Supersymmetric quantum mechanics: Difference between revisions

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Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Jain mathematician from Mysore, India.Template:Sfn Template:Sfn Template:Sfn He was the author of Gaṇitasārasan̄graha (or Ganita Sara Samgraha, c. 850), which revised the Brāhmasphuṭasiddhānta.Template:Sfn He was patronised by the Rashtrakuta king Amoghavarsha.Template:Sfn He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics.^[1] He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. There are several points worth noting about the work of Mahavira. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems.^[2] He is highly respected among Indian mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle.^[3] Mahāvīra's eminence spread in all South India and his books proved inspirational to other mathematicians in Southern India.Template:Sfn It was translated into Telugu language by Pavuluri Mallana as Saar Sangraha Ganitam.^[4]

He discovered algebraic identities like a³=a(a+b)(a-b) +b²(a-b) + b².Template:Sfn He also found out the formula for ⁿC_r as [n(n-1)(n-2)...(n-r+1)]/r(r-1)(r-2)...2*1.Template:Sfn He devised formula which approximated area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number.Template:Sfn He asserted that the square root of a negative number did not exist.Template:Sfn

Rules for decomposing fractions

Mahāvīra's Gaṇita-sāra-saṅgraha gave systematic rules for expressing a fraction as the sum of unit fractions.^[5] This follows the use of unit fractions in Indian mathematics in the Vedic period, and the Śulba Sūtras' giving an approximation of √2 equivalent to $1 + \frac{1}{3} + \frac{1}{3 \cdot 4} - \frac{1}{3 \cdot 4 \cdot 34}$ .^[5]

In the Gaṇita-sāra-saṅgraha (GSS), the second section of the chapter on arithmetic is named kalā-savarṇa-vyavahāra (lit. "the operation of the reduction of fractions"). In this, the bhāgajāti section (verses 55–98) gives rules for the following:^[5]

To express 1 as the sum of n unit fractions (GSS kalāsavarṇa 75, examples in 76):^[5]

31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape. 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.

1 = \frac{1}{1 \cdot 2} + \frac{1}{3} + \frac{1}{3^{2}} + \dots + \frac{1}{3^{n - 2}} + \frac{1}{\frac{2}{3} \cdot 3^{n - 1}}

To express 1 as the sum of an odd number of unit fractions (GSS kalāsavarṇa 77):^[5]

1 = \frac{1}{2 \cdot 3 \cdot 1 / 2} + \frac{1}{3 \cdot 4 \cdot 1 / 2} + \dots + \frac{1}{(2 n - 1) \cdot 2 n \cdot 1 / 2} + \frac{1}{2 n \cdot 1 / 2}

To express a unit fraction $1 / q$ as the sum of n other fractions with given numerators $a_{1}, a_{2}, \dots, a_{n}$ (GSS kalāsavarṇa 78, examples in 79):

\frac{1}{q} = \frac{a_{1}}{q (q + a_{1})} + \frac{a_{2}}{(q + a_{1}) (q + a_{1} + a_{2})} + \dots + \frac{a_{n - 1}}{q + a_{1} + \dots + a_{n - 2}) (q + a_{1} + \dots + a_{n - 1})} + \frac{a_{n}}{a_{n} (q + a_{1} + \dots + a_{n - 1})}

To express any fraction $p / q$ as a sum of unit fractions (GSS kalāsavarṇa 80, examples in 81):^[5]

Choose an integer i such that

\frac{q + i}{p}

is an integer r, then write

\frac{p}{q} = \frac{1}{r} + \frac{i}{r \cdot q}

and repeat the process for the second term, recursively. (Note that if i is always chosen to be the smallest such integer, this is identical to the greedy algorithm for Egyptian fractions.)

To express a unit fraction as the sum of two other unit fractions (GSS kalāsavarṇa 85, example in 86):^[5]

\frac{1}{n} = \frac{1}{p \cdot n} + \frac{1}{\frac{p \cdot n}{n - 1}}

where

p

is to be chosen such that

\frac{p \cdot n}{n - 1}

is an integer (for which

p

must be a multiple of

n - 1

).

