CHAPTER II BASIC IDEAS IN ARITHMETIC - Information...

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CHAPTER II

BASIC IDEAS IN ARITHMETIC

2.1 Numbers and Symbolism

Ganita literally means 'the science of calculations and is often taken as the

Hindu name for Mathematics. The word Samkhyana is found used for Ganita in

various old treatises. In Buddhist literature there are three classes of ganita as

Mudra (finger arithmetic), Ganana (mental arithmetic), Samkhyana

(higher arithmetic-in general)

The earliest enumeration to these three classes can be observed in the Digha

Nikaya, Vinaya Pitaka, Divyavadana and Milindapanho.1

It is believed that a renaissance of Indian Mathematics took place before the

beginning of the Christian era and its effect in the scope of Ganita was great.

Astronomy became separate and geometry came to be included within its scope.

Ganita thus came to mean Mathematics in general, while finger and mental

arithmetic were excluded from the scope of its meaning. For the calculation in

Ganita, writing and writing material became essential. Pati (board) or dhuli (sand)

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spread on pati or ground was used for writing with a piece of chalk (or hand). Thus

patiganita (board calculations) or dhuliganita (dust work) terms came into effect.

The Yajurveda introducing the numbers in decimal system, gave nine, after

dasa (ten), ekadasa (ten plus one), dvadasa (ten plus two) etc. While writing with

place value 2 comes in the unit’s place for twelve, one comes in the unit’s place for

eleven and there was no numeral to be given in the unit’s place for ten. The

necessity of a numerical symbol having no value arose. To sole this problem the

symbol of zero having nothing inside was originated. Thus ten was represented as

10. In this situation zero meant sunya having no value. When the negative numbers

were introduced, between one and negative one there is an interval of two, which is

irregular. Here zero as a number became a necessity.

It is in this context that one should step into the field of numeral

terminology and symbolism.

2.1.1 Numeral terminology

In India, from the most ancient times, 10 has been used as the base and

dealt with eighteen denominations as early as a date when Greeks had only upto

104 (myriad) and Romans upto 103 (Mille). Even in modern times, the numeral

language of no other nation is as scientific and perfect as that of the Indians

observe Datta and Singh.2

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In Valmikiramayana 6.28, 33-37 a number ‘ mahougha’ as great as 1057 is

seen3. In Lalitavistara,4 a Buddhist work of 1st century B.C, a clear dialogue between

mathematician Arjuna and Prince Gautama (Bodhisatva) highlights numerical

ideas like koti, Ayuta-100 kotis, Niyuta -100 Ayutas etc. – Kaccayana's5 Pali

grammar too has interesting series of number names.

In the Jain canonical work Anuyogadvarasutra (100 BC) numbers like Koti-

Koti are seen. Another number representing a period of time is known as

Sirsaprahelika6. It is stated to be as big as (8,400,000)28. But Tiloyapannatti vv

4.277-283 gives up to the number 8431, 1090, known as ‘acalatma’. Aryabhata,

Sridhara Bhaskara II, Narayana etc. also have their lists, with slight variations in

terminology.

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Table – A chronological representation of number values with respective terms.

Number

Value Vedic Period

Valmiki

Ramayana

Period

Purana

Period

5th

Century

(general)

Mahavira 12th Century

100 Eka Eka Eka Eka Eka Eka

101 Dasa Dasa Dasa Dasa Dasa Dasa

102 Sata Sata Sata Sata Sata Sata

103 Sahasra Sata Sata Sata Sata Sata

104 Ayuta Sata Sata Sata Dasasahasra Ayuta

105 Niyuta Sata Sata Sata Laksa Ayuta

106 Prayuta Sata Sata Sata Dasalaksa Prayuta

107 Arbuda Sata Sata Koti Dasalaksa Prayuta

108 Nyarbuda Sata Sata Arbuda Dasakoti Arbuda

109 Samudra Sata Kharbuda Vrnda Sata-koti Abda

1010 Madhya Sata Kharva - Arbuda Kharva

1011 Antya Sata Nikharva - Myarbuda Nikharva

1012 Par¡rdha Sanku Nikharva - Kharva Mahapadma

1013 - - Padma - Mahakharva Sanku

1014 - - Samudra - Padma Jaladhi

1015 - - Antya - Mahapadma Antya

1016 - - Madhya - Ksoni Madhya

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Number

Value Vedic Period

Valmiki

Ramayana

Period

Purana

Period

5th

Century

(general)

Mahavira 12th Century

1017 - Mahasanku Parardha - Maha-ksoni Parardha

1018 - - - - Sankha -

1019 - - - - Mahasankha -

1020 - - - - Ksitya -

1021 - - - - Mahaksitya -

1022 - Vrnda - - Ksobha -

1023 - - - - Mahaksobha -

1027 - Mahaarnda - - - -

1032 - Padma - - - -

1037 - Mahapadma - - - -

1042 - Kharva - - - -

1047 - Mahakharva - - - -

1052 - Okha - - - -

1057 - Mahaukha - - - -

2.1.2 Numbers, symbolism and the jainas

The numerical figures were seen in the seals and inscriptions of Mohenjo-

daro. Numbers 1-13 in initial strokes arranged side by side were seen. In Asoka

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inscription Brahmi and Kharosthi scripts are seen with different numerical

symbols.7 Cunningham believed that writing was known to India from earlier

times and that earliest alphabet was pictographic.8 Later, words were used to

represent numbers and system was known as Bhutasamkhya system.

