the numbers stand in the same place is called Tatstha-guṇana. It is as follows: after setting the multiplier under the multiplicand multiply unit by unit and the note the result underneath. Then as in vajrābhyāsa multiply unit by ten and ten by unit, add together and set down the result in the line. Next multiply unit by hundred, hundred by unit and ten by ten, add together and set down the result as before; and so on with the rest of the digits. This being done, the line of results is the product.".
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	Samskrita name for square - ''varga'' or ''kṛti .'' The word ''varga'' means "rows" or bunch of similar things. But in Mathematics it denotes the square power and also the square figure or its area. Aryabhaṭa I says : "A square figure of four equal sides and the (number representing its) area called varga. The product of two equal quantities is also varga."		Samskrita name for square - ''varga'' or ''kṛti .'' The word ''varga'' means "rows" or bunch of similar things. But in Mathematics it denotes the square power and also the square figure or its area. Aryabhaṭa I says : "A square figure of four equal sides and the (number representing its) area called varga. The product of two equal quantities is also varga."

	[[~~Bhaskara I\|~~Bhāskara I]] has given a method for finding square as follows:		[[Bhāskara I]] has given a method for finding square as follows:

	"According to the rule of squaring, square the last digit (leftmost), multiply by twice the last digit all the remaining digits, repeat the process by shifting one digit to the right (till the first digit is arrived)." Example : Square of 6387= 40793769		"According to the rule of squaring, square the last digit (leftmost), multiply by twice the last digit all the remaining digits, repeat the process by shifting one digit to the right (till the first digit is arrived)." Example : Square of 6387= 40793769

Admin at 09:17, 29 November 2022

2022-11-29T09:17:10Z

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 == References ==
 <references />
 [[Category:Mathematics]]
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+[[Category:Navigational boxes| ]]
-[[Category:Arithmetic]]
+[[Category:Navigational boxes without horizontal lists]]
 [[Category:Organic Articles English]]
-[[Category:Vigyan Ready]]
+[[Category:Pages with script errors]]

Indicwiki: 39 revisions imported from :alpha:Parikarmāṣṭaka_-_Fundamental_Operations

2022-11-28T10:08:23Z

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alpha>Abhishek: Abhishek moved page Parikarmastaka - Fundamental Operations to Parikarmāṣṭaka - Fundamental Operations without leaving a redirect

2022-11-28T09:43:23Z

Abhishek moved page Parikarmastaka - Fundamental Operations to Parikarmāṣṭaka - Fundamental Operations without leaving a redirect

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alpha>Ramamurthy S: Category added

2022-11-15T07:33:23Z

Category added

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

alpha>Ramamurthy S: Internal link added

2022-08-11T07:39:36Z

Internal link added

← Older revision		Revision as of 13:09, 11 August 2022
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	== Saṅkalana and Vyavakalana (Addition and Subtraction) ==		== Saṅkalana and Vyavakalana (Addition and Subtraction) ==
	[[File:Addition.svg\|alt=Addition\|thumb\|223x223px\|Addition]]		[[File:Addition.svg\|alt=Addition\|thumb\|223x223px\|Addition]]
	Addition is the first fundamental operation in mathematics.<ref>{{Cite web\|title=Fundamental-operations-integers\|url=https://www.aplustopper.com/fundamental-operations-integers/}}</ref> Subtraction is the reverse of addition.		Addition is the first fundamental operation in [[Development of Mathematics\|mathematics]].<ref>{{Cite web\|title=Fundamental-operations-integers\|url=https://www.aplustopper.com/fundamental-operations-integers/}}</ref> Subtraction is the reverse of addition.

	[[Aryabhata\|Āryabhaṭa II]] (950) defines addition as " The making into one of several numbers is addition".		[[Aryabhata\|Āryabhaṭa II]] (950) defines addition as " The making into one of several numbers is addition".

alpha>Ramamurthy S: Citation added

2022-07-09T14:33:32Z

Citation added

← Older revision		Revision as of 20:03, 9 July 2022
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	== Saṅkalana and Vyavakalana (Addition and Subtraction) ==		== Saṅkalana and Vyavakalana (Addition and Subtraction) ==
	[[File:Addition.svg\|alt=Addition\|thumb\|223x223px\|Addition]]		[[File:Addition.svg\|alt=Addition\|thumb\|223x223px\|Addition]]
	Addition is the first fundamental operation in mathematics. Subtraction is the reverse of addition.		Addition is the first fundamental operation in mathematics.<ref>{{Cite web\|title=Fundamental-operations-integers\|url=https://www.aplustopper.com/fundamental-operations-integers/}}</ref> Subtraction is the reverse of addition.

