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Mahāvīra

2022-12-01T10:13:02Z

Ramamurthy S: Heading updated

Mahāvīra or Mahāvīrācārya was a 9th-century Jain mathematician born in Mysore, in Southern India. He was born in the year 815 AD<ref>{{Cite web|title=Mahāvīra|url=https://vedicmathschool.org/mahavira/}}</ref>. ''[[Gaṇitasārasan̄graha]]'' was authored by Mahāvīra. He was in the royal court of king Amoghavarṣa of Rāṣṭrakūṭa dynasty.

''Gaṇitasārasan̄graha'' has the following chapters<ref>{{Cite web|title=Gaṇitasārasan̄graha|url=https://en.wikipedia.org/wiki/Ganita_sar_sangrah}}</ref>

# '''Saṃjñādhikāraḥ''' (Terminology)
# '''Parikarmavyavahāraḥ''' (Arithmetical operations)
# '''Kalāsavarṇavyavahāraḥ''' (Fractions)
# '''Prakīrṇakavyavahāraḥ''' (Miscellaneous problems)
# '''Trairāśikavyavahāraḥ''' (Rule of three)
# '''Miśrakavyavahāraḥ''' (Mixed problems)
# '''Kṣetragaṇitavyavahāraḥ''' (Measurement of Areas)
# '''Khātavyavahāraḥ''' (calculations regarding excavations)
# '''Chāyāvyavahāraḥ''' (Calculations relating to shadows)

: Mahāvīrācārya has praised mathematics in ''Gaṇitasārasan̄graha''
: लौकिके वैदिके वापि तथा सामयिकेऽपि यः।
:व्यापारस्तत्र सर्वत्र संख्यानमुपयुज्यते॥
: Meaning : Where there is business in worldly, Vedic and contemporary, only numbers are used everywhere.
:It was Mahāvīra who first treats the series in [[Geometric progressions]] and gives almost all the formulae required therein.
:गुणसङ्कलितान्त्यधनं विगतैकपदस्य गुणधनं भवति ।
:तद्गुणगुणं मुखोनं व्येकोत्तर भाजितं सारम् ॥
:अन्त्यधन - the value of the last term. गुण - common ratio.
:The verse states that <math>S_n =\frac{ar^{n-1} X \ { r-a } }{r-1}

</math>
:<math>=\frac{a(r^n-1)}{r-1}

</math> where a is the first term and r is the common ratio and Sn is the sum to n terms.
:Mahāvīra's work differs <ref>{{Cite book|last=Gurjar|first=L V|title=Ancient Indian Mathematics and Vedha|year=1947|location=Pune|pages=102-103}}</ref>from that of others in respect of the definitions he introduced of various figures. He has given the definitions of a triangle-equilateral, isosceles and scalene-a square, a rectangle, [[isosceles trapezium]], trapezium with three sides equal, a quadrilateral, a circle, a semicircle, an ellipse, a hollow hemisphere and the lune. It is true that the results he derived regarding the area of an ellipse and the length of the curve of an ellipse are not accurate; but as a pioneer in this line his place stands high. Almost all the properties of the [[cyclic quadrilateral]] derived by [[Brahmagupta]], have been more lucidly explained by him.

== Contributions of Mahāvīra in Mathematics ==

* Separated astrology from mathematics<ref>{{Cite web |title=Mahāvīra |url=https://vedicmathschool.org/mahavira/}}</ref>
* Created the terms equilateral and isosceles triangle, rhombus , circle and semicircle
* Created formula that calculated the area and perimeters of ellipses.
* devised methods to calculate the square of a number and cube roots of a number.
* worked on the works of Āryabhaṭa and refined it.

== External Links ==

* [https://mathshistory.st-andrews.ac.uk/Biographies/Mahavira/ Mahāvīra]
*[http://www.chaturpata-atharvan-ved.com/spiritual-books-section/spiritual-books/acharya-literature/scientist-acharya-of-ancient-india/Ganit-Sara-Sangraha-MahavirAcharya-Jain-EN.pdf Ganit-Sara-Sangraha-MahavirAcharya-Jain-EN.pdf]

== See Also ==
[[महावीर]]

== References ==
<references />

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Mahāvīra

2022-12-01T09:17:58Z

Ramamurthy S: content added

Mahāvīra or Mahāvīrācārya was a 9th-century Jain mathematician born in Mysore, in Southern India. He was born in the year 815 AD<ref>{{Cite web|title=Mahāvīra|url=https://vedicmathschool.org/mahavira/}}</ref>. ''[[Gaṇitasārasan̄graha]]'' was authored by Mahāvīra. He was in the royal court of king Amoghavarṣa of Rāṣṭrakūṭa dynasty.

''Gaṇitasārasan̄graha'' has the following chapters<ref>{{Cite web|title=Gaṇitasārasan̄graha|url=https://en.wikipedia.org/wiki/Ganita_sar_sangrah}}</ref>

# '''Saṃjñādhikāraḥ''' (Terminology)
# '''Parikarmavyavahāraḥ''' (Arithmetical operations)
# '''Kalāsavarṇavyavahāraḥ''' (Fractions)
# '''Prakīrṇakavyavahāraḥ''' (Miscellaneous problems)
# '''Trairāśikavyavahāraḥ''' (Rule of three)
# '''Miśrakavyavahāraḥ''' (Mixed problems)
# '''Kṣetragaṇitavyavahāraḥ''' (Measurement of Areas)
# '''Khātavyavahāraḥ''' (calculations regarding excavations)
# '''Chāyāvyavahāraḥ''' (Calculations relating to shadows)

: Mahāvīrācārya has praised mathematics in ''Gaṇitasārasan̄graha''
: लौकिके वैदिके वापि तथा सामयिकेऽपि यः।
:व्यापारस्तत्र सर्वत्र संख्यानमुपयुज्यते॥
: Meaning : Where there is business in worldly, Vedic and contemporary, only numbers are used everywhere.
:It was Mahāvīra who first treats the series in [[Geometric progressions]] and gives almost all the formulae required therein.
:गुणसङ्कलितान्त्यधनं विगतैकपदस्य गुणधनं भवति ।
:तद्गुणगुणं मुखोनं व्येकोत्तर भाजितं सारम् ॥
:अन्त्यधन - the value of the last term. गुण - common ratio.
:The verse states that <math>S_n =\frac{ar^{n-1} X \ { r-a } }{r-1}