For example

0 - Sunya, Kham, Akasa

1 - Adi, Indu, Sasi, Bhumi

2 - Netra, Locana

3 - Loka, Kala, Ratna (used by Jains only)

4 - Sagara, Veda, Yuga

5 - Bana, Indriya, Karaniya (used by Jains only), Ratna, tatva

6 - Rasa, Rtu, Sastra, Dravya (used by Jains only)

7 - Naga, Muni, Adri, Tatva (used by Mahavira)

8. - Gaja, Dik, Karman (used by Mahavira)

9 - Anka, Padartha (Used by Mahavira)

10 - Disa, Avatara, Anguli

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Vararuci the Keralite astronomer cum mathematician has originated a new

notational system called Katapayadi system wherein the Devanagari alphabets are

assigned to represent numbers. Using this system numbers could be represented

by suitable words and could be easily fitted in slokas of different metres9. The

numbers 1 to 9 are represented by the letters ‘ka’ to ‘jha’ and also by ‘ta’ to

‘dha’ taken in order and ‘nha’ and ‘na’ representing zero. The letters ‘pa’ to

‘ma’ represent 1 to 5 and ‘ya’ ‘ha’ represent 1 to 8. The numerals obtained as per

this rule are taken in the reverse order to get the intended number. For example

Putumana Somayaji in his book Karanapaddhati cites the number of years in

caturyuga by the word ‘nanajnanapragalbhah’. As per Katapayadi ststem this

becomes 4320000. M.D. Srinivas has pointed out that Candravakyas (of Vararuci)

are couched in the Katapayadi system10 Word numerals without place value, with

place value in inscriptions, etc. were later developments.11 The use of zero symbol

also came up. The jain canonical work Anuyogadvarasutra written before

Christian era is one of the earliest literary evidence of the use of the word notational

place (sthana). Puranas, Yogasutras of Patanjali, literary works like Subandhu's

Vasavadatta, etc. need mention in this context. Place value notation can be traced

since Agnipurana, the Bhakhshali Manuscript and Paulisasiddhanta. Pingala's

Cchandahsutra (200 BC) makes use of zero. Hindu Numerals were used in Arabia

and Europe. There is Syrian references and Arabian references as those of Dhahiz,

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Abul Hasan Al Masudi, Al Baruni, etc. and European references as those of Ralbi

Ben Ezra, Leonardo of Pisa, Maximus Planudes, etc.12

It is with these ideas that prevailed in India that one has to peep specially

into the jaina contribution to this field.

The jainas have classified numbers mainly as Samkhyata (countable),

Asamkhyata (uncountable) and Ananta (infinite) and these have been analysed

after a couple pages in the thesis. They did show some speciality while speaking of

counting numbers. To them the least countable number is 2 and not 1. But again 2

is not equal to one in representation. The rule for this is ‘ Eko gananasamkhya na

upeti13, which means that one is not considered as a ‘gananasamkhya’ or counting

number.

The author of Dhavla books, Virasena (9th century AD) has used some

methods of expressing numbers. E.g.,

1. 7999998 - is expressed as a number in which 7 in the beginning, 8 at the

end and 9 repeated six times in between.

2. 2,27,99,498 is expressed as 2 kotis twenty seven ninety nine thousand, four

hundred and ninety eight (1 koti means 1 crore). Here the biggest number is

written at first.

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3. The biggest number that is definitely stated is the number of developable

human souls. To be precise between koti-koti-koti and koti-koti-koti-koti.

i.e., 1,00,00,0003 and 1,00,00,0004

4. 46666664 is written as sixty four, six hundred, sixty six thousand, sixty

lakhs, and four kotis. Here small numbers are written first.

Some special products of numbers which remain the same when read from

left to right or vice-versa (ie., number palindromes). are given by Mahavira. Such

numbers are said to be in the form of a necklace. Eg:-

139 109 15151

152207 73 11111111

12345679 9 111111111

333333666667 33 11000011000011

Yativrsabha in his work Tiloyapannatti has also projected the ideas of

number system as well as of symbolism.14 It is to be noted that the jaina school

needed a quantitative analysis of Karmic events for which the new mathematical

infinity was to play an important role. Through a long process, 21 types of ranges

(or numbers) were produced (by Yativrsabha) by carrying out certain types of

multiplication, squaring as well as by adding or projecting various types of sets15

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There are 21 types of numbers

Numbers

Countable

(Samkhy¡ta)

Jaghanya sj

Madhyama sm

Utkrsta su

Infinite

(Ananta)

Par¢ta

Yukta

Asamkhyata

Uncountable

(Asamkhy¡ta)

Parita

Yukta

Asamkhyata

Jaghanya Apj

Madhyama Apm

Utkrsta Apu

Jaghanya Ayj

Madhyama Aym

Utkrsta Ayu

Jaghanya Aaj

Madhyama Aam

Utkrsta Aau

Jaghanya Ipj

Madhyama Ipm

Utkrsta Ipu

Jaghanya Iyj

Madhyama Iym

Utkrsta Iyu

Jaghanya Iij

Madhyama Iim

Utkrsta Iiu

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The ideas of 21 types of numbers are developed in different ways. 4 pits in

cylindrical form with diameter 1 lac yojana and depth one thousand yojanas are

dug. Out of these the 3 stable pits are salaka, pratisalaka and mahasalaka and the

fourth pit is unstable.

2 mustard seeds are dropped into the fourth pit which is the jaghanya

samkhyata Sj least or minimal finite. On dropping one more mustard seed it

become non-minimal and non-maximal. Or it becomes madhyama samkhyata.

Similarly mustard seeds are dropped till the pit is filled up. These mustard seeds

are dropped one by one into islands and seas which are situated concentrically

around the Jambu island successively. As soon as the pit has been exhausted a seed

is dropped into salaka pit and the diameter of the unstable pit is increased to that

of the island or ocean where the last seed is dropped. Again this new unstable pit is

filled up and exhausted in a similar way and a second seed is dropped in the salaka

pit. As before the diameter of the unstable pit is increased, filled up, exhausted and

a seed is dropped into salaka pit. This process is continued till the salaka pit is

completely filled up and then one seed is dropped into the pratisalaka pit. When

the pratisalaka pit is filled up in this way one seed is dropped into mahasalaka pit.