\frac{1}{a \cdot b} = \frac{1}{a (a + b)} + \frac{1}{b (a + b)}

To express a fraction $p / q$ as the sum of two other fractions with given numerators $a$ and $b$ (GSS kalāsavarṇa 87, example in 88):^[5]

\frac{p}{q} = \frac{a}{\frac{a i + b}{p} \cdot \frac{q}{i}} + \frac{b}{\frac{a i + b}{p} \cdot \frac{q}{i} \cdot i}

where

i

is to be chosen such that

p

divides

a i + b

Some further rules were given in the Gaṇita-kaumudi of Nārāyaṇa in the 14th century.^[5]

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

Bibhutibhusan Datta and Avadhesh Narayan Singh (1962). History of Hindu mathematics: a source book.
Template:DSB
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Template:MacTutor Biography
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Template:Indian mathematics

↑ The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88
↑ Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43
↑ Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122
↑ Census of the Exact Sciences in Sanskrit by David Pingree: page 388
↑ ^5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 ^5.6 ^5.7 ^5.8 Template:Harvnb

[1] The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88

[2] Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43

[3] Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122

[4] Census of the Exact Sciences in Sanskrit by David Pingree: page 388

[k497-5] 5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 ^5.6 ^5.7 ^5.8 Template:Harvnb

[1]

[2]

[3]

[4]

[5]