</math>
:<math>=\frac{a(r^n-1)}{r-1}

</math> where a is the first term and r is the common ratio and Sn is the sum to n terms.
:Mahāvīra's work differs <ref>{{Cite book|last=Gurjar|first=L V|title=Ancient Indian Mathematics and Vedha|year=1947|location=Pune|pages=102-103}}</ref>from that of others in respect of the definitions he introduced of various figures. He has given the definitions of a triangle-equilateral, isosceles and scalene-a square, a rectangle, [[isosceles trapezium]], trapezium with three sides equal, a quadrilateral, a circle, a semicircle, an ellipse, a hollow hemisphere and the lune. It is true that the results he derived regarding the area of an ellipse and the length of the curve of an ellipse are not accurate; but as a pioneer in this line his place stands high. Almost all the properties of the [[cyclic quadrilateral]] derived by [[Brahmagupta]], have been more lucidly explained by him.

== Contributions of Mahāvīra in Mathematics[edit | edit source] ==

* Separated astrology from mathematics<ref>{{Cite web |title=Mahāvīra |url=https://vedicmathschool.org/mahavira/}}</ref>
* Created the terms equilateral and isosceles triangle, rhombus , circle and semicircle
* Created formula that calculated the area and perimeters of ellipses.
* devised methods to calculate the square of a number and cube roots of a number.
* worked on the works of Āryabhaṭa and refined it.

== External Links ==

* [https://mathshistory.st-andrews.ac.uk/Biographies/Mahavira/ Mahāvīra]
*[http://www.chaturpata-atharvan-ved.com/spiritual-books-section/spiritual-books/acharya-literature/scientist-acharya-of-ancient-india/Ganit-Sara-Sangraha-MahavirAcharya-Jain-EN.pdf Ganit-Sara-Sangraha-MahavirAcharya-Jain-EN.pdf]

== See Also ==
[[महावीर]]

== References ==
<references />

[[Category:CS1 français-language sources (fr)]]
[[Category:CS1 maint]]
[[Category:CS1 Ελληνικά-language sources (el)]]
[[Category:Citation Style 1 templates|W]]
[[Category:Collapse templates]]
[[Category:Indian Mathematicians]]
[[Category:Mathematics]]
[[Category:Navigational boxes| ]]
[[Category:Navigational boxes without horizontal lists]]
[[Category:Organic Articles English]]
[[Category:Pages with script errors]]
[[Category:Sidebars with styles needing conversion]]
[[Category:Template documentation pages|Documentation/doc]]
[[Category:Templates based on the Citation/CS1 Lua module]]
[[Category:Templates generating COinS|Cite web]]
[[Category:Templates generating microformats]]
[[Category:Templates that are not mobile friendly]]
[[Category:Templates used by AutoWikiBrowser|Cite web]]
[[Category:Templates using TemplateData]]
[[Category:Wikipedia fully protected templates|Cite web]]
[[Category:Wikipedia metatemplates]]

Trairāśika (Rule of Three)

2022-11-30T08:43:58Z

Ramamurthy S: Internal links updated

==Introduction==
In the ancient Indian mathematical texts topics like ratio, proportion etc are dealt under the section rule of three. Ratio is used whenever comparison involving numbers.

For example: Cost of a bicycle is Rs. 10,000 and that of a motor bike is Rs. 1,00,000.

when we compare the cost of both the items. <math>\frac{100000}{10000} = \frac{10}{1} = 10:1</math>

Hence the cost the motorbike is ten times the cost of the bicycle. Ratio is the comparison by division. Ratio is denoted by ":" . Ratio expresses the number of times one quantity with the other. The two quantities must be in the same unit.

The two values are said to be in direct proportion when an increase/decrease in one results in an increase/decrease in the other by the same factor.<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>

Direct proportion is seen in the following instances.

#Cost of fuel increases as quantity of fuel increases
#Time taken increases with increase in pages to be typed.
#Cost of vegetable increases as weight of the vegetable increases.
#Number of units manufactured by a machine increases with the number of hours the machine works.

==Trairāśika (Rule of Three)==
[[File:Rule-of-3.png|alt=Rule of Three|thumb|Rule of Three]]
The Hindu name for the Rule of Three is called "''trairāśika''" (three terms, hence the rule of three)<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>. The term ''trairāśika'' occurs in [[Bakhshālī Manuscript|Bakshālī]] manuscript, Āryabhaṭīya. [[Bhāskara I]] (c. 525) remarked on the origin of this name as "Here three quantities are needed (in the statement and calculation) so the method is called ''trairāśika (''the rule of three terms)". A problem on the rule of three has this form : if ''p'' yields ''f'' , what will ''i'' yield ?. The three terms used are ''p,'' ''f'' , ''i'' . Hindus called the term p (''pramāṇa'' - argument), f(''phala'' - fruit) and i (''icchā -'' requisition'').'' Sometimes they are referred to simply as the first ,second and third respectively.

[[Aryabhata|Āryabhaṭa II]] gave different names as ''māna, vinimaya'' and ''icchā'' respectively to the three terms''.''

[[Brahmagupta]] gives the rule as "In the rule of three ''pramāṇa (''argument''),'' ''phala(''fruit'')'' and ''icchā(''requisition'')'' are the (given) terms; the first and the last terms must be similar. The ''icchā'' multiplied by the ''phala'' and divided by the ''pramāṇa'' gives the fruit (of the demand) ".

Bhāskara I in his Āryabhaṭīya-bhaṣya talks about the Trairāśika

त्रयो राशयः समाहृताः त्रिराशिः ।<ref>{{Cite book|last=Shukla|first=Kripa Shankar|title=Aryabhatiya of Aryabhata|publisher=The Indian National Science Academy|year=1976|page=116}}</ref> त्रिराशिः प्रयोजनमस्य गणितस्येति त्रैराशिकः । त्रैराशिके फलराशिः त्रैराशिकफलराशिः । ''(Āryabhaṭīya-bhaṣya by Bhāskara I on 11.26, p.116)''

"Trairāśi is the three quantities assembled . It is (called) Trairāśika because of this computation with these quantities. Trairāśika -phalarāśi is the desired result in the Rule of Three. "

Trairāśika involves three known quantities and one unknown quantity. The known quantities are ''pramāṇa (''known measure''),'' ''pramāṇaphala'' ''(''result related to known measure'')'' and ''icchā (''desired measure'').'' The term used for the unknown quantity is ''icchāphala (''result related to desired measure'').''