The process is stopped when the mahasalaka pit is filled up and the new unstable

pit with diameter as that of the sea or island where the last seed is dropped for

exhaustion is dug. This pit is completely filled up with seeds and that number of

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seeds will denote 'jaghanya parita asamkhyata'. If one is subtracted from this

‘utkrsta samkhyata’ is obtained.

i.e., Su = Apj -1

Then the vargita- samvargita of Apj is 'jaghanya yukta asamkhyata'

i.e., pjA

yj pjA A and

‘parita utkrsta asamkhyata’ 1pu yj

A A . Again 'jaghanya asamkhyata

asamkhayata' 2

aj yjA A and the maximal yukta asamkhyata 1

yu ajA A .

Again six innumerate sets, as per the karma theory of the jainas, are added

to three times vargita samvargita of Aaj and the operation of three times vargita

samvargita is done. To this 6 more innumerate sets are added and three times

vargita samvargita is done and jaghanya parita ananta or Ipj is obtained.

i.e., 3 3 3

former 6innumeratesets latter 6innumeratesetspj aj

I A

and 1au pj

A I .

Then the classification of infinities are obtained as follows.

1pjI

yj pj pjI I I

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1pu yj

I I

2

ij yjI I

1yu ij

I I

and 3 3 3

6infinitesets 2infinitesetsiu ij

I I the infinite major part

of Ominiscience. Here ananta means Gananananta.

It is worth admiration that Yativrsabha has highlighted in addition to

gananananta (compution infinity) 10 other types of ananta which can be enlisted

as follows: namananta, sthapanananta, dravyananta, sasvatananta,

apradesikananta, ekananta, ubhayananta. vistarananta, sarvananta, bhavananta16.

Thus there were various methods to represent numbers in ancient and

medieval India - by name of objects, name of alphabets, with and without

denomination etc. Katapayadi system, Bhuta Samkhya system, etc. were only some

of them. There existed certain specifications to write them

+ÆEòÉxÉÉÆ ´ÉÉ¨ÉiÉÉä MÉÊiÉ& (Ankanam vamato gatih)

It is a specification which means the numbers which represent the

respective alphabet or object (In Katapayadi and Bhuta Samkhya system

respectively) are written from right to left. Dipak Jadav has pointed out17 that

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Nemicandra used Katapayadi system to constitute 17 alpha numbers

(aksarasamkhya). The smallest number 32 used is represented by ®úÉMÉ (raga).

Nemicandra has used metric scale in Prakrt which is often largely found in

Sanskrit literature.

Eg:- ‘+b÷´ÉhhÉÉ ºÉkÉºÉªÉÉ ºÉkÉºÉ½þºÉÉ’- Eight fifty seven hundred seven thousand- 7578

+]Âõ]õÉäkÉ®ú ºÉnù¨É = Eight and Hundred.

Nemicandra has cited numbers in non metric scale notation. For example

>ðhÉiÉÒºÉºÉªÉÉ< BCEòÉhÉ=nùÒ - 1 less 30 hundred one ninety = 2989

Dipak Jadav has pointed out that 19, 29, 39, 49, etc. offer as Indian instances

of vinculum numbers. Vinculum is also seen in Gommatasara-Jivakanda of

Nemicandra

Ê¤É®úÊ½þªÉºÉªÉ - Two less One hundred - 98

>ðhÉiÉÒºÉ - One less Thirty - 29

hÉ´ÉªÉ´ÉÒºÉ - Nine Plus Twenty - 29

To express number under mathematical operations is an interesting

phenomenon in Indian Mathematics. In Tiloyapannatti and Nemicandra's works

are seen to have such examples,

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UôCEòÊn = six kriti = 62 = 36.

nÖùMÉÖhÉºÉÉä±É 2 16 32

In certain other cases each of the digits of the number is expressed without

its denomination

nÖùMÉSÉ=®ú]Âõ]õb÷ºÉMÉ<ÊMÉ = Two four eight eight seven one.

= 178842 (in reverse order)

In some jaina works certain numbers have been expressed as the

composition of different methods. e.g., ¤ÉÉ®ú JÉ UôCEÆò = 6012 (reverse order)

In Dhavala-3 p. 249 the countable numbers are divided into 2 namely,

+ÉäVÉ®úÉÊ¶É (odd numbers) and ªÉÖM¨É®úÉÊ¶É (even numbers).18 Odd numbers are again

divided into 2 namely, iÉäVÉÉäVÉ and EòÊ±É+ÉäVÉ. iÉäVÉÉäVÉ are those numbers when divided

by 4 gives remainder 3. EòÊ±É+ÉäVÉ are those when divided by 4 gives remainder 1.

Clearly if a 3 mod 4, a is iÉäVÉÉäVÉ and

a 1 mod 4, a is EòÊ±É+ÉäVÉ

Eg:- 15

4 remainder 3 15 is iÉäVÉÉäVÉ

25

4 remainder 1 25 is EòÊ±É+ÉäVÉ

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Even numbers are again divided into 2. namely EÞòiÉªÉÖM¨É and ¤ÉÉnù®úªÉÖM¨É

EÞòiÉªÉÖM¨É is a number which gives remainder 0 on division by 4. ¤ÉÉnù®úªÉÖM¨É is a

number which gives remainder 2 on division by 4.

Clearly if a 0 mod 4, a is EÞòiÉªÉÖM¨É and

if a 2 mod 4, a is ¤ÉÉnù®úªÉÖM¨É

Eg:-, 15

4 remainder 0 48 is EÞòiÉªÉÖM¨É

3

4 remainder 2 34 ¤ÉÉnù®úªÉÖM¨É

Now it is very interesting to see that these numbers EÞòiÉªÉÖM¨É, EòÊ±É+ÉäVÉ,

¤ÉÉnù®úªÉÖM¨É, iÉäVÉÉäVÉ can be called the residual class of 4.