	[[Aryabhata\|Āryabhaṭa II]] (950) defines addition as " The making into one of several numbers is addition".		[[Aryabhata\|Āryabhaṭa II]] (950) defines addition as " The making into one of several numbers is addition".
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	== Bhājana (Division) ==		== Bhājana (Division) ==
	[[File:Division 13-4.png\|alt=Division\|thumb\|Division]]		[[File:Division 13-4.png\|alt=Division\|thumb\|Division]]
	Division is considered as the inverse of multiplication.		Division is considered as the inverse of multiplication.<ref>{{Cite web\|title=Division-as-The-Inverse-of-Multiplication\|url=https://www.math-only-math.com/Division-as-The-Inverse-of-Multiplication.html}}</ref>

	Samskrita names for Division - ''bhāgahāra'' (divide) , ''bhājana'' (break), ''harana'' (take away), ''chedana'' (cut).		Samskrita names for Division - ''bhāgahāra'' (divide) , ''bhājana'' (break), ''harana'' (take away), ''chedana'' (cut).

alpha>Ramamurthy S at 14:15, 9 July 2022

2022-07-09T14:15:54Z

← Older revision		Revision as of 19:45, 9 July 2022
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	Addition is the first fundamental operation in mathematics. Subtraction is the reverse of addition.		Addition is the first fundamental operation in mathematics. Subtraction is the reverse of addition.

	Āryabhaṭa II (950) defines addition as " The making into one of several numbers is addition".		[[Aryabhata\|Āryabhaṭa II]] (950) defines addition as " The making into one of several numbers is addition".

	Āryabhaṭa II (950) defines subtraction as " The taking out (of some number) from the ''sarvadhana'' (total) is subtraction. What remains is called ''śeṣa'' (remainder) ".		Āryabhaṭa II (950) defines subtraction as " The taking out (of some number) from the ''sarvadhana'' (total) is subtraction. What remains is called ''śeṣa'' (remainder) ".
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	<references />		<references />
	[[Category:Mathematics]]		[[Category:Mathematics]]
			[[Category:Mathematics]]
			[[Category:Arithmetic]]

alpha>Ramamurthy S: /* External Links */

2022-06-17T11:17:52Z

External Links

← Older revision		Revision as of 16:47, 17 June 2022
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	* [https://archive.org/details/Patiganita Patiganita]		* [https://archive.org/details/Patiganita Patiganita]
	*[[:en:Aryabhatiya\|Aryabhatiya]]		*[[:en:Aryabhatiya\|Aryabhatiya]]
			*[[:en:Shulba_Sutras\|Shulba_Sutras]]

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

alpha>Ramamurthy S: /* Ghana (Cube) */

2022-06-17T11:11:14Z

Ghana (Cube)

← Older revision		Revision as of 16:41, 17 June 2022
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	Samskrita name for Cube is ''ghana, vṛnda.''		Samskrita name for Cube is ''ghana, vṛnda.''
	[[File:Cube chart nep.JPG\|alt=Cube - Cube Root\|thumb\|245x245px\|Cube - Cube Root]]		[[File:Cube chart nep.JPG\|alt=Cube - Cube Root\|thumb\|245x245px\|Cube - Cube Root]]
	[[Bhāskara II]] mentioned a rule for find the cube of a number as " Set down the cube of the last; then the square of the last multiplied by three times the succeeding; then the square of the succeeding multiplied by three times the last and then the cube of the succeeding; these placed so that there is difference of a place between one result and the next, and added give the cube. The given number is distributed into portions according to places, one of which is taken for the last and the next as the first and in like manner repeatedly (if there be occasion). Or the same process may be begun from the first place of figures for finding the cube."		[[Bhaskara II\|Bhāskara II]] mentioned a rule for find the cube of a number as " Set down the cube of the last; then the square of the last multiplied by three times the succeeding; then the square of the succeeding multiplied by three times the last and then the cube of the succeeding; these placed so that there is difference of a place between one result and the next, and added give the cube. The given number is distributed into portions according to places, one of which is taken for the last and the next as the first and in like manner repeatedly (if there be occasion). Or the same process may be begun from the first place of figures for finding the cube."

	Example: Cube of 1234 has four places as shown below. Initially we take the last digit 1 and succeeding digit 2 i.e 12 and apply the method of cubing		Example: Cube of 1234 has four places as shown below. Initially we take the last digit 1 and succeeding digit 2 i.e 12 and apply the method of cubing