Example: A car covers 30 kms with 2 litres of petrol. To cover 150 kms, how many litres of petrol are required.?

Solution: For 30 kms, petrol needed = 2 litres

For 150 kms, petrol needed = 'x' litres

Here ''pramāṇa ='' 30 ''; pramāṇaphala ='' 2 ''; icchā ='' 150 ''; icchāphala = '''x''<nowiki/>' litres''

''pramāṇa -> pramāṇaphala'' ( 30 -> 2)

''icchā -> (icchā X pramāṇaphala) / pramāṇa = icchāphala''

150 -> ( 150 x 2) / 30 = 300/30 = 10

x= 10 ; 10 litres of petrol are required to cover 150 kms.

Solution on Trairāśika by another mathematician Śrīdhara states: "Of the three quantities, the ''pramāṇa'' ("argument") and ''icchā'' ("requisition") which are of the same denomination are the first and the last; the phala ("fruit") which is of a different denomination stands in the middle; the product of this and the last is to be divided by the first."

Example from -''Līlāvatī vs.74,p.72'' : If <math>2\frac{1}{2}</math> palas (a weight measure) of saffron costs <math>\frac{3}{7}</math> niṣkas (a unit of money), O expert businessman , tell me quickly what quantity of saffron can be bought for <math>9</math> niṣkas.

Solution:

''pramāṇa and pramāṇaphala - <math>\frac{3}{7}</math>'' niṣkas and <math>2\frac{1}{2}</math> palas

''icchā and icchāphala - <math>9</math> niṣkas and x''

As per Rule of Three - place the quantities indicated by niṣkas in first (''pramāṇa )'' and third (''pramāṇaphala)'' column. place the remaining quantity in the middle column .
{| class="wikitable"
|+
!First - quantity (''pramāṇa)''
!Middle - quantity (''pramāṇaphala)''
!Last - quantity (''icchā)''
!
|-
|''<math>\frac{3}{7}</math>''
|<math>2\frac{1}{2} = \frac{5}{2}</math>
|''<math>9</math>''
|
|}
Result = ''<math>\frac{Middle\, quantity \ X \ Last\,quantity}{First\,quantity}</math>''

''icchāphalam ='' <math>\frac{\frac{5}{2}\ X\ 9}{\frac{3}{7}}</math>= <math>\frac{5 \ X \ 9\ X\ 7}{2\ X\ 3}= \frac{105}{2}</math> palas

Hence the quantity of saffron that can be bought for <math>9</math> niṣkas is <math>52\frac{1}{2}</math> palas.

== Inverse Rule of Three ==
[[File:Rule-of-3-3.png|alt=Inverse Rule of Three|thumb|Inverse Rule of Three]]
The Hindu name for the Inverse Rule of Three is ''Vyasta''-''trairāśika'' ("inverse rule of three terms").

In ''Trairāśika'' when the ''icchā'' increases, ''icchāphala'' also increases. In ''Vyasta-trairāśika when the icchā'' increases'', icchāphala'' decreases''.''

Two values are said to vary inversely when the increase in one results in a decrease in the other. Example: If 5 men can do a work in 10 days, then 10 men can do the work in lesser number of days. When number of men increases, the number of days decreases. Hence the number of persons and the time taken are said to vary inversely with each other.

Bhāskara II defines ''Vyasta''-''trairāśika'' as "When the desired measure increases, the fruit (result related to desired measure) decreases and when the desired measure decreases, the fruit (result related to desired measure) increases "

Examples related to inverse proportion are:

* if the speed of a vehicle is more, the time taken to cover the distance will be less.
* if more customer support agents are utilized, the time taken to serve a customer will be less.
Solution on ''Vyasta''-''trairāśika'' by another mathematician Śrīdhara states: "When there is change in the unit of measurement , the middle quantity multiplied by the first quantity and divided by the last quantity gives the result"

''<math>Result =\frac{Middle\, quantity \ X\ First\,quantity }{\ Last\,quantity}</math>''

In ''Trairāśika , pramāṇa'' and ''pramāṇaphala'' vary in such a way that <math>\frac{pramanaphala}{pramana}</math> is a constant .

Hence in Rule of Three (''Trairāśika) <math>\frac{icchapalam}{iccha}=\frac {pramanaphala}{pramana}</math>''

''<math>\frac{Result \ related \ to \ desired \ measure}{Desired \ measure}=\frac {Result \ related \ to \ known \ measure }{Known \ measure }</math>''

In ''Vyasta''-''trairāśika , pramāṇa'' and ''pramāṇaphala'' vary in such a way that ''pramāṇaphala X pramāṇa'' is a constant . Hence in Inverse Rule of Three (''Vyasta''-''trairāśika) icchāphala X icchā = pramāṇaphala X pramāṇa''

i.e Result related to desired measure X Desired measure = Result related to known measure X Known measure

Example: With a measure of 7 āḍhakas, a certain quantity of grain measures 100 units. How many units will there be if the measure is 5 āḍhakas?(āḍhakas is a unit of measure of grains.)

Solution: 7 āḍhakas ⇒ 100 units

5 āḍhakas ⇒ x units
{| class="wikitable"
|+
!First quantity
!Middle quantity
!Last quantity
|-
|''pramāṇa''
|''pramāṇaphala''
|''icchā''
|-
|7
|100
|5
|}
''<math>Result =\frac{Middle\, quantity \ X\ First\,quantity }{\ Last\,quantity}</math>''

''<math>Number\ of\ Units=\frac{100 \ X \ 7 }{5} = 140</math>''

Hence the number of units for the measure of 5 āḍhakas is 140 .