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+'''Mahāvīra''' (or '''Mahaviracharya''', "Mahavira the Teacher") was a 9th-century [[Jain]] [[mathematician]] from [[Mysore]], [[India]].{{sfn|Pingree|1970}}{{sfn|O'Connor|Robertson|2000}}{{sfn|Tabak|2009|p=42}} He was the author of ''Gaṇitasārasan̄graha'' (or ''Ganita Sara Samgraha'', c. 850), which revised the [[Brāhmasphuṭasiddhānta]].{{sfn|Pingree|1970}} He was patronised by the [[Rashtrakuta]] king [[Amoghavarsha]].{{sfn|Puttaswamy|2012|p=231}} He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics.<ref>The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ...  by Clifford A. Pickover: page 88</ref> He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. There are several points worth noting about the work of Mahavira. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems.<ref>Algebra: Sets, Symbols, and the Language of Thought  by John Tabak: p.43</ref>  He is highly respected among Indian mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle.<ref>Geometry in Ancient and Medieval India  by T. A. Sarasvati Amma: page 122</ref> Mahāvīra's eminence spread in all South India and his books proved inspirational to other mathematicians in [[Southern India]].{{sfn|Hayashi|2013}} It was translated into [[Telugu language]] by [[Pavuluri Mallana]] as ''Saar Sangraha Ganitam''.<ref>Census of the Exact Sciences in Sanskrit  by David Pingree: page 388</ref>
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+He discovered algebraic identities like a<sup>3</sup>=a(a+b)(a-b) +b<sup>2</sup>(a-b) + b<sup>2</sup>.{{sfn|Tabak|2009|p=42}} He also found out the formula for <sup>n</sup>C<sub>r</sub> as [n(n-1)(n-2)...(n-r+1)]/r(r-1)(r-2)...2*1.{{sfn|Tabak|2009|p=43}} He devised formula which approximated area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number.{{sfn|Krebs|2004|p=132}} He asserted that the [[square root]] of a [[negative number]] did not exist.{{sfn|Selin|2008|p=1268}}
+== Rules for decomposing fractions ==
+Mahāvīra's ''Gaṇita-sāra-saṅgraha'' gave systematic rules for expressing a fraction as the [[Egyptian fraction|sum of unit fractions]].<ref name=k497>{{Harvnb|Kusuba|2004|pp=497–516}}</ref> This follows the use of unit fractions in [[Indian mathematics]] in the Vedic period, and the [[Shulba Sutras|Śulba Sūtras]]' giving an approximation of √2 equivalent to <math>1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}</math>.<ref name=k497/>
+In the ''Gaṇita-sāra-saṅgraha'' (GSS), the second section of the chapter on arithmetic is named ''kalā-savarṇa-vyavahāra'' (lit. "the operation of the reduction of fractions"). In this, the ''bhāgajāti'' section (verses 55–98) gives rules for the following:<ref name=k497/>
+* To express 1 as the sum of ''n'' unit fractions (GSS ''kalāsavarṇa'' 75, examples in 76):<ref name=k497/>
+{{quote|rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ /<br/>
+dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //}}
+{{quote|When the result is one, the denominators of the quantities having one as numerators are [the numbers] beginning with one and multiplied by three, in order. The first and the last are multiplied by two and two-thirds [respectively].}}
+:: <math> 1 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}} </math>
+* To express 1 as the sum of an odd number of unit fractions (GSS ''kalāsavarṇa'' 77):<ref name=k497/>
+:: <math>1 = \frac1{2\cdot 3 \cdot 1/2} + \frac1{3 \cdot 4 \cdot 1/2} + \dots + \frac1{(2n-1) \cdot 2n \cdot 1/2} + \frac1{2n \cdot 1/2} </math>
+* To express a unit fraction <math>1/q</math> as the sum of ''n'' other fractions with given numerators <math>a_1, a_2, \dots, a_n</math> (GSS ''kalāsavarṇa'' 78, examples in 79):
+:: <math>\frac1q = \frac{a_1}{q(q+a_1)} + \frac{a_2}{(q+a_1)(q+a_1+a_2)} + \dots + \frac{a_{n-1}}{q+a_1+\dots+a_{n-2})(q+a_1+\dots+a_{n-1})} + \frac{a_n}{a_n(q+a_1+\dots+a_{n-1})}</math>
+* To express any fraction <math>p/q</math> as a sum of unit fractions (GSS ''kalāsavarṇa'' 80, examples in 81):<ref name=k497/>
+: Choose an integer ''i'' such that <math>\tfrac{q+i}{p}</math> is an integer ''r'', then write
+:: <math> \frac{p}{q} = \frac{1}{r} + \frac{i}{r \cdot q} </math>
+: and repeat the process for the second term, recursively. (Note that if ''i'' is always chosen to be the ''smallest'' such integer, this is identical to the [[greedy algorithm for Egyptian fractions]].)
+* To express a unit fraction as the sum of two other unit fractions (GSS ''kalāsavarṇa'' 85, example in 86):<ref name=k497/>
+:: <math>\frac1{n} = \frac1{p\cdot n} + \frac1{\frac{p\cdot n}{n-1}}</math> where <math>p</math> is to be chosen such that <math>\frac{p\cdot n}{n-1}</math> is an integer (for which <math>p</math> must be a multiple of <math>n-1</math>).
+:: <math>\frac1{a\cdot b} = \frac1{a(a+b)} + \frac1{b(a+b)}</math>
+* To express a fraction <math>p/q</math> as the sum of two other fractions with given numerators <math>a</math> and <math>b</math> (GSS ''kalāsavarṇa'' 87, example in 88):<ref name=k497/>
+:: <math>\frac{p}{q} = \frac{a}{\frac{ai+b}{p}\cdot\frac{q}{i}} + \frac{b}{\frac{ai+b}{p} \cdot \frac{q}{i} \cdot{i}}</math> where <math>i</math> is to be chosen such that <math>p</math> divides <math>ai + b</math>
+Some further rules were given in the ''Gaṇita-kaumudi'' of [[Narayana Pandit|Nārāyaṇa]] in the 14th century.<ref name=k497/>
+==Notes==
+{{reflist|2}}
+==References==
+*Bibhutibhusan Datta and Avadhesh Narayan Singh (1962). ''History of Hindu mathematics: a source book''.
+*{{DSB|first=David|last=Pingree|authorlink=David Pingree|title=Mahāvīra|url=http://www.encyclopedia.com/doc/1G2-2830902766.html|date=1970|ref=harv}}
+*{{citation|last=Selin|first=Helaine|title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures|url=http://books.google.com/books?id=kt9DIY1g9HYC&pg=PA1268|year=2008|publisher=Springer|isbn=978-1-4020-4559-2}}
+*{{citation|first=Takao|last=Hayashi||authorlink=Takao Hayashi|title=Mahavira|encyclopedia=Encyclopædia Britannica|url=http://www.britannica.com/EBchecked/topic/853508/Mahavira|year=2013}}
+*{{MacTutor Biography|id=Mahavira|title=Mahavira|date=2000}}
+*{{citation|last=Tabak|first=John|title=Algebra: Sets, Symbols, and the Language of Thought|url=http://books.google.com/books?id=h-zRieb7VbwC&pg=PA42|year=2009|publisher=Infobase Publishing|isbn=978-0-8160-6875-3}}
+*{{citation|last=Krebs|first=Robert E.|title=Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance|url=http://books.google.combooks?id=MTXdplfiz-cC&pg=PA131|year=2004|publisher=Greenwood Publishing Group|isbn=978-0-313-32433-8}}
+*{{citation|last=Puttaswamy|first=T.K|title=Mathematical Achievements of Pre-modern Indian Mathematicians|url=http://books.google.com/books?id=DAPLaxw-53IC&pg=PA231|year=2012|publisher=Newnes|isbn=978-0-12-397938-4}}
+* {{citation | last=Kusuba|first=Takanori | contribution=Indian Rules for the Decomposition of Fractions | year=2004 | title=Studies in the History of the Exact Sciences in Honour of David Pingree | publisher=[[Brill Publishers|Brill]] | isbn=9004132023 | issn=01698729 | editor1=Charles Burnett | editor2=Jan P. Hogendijk | editor3=Kim Plofker | editor4=Michio Yano}}
+<!-- The article by David Pingree in the DSB has a more complete list of sources -->
+{{Indian mathematics}}
+{{DEFAULTSORT:Mahavira}}
+[[Category:Indian mathematics]]
+[[Category:Medieval Indian mathematicians]]
+[[Category:9th-century mathematicians]]
+[[Category:Indian Jains]]
+[[Category:9th-century Indian people]]

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