2.1.3 Perfect numbers in jaina cosmology

As per the Karananuyoga texts one can find that the middle universe, a

part of cosmos containing the geography of our earth is divided into concentric

rings. It is said that at the centre Jambu Island of diameter D of 1 lakh yojana is

situated and is surrounded by a number of concentric circular rings of seas and

lands alternately. Here it can be seen that the width of each ring is double as that

of the proceeding ring so that the width of nth ring is 2n D.

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The diameter of the outer boundary of 1st ring

= 1

2D W

= 2 2D . D

= 5D

The diameter of the outer boundary of the 2nd ring

= 1 2

2 2D W W

= 5 2 4D . D

= 5 8D D

= 13D

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Fig. 1

J.I.

D

2 D

W1

4 D

W2

8 D

W3

16 D

W4

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So the diameter of the outer boundary of nth ring

n

D = 1 2

2 2 2n

D W W ... W

= 1 22

nD W W ... W

= 2 2 4 8 16 2nD D D D D ... D

= 2 2 1

22 1

n

D D

= 12 2 2nD D

= 21 2 4n D

= 22 3n D D

= 22 2 3n.D D

= 4 3n

W D

The inner diameter of the nth ring,

n

I = 1 2 1

2 2 2n

D W W ... W

= 1 2 12

nD W W ...W

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= 12 2 1

22 1

n

D D

= 12 4nD D

= 12 3n D

= 12 3n D D

= 2 2 3n. D D

= 2 3n

W D

The middle diameter of the nth ring

n

C = 1 2 1

2 2 2n n

D W W ... W W

= n n

I W

= 12 3 2n nD D

= 2 2 1 3n D D

= 2 3 3n. D D

= 3 3n

W D

The area of the nth ring,

48

n

A = 2 2

4n n

D I

= 4

n n n nD I D I

= 24

n n nC D I

= 2 3 34

n n n. W D D I

= 6 6 24

n nW D W

= 3n n

W D W

Tiloyapannatti v. 5.36 tries to enumerate the no. of areal pieces like Jambu

islands (in Kn) ie., any ring as

n

K = 2 2

2 2 2

6 6 2 3 4n n n nn n

W D W W D WD I

D D D

This result is found in 5.263 by taking D=1

i.e., n

K = 12 1 12 2 1 2n n

n nW W

So,

1

nK

K =

112 2 1 2

2 1 212 2 1 2

n n

n n

nP (Say)

49

n Kn Pn

1 24 1

2 144 6 1st perfect no.

3 672 28 2nd perfect no.

4 2880 120

5 11904 496 3rd perfect no.

6 48384 2016

7 195072 8128 4th perfect no.

A positive integer n is called a perfect number if it is the sum of its proper

divisors including 1. Thus ancient cosmography and the related calculations can be

seen to involve examples of perfect numbers.19

2.2 Operations

Hindu Ganita precisely and mainly consisted of the following at the early

renaissance period itself (Ref: Stananga Sutra, Sutra 747).20

1. Parikarma (Fundamental operations)

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2. Vyavahara (Determination)

3. Rajju (Rope measuring geometry)

4. Rasi (Rule of three)

5. Kalasavarna (Fractional operations)

6. Yavat Tavat (Simple equations)

7. Varga (Square meaning quadratic equations)

8. Ghana (cube meaning cubic equations)

9. Varga-Varga (Biquadratic equations)

10. Vikalpa and Prastara (Permutations and Combinations).

The twenty operations according to Prthudakasvami, the commentator of

Brahmasphutasidhanta, are Sankalita (addition), Vyavakalita (subtraction),

Gunana (multiplication), Bhagahara (division), Varga (square), Vargamula (square

root), Ghana (cube), Ghanamula (cube root), Pancajati (5 rules of reduction

relating to fractions), Trairasika, Vyasta-trairasika, Pancarasika, Saptarasika,

Navarasika, Ekadasarasika, Bhanda-pratibhanda and the 8 determinations

according to him are misraka (mixture), sredhi (progression or series) ksetra

(plane figures), khata (excavation), citi (stock), krakacika (saw), rasi (mound) and

chaya (shadow)21

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Brahmagupta says: "He who distinctly and severally knows the twenty

logistics and the eight determinations including measurement by shadow is a

mathematician"22.

The eight fundamental operations seen in Hindu mathematics are addition,

subtraction, multiplication, division, square, square root, cube and cube root.

Mahavira also considered these eight as fundamental operations. In many jaina

books we can see in addition to these fundamental operations other operations,

namely vargita-samvargita, viralana-deya etc...

2.2.1. Multiplication (Pratyutpannah)

The method given in G.S.S 2.1 can be understood by the following example.23

Multiply 1998 by 27.

1998

27

2x1 2

2x9 1 8

2x9 1 8

2x8 1 6 Result = 53946

52

7x1 7

7x9 6 3

7x9 6 3

7x8 5 6

5 3 9 4 6

This is a simple method which is almost similar to modern method.

Sridhara gives different methods for multiplication.24

(1) Sthana vibhaga

In this method digits are written in different types. For example to find 125 x 13

(a) 125 (b) 13 (c) 125

13 2 3

--------- ---------- ---------

13 1365 375

26 26 125

65 ---------- ---------

---------- 1625 1625

1625

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(2) Rupa vibhaga

Here multiplication is done by breaking the multiplicand or the multiplier

into 2 or more parts whose sum or product is equal to it.

Eg: 1227x32 = (1227x30) + (1227x2) or (1200x32) + (27x32)

(3) Utkrama vidhi (Viloma vidhi)

In this method the multiplicand the multiplier are written one below the

other.

Eg: 12

135

The multiplication starts with the last digit of the multiplier. So 1x2=2.

Then one is replaced by 2 and 1x1 = 1. This one is written on the left side of 2. Then 12

is moved one place to the right. Now the numbers on the board are as below:

12

1235

Now 3x2 = 6 and 3 is replaced by 6

3x1 = 3 and 3+2 = 5 and 2 is replaced by 5.