== Pañca-rāśika (Rule of Five) ==
''Trairāśika'' given by Āryabhaṭa is the basis for ''Pañca-rāśika'' (Rule of Five) , ''Sapta-rāśika'' (Rule of Seven) , ''Nava-rāśika'' (Rule of Nine) and

''Ekādaśa-rāśika'' (Rule of eleven).

''Pañca-rāśika'' (Rule of Five) involves finding an unknown quantity with five known quantities.

''Sapta-rāśika'' (Rule of Seven) involves finding an unknown quantity with seven known quantities.

''Nava-rāśika'' (Rule of Nine) involves finding an unknown quantity with nine known quantities.

''Ekādaśa-rāśika'' (Rule of eleven) involves finding an unknown quantity with eleven known quantities.

These problems involves two sets of data. The first set is pramāṇa-pakṣa (known measure side) where all the quantities are given. The second set is icchā-pakṣa (desired measure side) where one quantity is to be found out.

''Trairāśika'' comes under the Rule of Odd Terms.

Śrīdhara has given the Rule of Odd Terms as "After transposing the fruit from one side to the other , and then having transposed the denominators (in like manner) and having multiplied the numbers (so obtained on either side), divide the side with larger number of quantities (numerators) by the other."

Example: if a rectangular piece of stone with length, breadth, and thickness equal to 9, 5 and 1 cubits (respectively) costs 8 , what will two other rectangular pieces of stone of dimensions 10, 7 and 2 cubits cost ?

Solution: This problem belongs to ''Nava-rāśika'' (Rule of Nine) involving nine known quantities.

pramāṇa-pakṣa (known measure side) : 1,9,5,1,8

icchā-pakṣa (desired measure side): 2,10,7,2,x
{| class="wikitable"
|+
!
!pramāṇa-pakṣa
!icchā-pakṣa
|-
|Number of stones
|1
|2
|-
|Length
|9
|10
|-
|Breadth
|5
|7
|-
|Thickness
|1
|2
|-
|Cost
|8
|x
|}
Interchange the row containing fruit (cost) as shown below.
{| class="wikitable"
!
!pramāṇa-pakṣa
!icchā-pakṣa
|-
|Number of stones
|1
|2
|-
|Length
|9
|10
|-
|Breadth
|5
|7
|-
|Thickness
|1
|2
|-
|Cost
|x
|8
|}
Divide the numbers in the side with larger number of known quantities by the numbers of the other side. Here 2nd column has large number of known quantities.

<math>x = \frac{2 \ X \ 10 \ X \ 7 \ X \ 2 \ X \ 8} {1 \ X \ 9 \ X \ 5 X \ 1} = \frac {448} {9} =49\frac{7}{9} </math>

The Cost of two rectangular pieces of stone of dimensions 10, 7, and 2 cubits is <math>49\frac{7}{9} </math>

== Simple Interest ==
In ancient Indian Mathematical works miśraka-vyavahāra dealt with the problems related to find interest, principal or time.

Interest - fee paid for a loan received.

Principal - the amount borrowed

Interest will be expressed as a percentage of the principal for a given time duration. In ancient Indian Mathematical works simple interest , not the compound interest was dealt.

Here are the samskrit terms used :
{| class="wikitable"
|+
!
!
|-
|Principal (P)
|मूलधनम्
|-
|Period (N)
|कालः
|-
|Interest (I)
|वृद्धिः
|-
|Amount (A) = Principal (P) + Interest (I)
|मूलवृद्धिधनम्
|}
Example: If a principal of 1000 rupees gets an interest of R rupees for one month , then what will be the interest received by the principal of P rupees for a period of N months.

This belongs to pañca-rāśika
{| class="wikitable"
|+
!
!pramāṇa-pakṣa
!icchā-pakṣa
|-
|Principal
|100
|P
|-
|Months
|1
|N
|-
|Interest
|R
|x
|}
'''↓'''
{| class="wikitable"
!
!pramāṇa-pakṣa
!icchā-pakṣa
|-
|Principal
|100
|P
|-
|Months
|1
|N
|-
|Interest
|x
|R
|}
Formula for Simple Interest is

<math>x = \frac{PNR}{100 \ X \ 1}=\frac{PNR}{100} </math>

Śrīdhara has stated the formula for Simple interest as "Multiply the argument (Po) by its time (No) and the other time (N) by the fruit (R) ; divide each of those (products) by their sum and multiply by the amount (A) (i.e. capital plus interest). The results give the capital and the interest (respectively)." Po PO P0

Principal <math>P = \frac{A \ X \ Po \ X \ No}{(Po \ X \ No) \ + \ (R \ X \ N)} </math>

Interest <math>I = \frac{A \ X \ R \ X \ N}{(Po \ X \ No) \ + \ (R \ X \ N)} </math>

Interest = Amount - Principal

Here
{| class="wikitable"
|+
|Po
|Standard Principal (Usually 100)
|-
|No
|Standard Period (Usually 1 month in Indian mathematical texts)
|-
|P
|Principal (Capital)
|-
|I
|Interest
|-
|A
|Amount = Principal + Interest
|-
|N
|Period (Time)
|-
|R
|Rate of interest (Fruit) or interest on Po for period No
|}
If Po = 100 and No= 1 month

<math>P = \frac{A \ X \ 100 \ X \ 1}{(100 \ X \ 1) \ + \ (R \ X \ N)} = \frac{100 \ X \ A}{100 \ + \ (R \ X \ N)} </math>

Example: If 1½ units is the interest on 100½ units for one third of a month, what will be the interest on 60¼ units for 7½ months?

Solution :

This belongs to pañca-rāśika

pramāṇa-pakṣa (known measure side) : 100½ units , ⅓ months , 1½ interest . converting this mixed fraction to improper fraction.

