Parikarmāṣṭaka - Fundamental Operations: Difference between revisions

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== Introduction ==
== Introduction ==
Arithmetic deals with calculations using numbers. Pāṭīgaṇita is the Samskrit word for arithmetic and geometry .Pāṭīgaṇita is formed by combining the words Pāṭī (slate) and gaṇita (mathematics). Since gaṇita was done using a board of a slate , it was called Pāṭīgaṇita. For all transactions using numbers will require the basic operations of addition, subtraction, multiplication, division, squaring etc. Ancient Indian Mathematicians mentioned eight fundamental operations together called as ''Parikarmāṣṭaka''.
Arithmetic deals with calculations using numbers. Pāṭīgaṇita is the Samskrit word for arithmetic and geometry .Pāṭīgaṇita  
 
{{Infobox person
| name              = Mathematical Operations
| image              = [[File:Arithmetic symbols.svg|Arithmetic_symbols|150px]]
}}
 
is formed by combining the words Pāṭī (slate) and gaṇita (mathematics). Since gaṇita was done using a board of a slate , it was called Pāṭīgaṇita. For all transactions using numbers will require the basic operations of addition, subtraction, multiplication, division, squaring etc. Ancient Indian Mathematicians mentioned eight fundamental operations together called as ''Parikarmāṣṭaka''.


== Definition ==
== Definition ==
''Parikarma'' means arithmetic operations and ''aṣṭaka'' means group of eight. ''Parikarmāṣṭaka'' signifies eight basic operations.  
''Parikarma'' means arithmetic operations and ''aṣṭaka'' means group of eight<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|location=New Delhi}}</ref>. ''Parikarmāṣṭaka'' signifies eight basic operations.  


The eight fundamental operations are  :
The eight fundamental operations are  :
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== Saṅkalana and Vyavakalana (Addition and Subtraction) ==
== Saṅkalana and Vyavakalana (Addition and Subtraction) ==
Addition is the first fundamental operation in mathematics. Subtraction is the reverse of addition.  
[[File:Addition.svg|alt=Addition|thumb|223x223px|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.  


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) ".  


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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==== Sthāna-vibhāga-guṇana - Place wise multiplication : ====
==== Sthāna-vibhāga-guṇana - Place wise multiplication : ====
[[File:Poser-une-multiplication.gif|alt=Multiplication|thumb|195x195px|Multiplication]]
Multiply the multiplicand by each digit of the multiplier separately. Place them appropriately one below the other . Add those numbers. This method is the standard method of doing multiplication.
Multiply the multiplicand by each digit of the multiplier separately. Place them appropriately one below the other . Add those numbers. This method is the standard method of doing multiplication.


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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, 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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== Bhājana (Division) ==
== Bhājana (Division) ==
Division is considered as the inverse of multiplication.  
[[File:Division 13-4.png|alt=Division|thumb|Division]]
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).
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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."


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  
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== Varga-mūla (Square Root) ==
== Varga-mūla (Square Root) ==
Samskrita name for Square root is ''Varga-mūla''.  ''mūla , pada''  means root  in Hindu terminology.  The word karanī is found in the Śulbasūtras as a term for the square root.  
[[File:Square root of 9.svg|alt=Square and Square root|thumb|200x200px|Square and Square root]]
Samskrita name for Square root is ''Varga-mūla''.  ''mūla , pada''  means root  in Hindu terminology.  The word karanī is found in the Śulbasūtras as a term for the square root.  