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Again the multiplicand is moved one place to the right and the numbers on

the board are as given below:

12

1565

Now 5x2 = 10 and 5 is replaced by 0.

Then 5x1 = 5 and 5+1 = 6 and 6+6 = 12.

Now 6 is replaced by 2 and 5+1 = 6. So 5 is replaced by 6. Now the

multiplication is over and number on the board is 1620 which is the required result.

(4) Krama Vidhi (Anuloma Vidhi)

Here to find 135 x 12 instead of writing 12

135, we have to write

12

135.

Multiply the digit in the unit place of 135 with digits in 12. i.e., 5x2=10 write '0'

below 2. Then 5x1=5, 5+1=6 replace 5 by 6 and the numbers are written as

12

1360

Now 3x2=6, 6 added to 6, gives 12. Instead of 6 in 1360 write 2. Then 3x1=3 and

3+1=4. Replace '3' in 1360 by 4 and shift the number one more place towards the

right. Then the numbers on the board is

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12

1420

Now 1x2 = 2. This '2' added with the digit '4' below 2 gives 6. Replace 4 by 6. 1x1=1.

Then 135x12 = 1620.

But in Lilavati 14 16 different methods are given for multiplication.25

(a) 1998 x 27

27 x 8 = 216

27 x 90 = 2430

27 x 900 = 24300

27 x 1000 = 27000

--------

53946

=====

(b) 1998 x 27 1998 x 7 = 13986

1998 x 20 = 39960

--------

53946

=====

(c) 1998 x 27 1998 x 20 = 39960 +

1998 x 7 = 13986

--------

53946

=====

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Here 27 = 20 + 7

(d) 1998 x 27 1998 x 30 = 59940 -

1998 x 3 5994

-------

53946

=====

Here 27 = 30 - 3.

(e) 1998 x 27 1998 x 9 = 17982

17982 x 3 = 53946

=====

Here 27 = 9 x 3.

2.2.2. Division (Bhagahara)

The modern method of division is clearly seen in Ganitasarasangraha of

Mahavira v. 2.18. The rule is "Having removed the common factor, if any, from the

divisor and the dividend, divide (by the divisor the digits of the dividend) one after

another in the reverse order".

The same rule is given in Patiganita of Sridhara V.22.26

The rule for multiplication and division of positive and negative numbers is given

in G.S.S 1.50.

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@ñhÉªÉÉävÉÇxÉªÉÉäPÉÉÇiÉä ¦ÉVÉxÉä SÉ ¡ò±ÉÆ PÉxÉ¨ÉÂ

@ñhÉÆ vÉxÉhÉÇªÉÉäºiÉÖ ºªÉÉiº´ÉhÉÇªÉÉäÌ´É´É®Æú ªÉÖiÉÉè

This verse means that in multiplying as well as dividing two negative or

two positive quantities the result is a positive quantity. But it is a negative quantity

in relation to two quantities of which one is positive and other is negative.

2.2.3. Square (Varga)

In G.S.S Mahavira gives 2 methods for squaring.27

(i) The square of numbers consisting of 2 or more places is equal to the sum

of the squares of all the numbers (in all the places) combined with twice

the product of those taken (two at a time) in order. This is clearly the rule

(a+b+c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac.

58

(ii) This method can be explained by suitable example.

To find (152)2

12 1

2x1x5 10

2x1x2 4

52 25

2x5x2 20

22 4

(1) (3) (2)

2 3 1 0 4

To find the square of a number Bhaskara gives 3 methods in Lilavati-20.28

These methods depend on the following:

(i) (a+b)2 = a2 + b2 + 2ab

(ii) (a+b)2 = (a-b)2 + 4ab

(iii) a2 = (a+b) (a-b) + b2

59

In Patiganita V.23 Sridhara gives a method29 which is similar to the 1st

method of Bhaskara. Write the square of the last digit multiply the remaining

digits by twice the last moving it from place to place towards the right and set down

the resulting products over the respective digits then rub out the last digit and

move the remaining digit one place to the right.

In Dhavala square of a number is given as first square.30

i.e., square of a = a2

2nd square of a = (a2)2 = a4

3rd square of a = (a2)3 = a6

nth square of a = (a2)n = a2n

2.2.4. Square root(Vargamula)

The rule to find square root of a number is given in G.S.S. 2.36 as

+xiªÉÉèVÉÉnù{É¾þiÉEÞòÊiÉ¨ÉÚ±ÉäxÉ ÊuùMÉÖÊhÉiÉäxÉ ªÉÖM¨É¾þiÉÉè*

±É¤vÉEÞòÊiÉºiªÉÉVªÉÉèVÉä ÊuùMÉÖhÉnù±ÉÆ ´ÉMÉÇ̈ ÉÚ±É¡ò±É¨ÉÂ** 31

This rule can be illustrated by the following example

To find the square root of 65536.

60

Square root of 256512

65536 =2

The same formula is given by Aryabhata and Sridhara.

But Virasena in his work Dhavala 3 pp.136-139 defines the square root in the

following way

Ist square root of a = a1/2

2nd square root of a = (a1/2) 1/2 = 2

1

2a

3nd square root of a 2 3

11 12

2 2a a

nth square root of a = n

1

2a

Inorder to find the square and square root of negative numbers, Mahavira

gives the following rule.

61

@ñhÉªÉÉävÉÇxÉªÉÉäªÉÉæMÉÉä ªÉlÉÉºÉRÂóJÉªÉ¨ÉÞhÉÆ vÉxÉ¨ÉÂ*

¶ÉÉävªÉÆ vÉxÉ¨ÉÞhÉÆ ®úÉú¶Éä& @ñhÉÆ ¶ÉÉävªÉÆ vÉxÉÆ ¦É´ÉäiÉÂ**

vÉxÉÆ vÉxÉhÉÇªÉÉä´ÉÇMÉÉæ ¨ÉÚ±Éä º´ÉhÉæ iÉªÉÉä& Gò¨ÉÉiÉÂ*

@ñhÉÆ º´É°ü{ÉiÉÉä%´ÉMÉÉæ ªÉiÉºiÉº¨ÉÉzÉ iÉi{Énù¨É**Â** 32

which means the square of a positive as well as of a negative (quantity) is positive

and the square roots of those (square quantities) are positive and negative in order.