<math>\frac{201}{2}</math> units , <math>\frac{1}{3}</math> months , <math>\frac{3}{2}</math> interest

icchā-pakṣa (desired measure side): 60¼ units , 7½ months , x interest

<math>\frac{241}{4}</math> units , <math>\frac{15}{2}</math> months , <math>\frac{x}{1}</math> interest
{| class="wikitable"
!
!pramāṇa-pakṣa
!icchā-pakṣa
|-
|Principal
|201
2
|241
4
|-
|Months
|1
3
|15
2
|-
|Interest
|3
2
|x
1
|}
↓ Interchange the row containing Interest (fruit) as shown below.
{| class="wikitable"
!
!pramāṇa-pakṣa
!icchā-pakṣa
|-
|Principal
|201
2
|241
4
|-
|Months
|1
3
|15
2
|-
|Interest
|x
1
|3
2
|}
↓ Interchange the denominators as shown below. this is required for the terms which are fractions.
{| class="wikitable"
!
!pramāṇa-pakṣa
!icchā-pakṣa
|-
|Principal
|201
4
|241
2
|-
|Months
|1
2
|15
3
|-
|Interest
|x
2
|3
1
|}
Divide the 2nd column (large number of known quantities) by 1st column (numbers of the other side).

<math>x = \frac{ 241 \ X \ 2 \ X \ 15 \ X \ 3 \ X \ 3 \ X \ 1} {201 \ X \ 4 \ X \ 1 \ X \ 2 \ X \ 2 } = \frac{10845}{536}= 20 \frac{125}{536} </math>

Hence Interest on 60¼ units , 7½ months = <math>20 \frac{125}{536} </math>

====== Amount becoming 'n' times the principal : ======
[[Sridhara|Śrīdhara]] has stated the formula to find when the principal will double or triple or quadruple after 'N' months at R% per month.

कालप्रमाणघातः फलभक्तो व्येकगुणहतः कालः ।<ref>{{Cite book|last=Shukla|first=Kripa Shankar|title=The Patiganita Of Sridharacharya|publisher=Lucknow University|year=1959|location=Lucknow|pages=60}}</ref> (Pāṭīgaṇita III R.52, p.60)

"The product of the time and the argument divided by the fruit and (then) multiplied by the multiple minus one, gives the required time."

Here time is standard time, argument is standard principal and fruit is rate of interest.

The formula will be

<math>Time \ N = \frac{Standard \ principal \ X \ Standard \ time \ X \ (n -1)}{R} </math>

Here Standard Principal = 100 ; Standard time = 1 month ; Rate of Interest = R

Example: If 6 ''drammas'' is the interest in 200 (''drammas'') per month, when will the sum be three times?

Solution:

Given: P = 200 ''drammas'', N = 1 month, I = 6 ''drammas''

<math>I = \frac{P \ X \ N \ X \ R}{100}</math>

<math>6 = \frac{200 \ X \ 1 \ X \ R}{100}</math>

R= 3%

To calculate the period in which the sum becomes three times the principal

<math>Time \ N = \frac{Standard \ principal \ X \ Standard \ time \ X \ (n -1)}{R} </math>

Here Standard principal = 100 ; Standard tine = 1 month ; n = 3 times

<math>Time \ N = \frac{(100 \ X \ 1) \ X \ (3 - 1)}{3} = \frac{200}{3} = 66 \frac{2}{3}
</math> months

Hence the sum becomes three time after <math>66\frac{2}{3} </math> months i.e 5 years <math>6\frac{2}{3} </math>months

== See Also ==
[[त्रैराशिक (तीन का नियम)]]

== External Links ==

* [https://www.matematicas18.com/en/tutorials/arithmetic/rule-of-three/ Rule-of-three]
* [http://www.mathspadilla.com/2ESO/Unit4-ProportionalityAndPercentages/rules_of_three.html Invers_rule_of_three.html]

== References ==
<references />

[[Category:Arithmetic]]
[[Category:Mathematics]]
[[Category:Organic Articles English]]
[[Category:Pages with broken file links]]

Parikarmāṣṭaka - Fundamental Operations

2022-11-30T08:40:12Z

Ramamurthy S: Internal links updated

== Introduction ==
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 ==
''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 :

# Saṅkalanam (addition)
# Vyavakalanam (subtraction)
# Guṇana (multiplication)
# Bhājana (division)
# Varga (squaring)
# Varga-mūla (square root)
# Ghana (cubing) and
# Gana-mūla (cube root)

Addition and subtraction form the basis of all calculations. Bhāskara I mentions in the below shloka

संयोगभेदा गुणनागतानि शुद्धेश्च भागो गतमूलमुक्तम् ।

व्याप्तं समीक्ष्योपचयक्षयाभ्यां विद्यादिदं द्व्यात्मकमेव शास्त्रम् ॥ (Āryabhaṭīya-bhāṣya in Gaṇitapāda, p.43)

"All arithmetical operations resolve into two categories though usually considered to be four. The two main categories are increase and decrease. Addition is increase and subtraction is decrease. These two varieties of operations permeate the whole of mathematics. Multiplication and evolution ( square etc,) are particular kinds of addition; and division and involution( square root etc) are particular kinds of subtraction. Indeed every mathematical operation will be recognised to consist of increase or decrease. Hence the whole of this science should be known as consisting truly of these two only."<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>

== Saṅkalana and Vyavakalana (Addition and Subtraction) ==
[[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.

[[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) ".

[[Bhāskara II]] has mentioned about these operations in his work on Līlavatī.

कार्यः क्रमादुत्क्रमतोऽथवाऽङ्कयोगो यथास्थानकमन्तरं वा ॥ (Līlavatī , vs.12, p.12)

"The addition or subtraction (of digits in the given numbers) is to be done place wise right to left or left to right."

Write the given numbers one below the other so that the digits are aligned to their place value. Then starting with the units place add or subtract the digits, later move to tens and so on.

Samskrita names for Addition - ''yoga'' (addition), ''saṃyoga'' (sum), ''saṃyojana'' (joining together) , ''saṃyuti'' (sum), ''saṃyuti'' (sum), ''saṇkalana'' (making together).

Samskrita names for Subtraction - ''vyutkalita''(made apart), ''vyutkalana''(making apart), ''śodhana'' (clearing), ''pātana'' (causing to fall), ''viyoga'' (separation) , ''śeṣa'' (residue) and ''anatara'' (difference) have been used for the remainder.