In Āryabhaṭīya the method of finding the square root is given as " Always divide the even place by twice the square-root (upto the preceding odd place); after having subtracted from the odd place the square (of the quotient) , the quotient put down at the next place (in the line of the root) gives the root "
In Āryabhaṭīya the method of finding the square root is given as " Always divide the even place by twice the square-root (upto the preceding odd place); after having subtracted from the odd place the square (of the quotient) , the quotient put down at the next place (in the line of the root) gives the root "
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| colspan="7" |
| colspan="7" |
|0
|0
|}
== Ghana (Cube) ==
Samskrita name for Cube  is ''ghana, vṛnda.''
[[File:Cube chart nep.JPG|alt=Cube - Cube Root|thumb|245x245px|Cube - Cube Root]]
[[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
{| class="wikitable"
|+
|1
|2
|3
|4
|}
{| class="wikitable"
|+
|
|
|
|1
|2
|
|
|-
|Cube of last digit
|
|
|1
|
|
|
|-
|Thrice the square of the last digit
(3 x 1<sup>2</sup>) muliplied by the succeding
digit (2) is 2 x 3 x 1<sup>2</sup>  and placing at the next place
|
|
|
|6
|
|
|-
|Thrice the square of the succeding digit (2)multiplied
by the last digit is 3 x 2<sup>2</sup> x 1 and placing at the next place
|
|
|
|1
|2
|
|-
|Cube of succeeding  digit (2<sup>3</sup>)
|
|
|
|
|
|8
|-
|Cube of 12 = sum of the above digits
|
|
|'''1'''
|'''7'''
|'''2'''
|'''8'''
|}
Next we will take the next digit 3 i.e the number is 123. Here 12 is the last digit and 3 is succeeding digit. The method continues thus.
{| class="wikitable"
|
|
|
|12
|3
|
|
|
|-
|Cube of last digit -12 (already obtained)
|1
|7
|2
|8
|
|
|
|-
|Thrice the square of the last digit
(3 x 12<sup>2</sup>) muliplied by the succeding
digit (3) is 3 x 3 x 12<sup>2</sup>  and placing at the next place
|
|1
|2
|9
|6
|
|
|-
|Thrice the square of the succeding digit (3)multiplied
by the last digit is 3 x 3<sup>2</sup> x 12 and placing at the next place
|
|
|
|3
|2
|4
|
|-
|Cube of succeeding  digit (3<sup>3</sup>)
|
|
|
|
|
|2
|7
|-
|Cube of 123 = sum of the above digits
|'''1'''
|'''8'''
|'''6'''
|'''0'''
|'''8'''
|'''6'''
|'''7'''
|}
Now the remaining digit 4 is taken  so that the number is 1234 of which 123 is the last digit and 4 is the succeeding digit. The method continues thus.
{| class="wikitable"
|
|
|
|123
|4
|
|
|
|
|
|
|-
|Cube of last digit -123 (already obtained)
|1
|8
|6
|0
|8
|6
|7
|
|
|
|-
|Thrice the square of the last digit
(3 x 123<sup>2</sup>) muliplied by the succeding
digit (4) is 4 x 3 x 123<sup>2</sup>  and placing at the next place
|
|
|1
|8
|1
|5
|4
|8
|
|
|-
|Thrice the square of the succeeding digit (4)multiplied
by the last digit is 3 x 4<sup>2</sup> x 123 and placing at the next place
|
|
|
|
|
|5
|9
|0
|4
|
|-
|Cube of succeeding  digit (4<sup>3</sup>)
|
|
|
|
|
|
|
|
|6
|4
|-
|Cube of 1234 = sum of the above digits
|'''1'''
|'''8'''
|'''7'''
|'''9'''
|'''0'''
|'''8'''
|'''0'''
|'''9'''
|'''0'''
|'''4'''
|}
== Ghana-mūla (Cube Root) ==
Samskrita name for Cube root  is ''ghana-mūla , ghana-pada.''
In Āryabhaṭīya the description of the operation of the cube-root is given as "Divide the second ''aghana'' place by thrice the square of the cube-root; subtract from the first ''aghana'' place the square of the quotient multiplied by thrice the preceding cube-root); and (subtract) the cube (of the quotient) from the ''ghana'' place; (the quotient put down at the next place (in the line of the root) gives the root)" . Cube root of 2628072 is 138
{| class="wikitable"
|+
| colspan="2" rowspan="2" |
|Ghana
|Aghana
|Aghana
|Ghana
|Aghana
|Aghana
|Ghana
| rowspan="15" |
|Root
|-
|2
|6
|2
|8
|0
|7
|2
|
|-
|Subtract 1<sup>3</sup>
|
|1
|
|
| colspan="4" rowspan="5" |
|1
|-
|Divide by 3 x 1<sup>2</sup>
|''3''
|1
|6
|''3 - Quotient''
|13
|-
|
| rowspan="5" |
| rowspan="4" |
|9
|
| rowspan="5" |
|-
|
|7
|2
|-
|Subtract 3 x 1 x 3<sup>2</sup>
|2
|7
|-
|
|4
|5
|8
| colspan="3" rowspan="2" |
|-
|Subtract 3<sup>3</sup>
| rowspan="7" |
|
|2
|7
|-
|Divide by 3 x 13<sup>2</sup>
|507
|4
|3
|1
|0
|8 -Quotient
|
|138
|-
|
| rowspan="5" |
|4
|0
|5
|6
| colspan="2" |
| rowspan="5" |
|-
|
| rowspan="4" |
|2
|5
|4
|7
| rowspan="2" |
|-
|Subrract 3 x 13 x 8<sup>2</sup>
|2
|4
|9
|6
|-
|
| colspan="2" rowspan="2" |
|5
|1
|2
|-
|Subtract 8<sup>3</sup>
|5
|1
|2
|-
| colspan="8" |
|0
| colspan="2" |
|}
|}


== See also ==
== See also ==
[[परिकर्माष्टक- मूल संक्रिया]]
[[परिकर्माष्टक- मूल संक्रिया]]
== External Links ==
* [https://archive.org/details/Patiganita Patiganita]
*[[:en:Aryabhatiya|Aryabhatiya]]
*[[:en:Shulba_Sutras|Shulba_Sutras]]


== References ==
== References ==
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