As in the nature of things a negative (quantity) is not a square (quantity), so it has

no square root.

Mahavira never considered the square root of a negative number as real.

This is the first step of the set of imaginary numbers.

2.2.5. Cube (Ghana)

The continued product of three equal numbers are called ghana by

Mahavira and he gives different rules for finding the cube of a number.

As per G.S.S.vv43-45.

3a = a a a

a3 = a (a+b) (a-b) + b2 (a-b) + b3

a3 = a + 3a + 5a + . . . . . upto a terms

a3 = a2 + (a-1) (1+3+5+ . . . . . upto a terms)

62

a3 = 3 [1 x 2 + 2x3 +3x4 + .......... + (a-1) x a] + a

But G.S.S v. 2.46 clearly gives the identity (a+b)3 = 3a2b + 3ab2 + a3+b3

Bhaskara gives a more explicit method which can be illustrated by the

following example.

To cube 1543:

First take 15 and apply the above method of cubing by considering 1 as last

digit and 5 as the remaining.

13 = 1

5 x 3 x 12 = 15

3 x 52 x 1 = 75

53 = 125

_______

3375

Now consider 154 with 15 as last and 4 as the remaining

(15)3 = 3375

4 x 3 x 152 = 2700

3 x 42 x 15 = 720

63

43 = 64

________

3652264

Now consider 1543 with 154 as last and 3 as the remaining.

(154)3 = 3652264

3 x 3 x (154)2 = 213444

3 x 32 x 154 = 4158

33 = 27

_____________

3673650007

2.2.6. Cube root (Ghanamula)

In G.S.S 2.54 Mahavira gives a method to find the cube root which can be

illustrated by an example.

To find the cube root of 77308776

64

Cube root of 77308776 = 426

2.2.7. Operations by Zero

The different operations by zero is also discussed by Mahavira.

iÉÉÊb÷iÉ& JÉäxÉ ®úÉÊ¶É& JÉÆ ºÉÉäÊ´ÉEòÉ®úÒ ¾þiÉÉä {ÉÖiÉ&

½þÒxÉÉäÊ{É JÉ´ÉvÉÉÊnù& JÉÆ ªÉÉäMÉä JÉÆ ªÉÉäVªÉ°ü{ÉEò¨ÉÂ 33

which means 'a number multiplied by zero is zero and that (number) remains

unchanged when it is divided by combined with (or) diminished by zero.

Multiplication and other operations in relation to zero give rise to zero and in the

operation of addition, the zero becomes the same as what added to it.

65

i.e., A 0 = 0

A + 0 = A

A - 0 = A

A 0 = A

Here the rule of division by zero is not true. This might have originated

from the logic that when a sum is required to be divided among zero persons then

the sum definitely remains the same but all the other rules are same as that of

modern mathematics.

But Brahmagupta in Brahmasphutasiddhanta gives the correct rule for

operations by zero. He said that dividing a positive quantity or negative quantity

by zero we get a fraction with that quantity as numerator and zero as denominator.

He called such fractions with denominator 0 as 'Khaccheda'.

But Bhaskara II used the word ‘Khahara’ to denote such fractions. He used

the word kha-guna to denote the product of a number by zero.

Brahmagupta made an incorrect statement 0

00 . But Bhaskara gave the

result a

a a0 in this regard.

66

2.2.8. Vargita-samvargita

Another important operation seen in jaina works is ‘vargita-samvargita’34

i.e., raising of numbers to the given number. This is a great contribution of Acarya

Virasena. He used this for expressing very big numbers.

The first vargita- samvargita of a number is simply known as vargita-

samvargita and can be written as 1

x and so 1

xx x .

The second vargita –samvargita of x is defined as the vargita- samvargita of

the first vargita- samvargita of x and can be written as 2

x .

so

1xx

2 1 xxx x x

Similarly third vargita-samvargita of x is defined as the vargita- samvargita

of the second vargita-samvargita of x and can be written as

2x

3 2

x x

Eg:- 1st vargita-samvargita of 2 = 2

= 22 = 4.

2nd vargita-samvargita of

122

2 1 22 42 2 2 2 4 256

3rd vargita samvargita of 2 = 43

4 4 2562 (4 ) (256) .

By the operation of vargita-samvargita on 2 it becomes 4. By the operation

of second vargita-samvargita on 2 or by the operation of

67

vargita-samvargita on 4 the number becomes 44 256 . By the operation of third

vargita-samvargita on 2 or by the operation of vargita-samvargita on 44 the

number becomes 256

256 .

i.e., st nd rd1 vs 2 vs 3 vs2 4 2562 2 4 256 .

In general the pth vargita-samvargita of x is

p 1x

p p 1

x x p 2,3,.... To

include 1 define 0

x x .

One interesting property associated with the above concept and notation is

the relation 1 2 1

log log .

Lp p

x x x x x x . which is true because

68

p 1

p 2

p 3

xp 1

p 1 p 1

xp 1 p 2

p 1 p 2 p 1

xp 1 p 2 p 3

p 1 p 2 p 3 p 3

p 1 p 2 p 3 1 1

p 1 p 2 p 3 1x

p 1 p 2 p 3

x

x .logx

x . log x

x . x . logx

x . x .log x

x . x . x .logx

x . x . x ................x . logx

x . x . x ................x . logx

x . x . x .....