== Guṇana (Multiplication) ==
The multiplication of whole numbers is repeated addition. For example :

<math>2\quad X\quad 4 = 2+2+2+2 = 8</math>

Samskrita names for Multiplication - āhati (multiplication), ghāta (product), [guṇana , hanana, hati, vadha] (multiplication).
{| class="wikitable"
|+
|2
|X
|4
|=
|8
|-
|↑
|
|↑
|
|↑
|-
|guṇya
(multiplicand)
|
|guṇaka
(multiplier)
|
|guṇana-phala
(Result of Multiplication)
|}

=== Methods of Multiplication : ===

* Rūpa-guṇana - Direct Method
* Khaṇḍa-guṇana - Split Method
* Bhakta-guṇana - Factor Method
* Sthāna-vibhāga-guṇana - Place wise multiplication
* Iṣṭonayug-guṇana (Adding or subtracting a desired number)

==== Rūpa-guṇana - Direct method : ====
Here the tables of multiplier should be known. Multiplier is taken as a whole.Each digit of multiplicand is multiplied by the multiplier to get the product. In this method, the multiplier is taken a whole since it is small.

Example: 234 X 5 =

(1) (2)

2 3 4

x 5 =

1 1 7 0

==== Khaṇḍa-guṇana - Split Method : ====
Here the multiplier is split into sum of two numbers . This is represented as below.

a X b = a X (c + d) = (a X c) + (a X d) where b = c + d.

This is the distributive property of multiplication over addition.

Example: 234 X 16 = 234 X (10 + 6 ) = (234 X 10) + (234 X 6) = 2340 + 1404 = 3744

==== Bhakta-guṇana - Factor Method : ====
Here the multiplier is split into product of two numbers. This is represented as below.

a X b = a X (c X d) = (a X c) X d where b = c X d

Example: 234 X 16 = 234 X (8 X 2) = (234 X 8) X 2 = 1872 X 2 = 3744

==== 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.

Example: 234 X 16

2 3 4

X 1 6 =

1 4 0 4

+ 2 3 4 =

3 7 4 4

==== Iṣṭonayug-guṇana (Adding or subtracting a desired number) : ====
The Samskrit word Iṣṭonayug is a compound word consisting of ''iṣṭa , ūna , yuk'' which means respectively 'desired, minus and plus'.

इष्टोनयुक्तेन गुणेन निघ्नोऽभीष्टघ्नगुण्यान्वितवर्जितो वा । (Līlāvatī, vs.16, p.15)

"Add or subtract any convenient number to the multiplier and multiply it. Then multiply by the added or subtracted number and subtract or add this product from the previous one."

Add any desired number to the multiplier to get a convenient round figure. Then multiply the multiplicand with the round figure and the added number. Then subtract the products to get the final answer.

or

Subtract any desired number from the multiplier to get a convenient round figure. Then multiply the multiplicand with the round figure and the subtracted number. Then add the products to get the final answer.

Example:

234 X 16 = 234 X (20 - 4) = (234 X 20) - (234 X 4) = 4680 - 936 = 3744

234 X 16 = 234 X (10 + 6) = (234 x 10) + (234 x 6) = 2340 + 1404 = 3744

==== 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.

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.".

This method was known to the Hindu scholars of the 8th century, or earlier. The method seems to have travelled to Arabia and thence was transmitted to Europe, where it occurs in Pacioli's ''Suma'' and is stated to be "more fantastic and ingenious than the others."

Gaṇeśa has also remarked that "this (method) is very fantastic and cannot be learnt by the dull without the traditional oral instructions."

'''Example''' : Multiply 234 and 15

2 3 5

0 1 5 X
{| class="wikitable"
|+
|Hundreds
|Tens
|Unit
|-
|2
|3
|4
|-
|0
|1
|5
|}

# Multiply unit digit by unit digit. 4 X 5 = 20
# Multiply the unit digit by tens digit and tens digit by unit digit and add them. (3 X 5) + (4 X 1) = 15 + 4 = 19
# Multiply unit digit by hundreds digit, hundreds digit by unit digit and tens digits by tens digit and add them. (2 x 5) + (4 X 0) + (3 X 1) = 10 + 0 + 3 = 13
# Multiply hundreds digit by tens digit and tens digit by hundreds digit and add them. (2 X 1) + (3 X 0) = 2 + 0 = 2 = 02
# Multiply hundred digit by hundreds digit. 2 X 0 = 0 = 00
# Place the results of each step as shown and add.
{| class="wikitable"
|+
|1.
|
|
|
|
|2
|0
|-
|2.
|
|
|
|1
|9
|
|-
|3.
|
|
|1
|3
|
|
|-
|4.
|
|0
|2
|
|
|
|-
|5.
|0
|0
|
|
|
|
|-
|
|'''0'''
|'''0'''
|'''3'''
|'''5'''
|'''1'''
|'''0'''
|}
The result is 3510.

== Bhājana (Division) ==
[[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).

The dividend is termed as ''bhājya'' or ''hārya'', the divisor is called ''bhājaka'', ''bhāgahara'' or ''hara.'' The quotient is called ''labhdi'' (the obtained) or ''labdha'' .Bhāskara II has mentioned the rule for division as:

भाज्याद्धरः शुद्ध्यति यद्गुणः स्यादन्त्यात्फलं तत्खलु भागहारे। समेन केनाप्यपवर्त्य हारभाज्यौ भवेद्वा सति सम्भवे तु ॥ (Līlāvatī, vs.18,p.18)

"Starting from the last digit of the divided, the (maximum) number of times by which divisor can be subtracted,that indeed is the quotient (result of division). If possible, divide after cancelling some common factor in the divisor and the dividend."

Bhāskara II mentioned along with the regular method of division, he has described the method of removing the common factors of the divisor and dividend to obtain the result.

Example <math>\frac{748}{108} = \frac{748/4}{108/4} = \frac{187}{27} = \frac{Bhajya }{Bhajaka}= Labdhi(6)</math>

== Varga (Square) ==
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:

"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

After step 4.1 add the numbers on each column. wherever there are two digits. unit digit to be retained. digit at tenth place to be carried to next column on the left side and added . Here also unit digit to be retained. digit at tenth place to be carried to next column on the left side and added ...... so on.
{| class="wikitable"
|+
!
!
!40
!7
!9
!3
!7
!6
!9
|-
!Step
!
!39
!15
!27
!23
!7
!6
!9
|-
|4.1
|72
|
|
|
|
|
|4
|9
|-
|3.2
|2 x 8 x 7
|
|
|
|1
|1
|2
|
|-
|3.1
|82
|
|
|
|6
|4
|
|
|-
|2.3
|2 x 3 x 7
|
|
|
|4
|2
|
|
|-
|2.2
|2 x 3 x 8
|
|
|4
|8
|
|
|
|-
|2.1
|32
|
|
|9
|
|
|
|
|-
|1.4
|2 x 6 x 7
|
|
|8
|4
|
|
|
|-
|1.3
|2 x 6 x 8
|
|9
|6
|
|
|
|
|-
|1.2
|2 x 6 x 3
|3
|6
|
|
|
|
|
|-
|1.1
|62
|36
|
|
|
|
|
|
|-
|1
|Given Number
|6
|3
|8
|7
|
|
|
|-
|2
|Shift Number to the right
|
|<s>6</s>
|3
|8
|7
|
|
|-
|3
|Shift Number to the right
|
|
|<s>6</s>
|<s>3</s>
|8
|7
|
|-
|4
|Shift Number to the right
|
|
|
|<s>6</s>
|<s>3</s>
|<s>8</s>
|7
|}

== Varga-mūla (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 "

Example : Square root of 956484 = 978
{| class="wikitable"
|+
| colspan="2" rowspan="2" |
|'''Avarga'''
|'''Varga'''
|'''Avarga'''
|'''Varga'''
|'''Avarga'''
|'''Varga'''
| rowspan="2" |
|-
|9
|5
|6
|4
|8
|4
|-
|Subtract by the square = '''9'''2
|
|8
|1
| colspan="4" |
|Root = '''9'''
|-
|Divide by twice the root = 2 x 9 =18
|18
|1
|4
|6
|'''7'''
| colspan="2" rowspan="4" |
|Root = 9'''7'''
|-
| rowspan="2" |
| rowspan="3" |
|1
|2
|6
|
| rowspan="3" |
|-
| rowspan="2" |
|2
|0
|4
|-
|Subtract by square of the quotient = 72 = 49
|
|4
|9
|-
|Divide by twice the root = 2 x 97 = 194
|194
| rowspan="3" |
|1
|5
|5
|8
|'''8'''
|Root = 97'''8'''
|-
| rowspan="2" |
| rowspan="2" |
|1
|5
|5
|2
| colspan="2" |
|-
| colspan="3" |
|6
|4
| rowspan="3" |
|-
|Subtract by square of the quotient = 82 = 64
|64
| colspan="4" |
|6
|4
|-
| colspan="7" |
|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 12) muliplied by the succeding

digit (2) is 2 x 3 x 12 and placing at the next place
|
|
|
|6
|
|
|-
|Thrice the square of the succeding digit (2)multiplied

by the last digit is 3 x 22 x 1 and placing at the next place
|
|
|
|1
|2
|
|-
|Cube of succeeding digit (23)
|
|
|
|
|
|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 122) muliplied by the succeding

digit (3) is 3 x 3 x 122 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 32 x 12 and placing at the next place
|
|
|
|3
|2
|4
|
|-
|Cube of succeeding digit (33)
|
|
|
|
|
|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 1232) muliplied by the succeding

digit (4) is 4 x 3 x 1232 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 42 x 123 and placing at the next place
|
|
|
|
|
|5
|9
|0
|4
|
|-
|Cube of succeeding digit (43)
|
|
|
|
|
|
|
|
|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 13
|
|1
|
|
| colspan="4" rowspan="5" |
|1
|-
|Divide by 3 x 12
|''3''
|1
|6
|''3 - Quotient''
|13
|-
|
| rowspan="5" |
| rowspan="4" |
|9
|
| rowspan="5" |
|-
|
|7
|2
|-
|Subtract 3 x 1 x 32
|2
|7
|-
|
|4
|5
|8
| colspan="3" rowspan="2" |
|-
|Subtract 33
| rowspan="7" |
|
|2
|7
|-
|Divide by 3 x 132
|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 82
|2
|4
|9
|6
|-
|
| colspan="2" rowspan="2" |
|5
|1
|2
|-
|Subtract 83
|5
|1
|2
|-
| colspan="8" |
|0
| colspan="2" |
|}

== See also ==
[[परिकर्माष्टक- मूल संक्रिया]]

== External Links ==

* [https://archive.org/details/Patiganita Patiganita]
*[[:en:Aryabhatiya|Aryabhatiya]]
*[[:en:Shulba_Sutras|Shulba_Sutras]]

== References ==
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Bhāskara II

2022-11-30T08:34:07Z

Ramamurthy S: Internal links updated

{{Infobox person
| name = Bhāskara II
| birth_date = c 1114 AD
| death_date = c 1185 AD
| era = Shaka era
| notable_works = Siddhānta Shiromani (Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya), Karaṇa-Kautūhala
}}

Bhāskara II (c. 1114–1185)<ref>{{Cite web|title=Bhāskara_II|url=https://en.wikipedia.org/wiki/Bhāskara_II}}</ref>, also known as '''Bhāskarāchārya''' and as Bhāskara II to avoid confusion with [[Bhāskara I]], was an Indian mathematician and astronomer. His main work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided into four parts called Līlāvatī, [[Algebra|Bījagaṇita]], Grahagaṇita and Golādhyāya which are also sometimes considered four independent works. These four sections deal with arithmetic, [[algebra]], mathematics of the planets, and spheres respectively. He also wrote another treatise named Karaṇā Kautūhala.

Some of Bhāskara's contributions to mathematics include the following:

[[File:Bhaskaracharya proof of pythagorean Theorem.png|alt=Bhaskaracharya proof of Pythagorean Theorem|thumb|Bhaskaracharya proof of Pythagorean Theorem]]

* A proof of the Pythagorean theorem by calculating the same area in two different ways and then cancelling out terms to get ''a''2 + ''b''2 = ''c''2.

* In ''Līlāvatī'', solutions of quadratic, cubic and quartic indeterminate equations are explained.