1

1 2 p 1

...........x . x. logx

logx. x. x .x ................x

It is clear that we can obtain rapidly increasing divergent sequence by

applying the above definition.

e.g., 1 2 3 25642 4,2 4 256,2 256 L

When talking about the classification of infinity in Tiloyapannatti

Yativrsabha introduces one more operation ‘first time vargita-samvargita’ and the

first time vargita-samvargita of x and it can be denoted by 1x

As explained above 1x

x x a say

so log p

x

p

x

69

Then 1 2a

a a x b say

1 2 3b

1 2 3 4c

b b a x c say

c c b a x d say

Continuing like this we arrive at a stage when x

x y . Then y is called the first

time vargita-samvargita of x and we will write 1y x

Similarly if 1 1

y1y x y k say

2 2

k1k x y l say

Continuing like this we arrive at a stage when y y

1x y p then p is called the

second time vargita-samvargita of x and so on.

2.2.9. Viralana-deya

In jaina works we can see another operation called viralana-deya means

spread and give.35 Viralana means the spreading of a number into its unities. Deya

means the substitution of 'n' in place of 'l'. So viralana of n is 1 1 1 . . . . up to n

times viralana-deya of n is n n n . . . n times. So vargita-samvargita of 'n' is

obtained by multiplying the n's obtained by the viralana-deya of n.

70

Similarly 2nd vargita-samvargita of n i.e, nn

nn is obtained by viralana-

deya of nn and so on.

2.3. Laws of Indices

In Dhaval¡- 3, p. 136-137 it is given that dividing the 1st square root of

sucyangula by the 2nd square root and multiplying the result by the 1st square root,

viskambhasuci is obtained36

Sucyangula is a measure of space and it is given as 1

32 2 in

Jainendrasiddhantakosa, Vol. IV

nth square root of a = n2

1

a .

sucyangula = 1 43 32 2 2

1st square root of sucyangula = 1

4 223 32 2

2nd square root of sucyangula = 1

2 123 32 2 .

Here the Viskambhasuci = 2

So according to the verse

23 1 1 2

3 3 3

13

22 and 2 2 2

2

This is true according to modern laws of indices.

71

In Dhavala - 3 p. 137 it is also given that

The 1st square root of sucyangula is multiplied by the 2nd square root. Then

1st square root of ghanangula is divided by the result then also viskambhasuci is

obtained.

sucyangula = 432

First square root of sucyangula = 43

1 22 32 2

Second square root of sucyangula = 23

1 12 32 2

ghanangula = (s£cyangula)3

= 43

342 2

First square root of ghanangula = 1

4 222 2

So by the verse 2

2 1

3 3

22

2 2

This is also true according to modern mathematics .

From the above verses it is clear that Virasena was very much familiar with

the laws of indices like

m n m n

m n m n

nm mn

a .a a

a a a

a a

72

2.4 Fractions

In ancient Indian mathematics we can see the idea of fractions and its

operations. In Jaina works the discussion of fractions is made under the name

Kalasavarna. Kala denoted the sixteenth part. So Kalasavarna literally means parts

resembling 1

16 . So Kalasavarna come to signify fractions in general.

Fractions are discussed by Virasena in his work Dhavala. Some of the ideas

given in his are37

1. Let a number 'm' be divided by the divisors d and d' and let q and q' be the

quotients then the following formula gives the result when m is divided by

d d' m q' q

qd d' q' 11q'q

.

2. If a

qb then

a qq

b n 1b

n

for any ‘n’.

3. a

If qb then

a qq

bb c1

c

and

a qq

bb c1

c

4. If a

qb and

a bq c then b' b

qb'1

c

5. a a a a b' b

If q and isanother fraction then qxb b' b b' b'

73

6. a a bc

If q and q c then xb b x q c

7. a a bc

If q and q c then xb b x q c

8. a a qc

If q and q' then q' qb b c b c

9. a a qc

If q and q' then q' qb b c b c

10. If a and b are two positive numbers then 2a a

aa b 1

ab

11. a a

If x and y thenb c

aa a bx

b byx y1 1x 1

c cx

and

a

a a cy

x cx y x 1 1y 1y by

12. If a c

x and y wherea,b,c, x and yarepositive numbers thenb b

b x y c bx a

2.4.1. Types of Fractions.

Different types of fractions are studied by Mahavira.

¦ÉÉMÉ|É¦ÉÉMÉÉ´ÉlÉ ¦ÉÉMÉ¦ÉÉMÉÉä ¦ÉÉMÉÉxÉÖ¤ÉxvÉ& {ÉÊ®úEòÒÌiÉiÉÉä%iÉ&*

¦ÉÉMÉÉ{É´ÉÉ½þººÉ½þ ¦ÉÉMÉ¨ÉÉjÉÉ ¹ÉbÂ÷VÉÉiÉªÉÉä%¨ÉÖjÉ Eò±ÉÉºÉ´ÉhÉæ**38

74

He classified the fractions into 6 types namely

Simple fractions (Bhaga), Fractions of fractions (Prabhaga) Complex

fractions (Bhagabhaga), Fractions in association Bhaganubandha Fractions in

dissociation (Bhagapavaha), Fractions consisting of 2 or more of the above

mentioned fractions (Bhagamatra)

Examples

1. Bhaga - simple fraction - 1

2,

3

4, etc.

2. Prabhaga - fraction of fractions - 1

2 of

3

4,

1

4 of

2

7, etc.

G.S.S 3.108 says39 that the sum of 1

8,

1

4 of

1

3,

1

5 of

1

2,

1

5 of

3

4 of

1

6 of a

certain quantity is 1

2. What is this unknown quantity?

The algebraical representation of this problem is if

1 1 1 1 1 3 1 1x. . x. . x. . . ,

4 3 5 2 5 4 6 2 then what is x?

3. Bhagabhaga - complex fractions

G.S.S 3-11 says40 that find the sum of 1 1 11

6 3 34, , ,7 8 8 4

All these fractions

are examples of Bhagabhaga variety.

75

4. Bhaganubandha-associated fractions.

In this type 2 different kinds of fractions are included

First kind is a fraction associated with one integer and the second kind is a

fraction associated with another fraction.