[[Līlāvatī]] (meaning a beautiful woman) is based on Arithmetic<ref>{{Cite web|title=Bhāskara II|url=https://www.booksfact.com/science/ancient-science/bhaskaracharya-greatest-mathematician-introduced-concept-infinity.html}}</ref>. It is believed that Bhāskara named this book after his daughter Līlāvatī. Many of the problems in this book are addressed to his daughter. For example “Oh Līlāvatī, intelligent girl, if you understand addition & subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10 & 100, as well as [the remainder of] those when subtracted from 10000.” The book contains thirteen chapters, mainly definitions, arithmetical terms, interest computation, arithmetical & geometric progressions. Many of the methods in the book on computing numbers such as multiplications, squares & progressions were based on common objects like kings & elephants, which a common man could understand.
* Solutions of [[Indeterminate Quadratic Equation|indeterminate quadratic equations]] (of the type ''ax''2 + ''b'' = ''y''2).<ref>{{Cite web|title=Bhāskara II|url=https://www.newworldencyclopedia.org/entry/Bh%C4%81skara_II}}</ref>
* The first general method for finding the solutions of the problem ''x''2 − ''ny''2 = 1 (so-called "Pell's equation") was given by Bhāskara II.
* Preliminary concept of mathematical analysis.
* Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.
* Calculated the derivatives of trigonometric functions and formulae.
* In ''Siddhānta-Śiromaṇi'', Bhāskara developed spherical trigonometry along with a number of other trigonometric results<ref>{{Cite web|title=Bhaskara’s knowledge of trigonometry|url=https://speak2world.wordpress.com/2014/10/13/bhaskaras-knowledge-of-trigonometry/}}</ref>.
The [[Siddhānta Śiromaṇi|''Siddhānta'' ''Śiromaṇi'']] (written in 1150) demonstrates Bhāskara 's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed spherical trigonometry, along with other interesting trigonometrical results. In particular Bhāskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhāskara , results found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for sin(a+b) and sin(a-b).
== See Also ==

* [[भास्कर द्वितीय]]

== External Links ==

* [https://mathshistory.st-andrews.ac.uk/Biographies/Bhaskara_II/ Bhāskara_II]
* [https://web.archive.org/web/20110707064659/http://www.4to40.com/legends/index.asp?p=Bhaskara Biography]

== References ==

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Āryabhaṭa

2022-11-30T08:32:08Z

Ramamurthy S: Internal links updated

Āryabhaṭa (476–550 CE) <ref>{{Cite web|title=Āryabhaṭa|url=https://en.wikipedia.org/wiki/Aryabhata}}</ref>was born in Pataliputra (Patna). He was an Indian mathematician and astronomer of the classical age

{{Infobox person
| name = Āryabhaṭa
| image = [[File:2064_aryabhata-crp.jpg|150px]]
| birth_date = 476 CE
| birth_place = Kusumapura (Pataliputra)
| death_date = 550 CE
| death_place = Pataliputra
| era = Gupta Era
| notable_works = Āryabhaṭīya, Arya-siddhanta
}}

of Indian mathematics and Indian astronomy. He flourished in the Gupta Era<ref>{{Cite web|title=Achievements of the Gupta Empire|url=https://www.studentsofhistory.com/the-gupta-empire}}</ref> and produced works such as the [[Āryabhaṭīya]] (which mentions that in 3600 Kali Yuga, 499 CE, he was 23 years old) and the Ārya-siddhānta<ref>{{Cite web|title=Āryabhaṭa|url=https://www.newworldencyclopedia.org/entry/Aryabhata}}</ref>.

Āryabhaṭīya deals with both mathematics and astronomy. It contains 121 stanzas and the subject matter is divided into 4 chapters, called Pāda (section).

Pāda -1 (Gītikā-pāda):Consists of 13 stanzas sets forth the basic definitions and important astronomical parameters and tables. It gives the definitions of

- [[Kalpa]], Manu and Yuga which are the larger units of time

- Sign, degree and minute which are the circular units

- Linear units yojana, hasta, aṅgula

Pāda - 2 (Gaṇita-pāda) :Consists of 33 stanzas talks about Mathematics. The topics covered are Geometrical figures, their properties and mensuration ; problems on the shadow of the gnomon ; simple, simulatenous, quadratic and linear indeterminate equations. Methods to extract square root and cube root.

Pāda - 3 (Kālakriyā-pāda) :Consists of 25 stanzas dealing with various unit of time and the determination of true positions of the Sun, Moon and the planets. Methods to compute the true longitudes of the Sun, Moon and the planets.

Pāda - 4 (Gola-pāda) :Consists of 50 stanzas dealing with motion of Sun, Moon and the planets on the celestial sphere. Calculation and graphical representation of the eclipses and visibility of the planets.

Āryabhaṭīya is generally supposed to be a collection of two compositions<ref>{{Cite book|last=Shukla|first=Kripa Shankar|title=Āryabhaṭīya of Āryabhaṭa|publisher=The Indian National Science Academy|year=1976|location=New Delhi|page=XXV}}</ref> : 1.Daśagītikā-sūtra: consists of pāda -1 stating the astronomical parameters in 10 stanzas in [[gītikā]] metre and 2.Āryāṣṭaśata : consist of second, third and fourth pādas having 108 stanzas in [[āryā]] metre).

Here are the notable features of Āryabhaṭīya :

# Alphabetical system of numeral notation defined by Āryabhaṭa is different from [[Systems of Numerations|Kaṭapayādi]] system but much more effective in expressing number briefly in verse.
# Circumference to diameter ratio π = 3.1416.
# The table of sine differences
#Solution of indeterminate equations
#Theory of Earth's rotation
#The astronomical parameters
#Time and divisions of time
#Theory of planetary motion
#Celestial latitudes of the planets
#Use of the radian measure in minutes
The commentary on Āryabhaṭīya was written by [[Bhāskara I]] , Prabhākara, Someśvara, Sūryadeva , Parameśvara, Nīlakaṇṭha Somayāji, Mādhava.

== See Also ==
[[आर्यभट्ट]]

== External Links ==

* [https://mathshistory.st-andrews.ac.uk/Biographies/category-indians/ Ancient Indian mathematics - Biographies]
* [https://www.storyofmathematics.com/indian.html/ Indian Mathematics and Mathematicians]

== References ==

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