Example for first kind is given in G.S.S 3-11641 as subtract from 20: 18

8

16

6 112

2 and 58

3 .

Example for 2nd kind is given in G.S.S 3-12142 as give out the result after

adding 1

4 associated with

1

2 of itself

1

10 associated with

1

4 of itself,

1

6

associated with 1

2 of itself,

1

2 associated with

1

3 of itself and

1

3 associated

with 1

3 of itself.

5. Bhagapavaha - fractions in dissociation

An example for such fraction is given in G.S.S 3-12943 as add 2

9,

1

9,

1

3,

1

8

and 2

7 which are respectively diminished by

1

2,

1

6,

1

4,

1

3 and

1

8 of

themselves in order and then give out the result.

6. Bhagamatra - Fractions consisting of 2 or more of the above mentioned

fractions.

76

Bhagamatra is of 26 types.

When discussing about problems in different types of fractions Mah¡v¢ra

gives the method to find the numerators of certain fractions when its

denominators and sum are given. This rule is applicable to those fractions for

which the numerators increases by one and in the sum the denominator is greater

than the numerator.

ºÉ¨É½þÉ®èúEòÉÆ¶ÉEòªÉÖÊiÉ¾þiÉªÉÖiªÉ¶ÉÉä%¶É BEò´ÉÞrùÒxÉÉ¨ÉÂ*

¶Éä¹ÉÊ¨ÉiÉ®úÉÆ¶ÉªÉÖÊiÉ¾þiÉ¨ÉxªÉÉÆ¶ÉÉä%ºiªÉä´É¨ÉÉ SÉ®ú¨ÉÉiÉä**44.

As per the sloka first let the numerators be 1 for all the fractions and make

their denominators equal and find the sum. Divide the given sum by this sum. Let

the quotient be 'x' which is the numerator for the first fraction. The remainder

obtained after this division is divided by the sum of the remaining provisional

numerators. The quotient obtained combined with the first numerator is the

numerator of the 2nd fraction. The remainder in this second division is divided by

the sum of the remaining provisional numerators. The quotient obtained added to

the 2nd numerator is the 3rd numerator and so on.

Eg:- The sum of numbers which are divided by 9, 10 and 11 is 877

990. Give out the

numerators.

77

Consider fractions 1

9

1

10 and

1

11.

i.e., 110

990

99

990 and

90

990.

110 99 90 299

990 990 990 990

Divide the given sum 877

990 by

299

990. Then the quotient is 2 and remainder

is 279. Then the first numerator is 2.

Divide the remainder 279 by 99+90=189. The quotient is 1 and the

remainder is 90. So 1+2=3 is the second numerator.

Divide the remainder in the second division i.e., 90, by the sum of the

remaining provisional numerators i.e., 90. Then the quotient is 1. So the third

numerator is 1 + 3 = 4.

2

9,

3

10 and

4

11 are the fractions.

Mahavira also gives the method to find the denominators of 2 fractions with

given numerators and sum:

<¹]õMÉÖhÉÉÆ¶ÉÉä%xªÉÉÆ¶É|ÉªÉÖiÉ& ¶ÉÖrÆù ¾þiÉ& ¡ò±ÉÉÆ¶ÉäxÉ*

<¹]õÉ{iÉªÉÖÊiÉ½þ®úPxÉÉä ½þ®ú& {É®úºªÉ iÉÖ iÉÊnù¹]õ½þÊiÉ&**45

78

The above rule states that multiply one of the given numerators by a chosen

quantity, ‘p’. Combine it with the other numerator. Then divide this quantity by

the numerator of the given sum. ‘p’ should be chosen so that there is no

remainder by this division. Now this quotient is divided by the chosen number and

multiplied by the denominator of the given sum. Denominator of the other fraction

is this denominator multiplied by the chosen quantity.

ie., if m

n is the sum of 2 intended fractions with 'a' and 'b' as their

numerators then the fractions are a b

andap b n ap b n

x x xpm p m p

where p is

any chosen quantity so that ap + b is divisible by m.

Eg:- If the sum of 2 fractions with numerators 2 and 5 are 11

9. Find the fractions.

Given m 11

n 9 ., a = 2 and b = 5. Take p = 14. Then 2 14 + 5 = 33 becomes

divisible by m = 11.

first fraction 2 2 28

9 27 273

14 14

.

second fraction = 5 5

9 273 14

14

79

2.4.1.1. Unit fractions

Another important contribution of Mahavira is the idea about unit

fractions. The word used by him to represent unit fraction is ‘rupamsaka rasi’. He

was very much interested in problems connected with unit fractions. Some

examples are the following.

a. To decompose a unit fraction into 2 unit fractions

´ÉÉ\UôÉ½þiÉªÉÖÊþiÉ½þÉ®ú¶Uäônù& ºÉ ´ªÉäEò´ÉÉ\UôªÉÉ{iÉÉä%xªÉ&*

¡ò±É½þÉ®ú½þÉ®ú±É¤vÉä º´ÉªÉÉäMÉMÉÖÊhÉiÉä ½þ®úÉè ´ÉÉ ºiÉ&**46

This sloka means that the denominator of the given sum multiplied by a

chosen number is the denominator of one of the intended fractions and this

denominator divided by the previously chosen number as lessened by one gives the

other denominator, or factorise the given denominator and multiply one of these

factors and the sum of the factors. This is the first required denominator. Similarly

multiplying the other factor and the sum of the factors will give the second

denominator.

ie., if 1

n is the given fraction

1 1 1

npn np

p 1

where p is any chosen quantity. Or if

1 1

n ab , b 1 then

1 1 1

n a a b b a b

.

80

ºÉä¹]õÉä ½þÉ®úÉä ¦ÉHò& º´ÉÉÆ¶ÉäxÉ ÊxÉ®úOÉ¨ÉÉÊnù¨ÉÉÆ¶É½þ®ú&*

iÉtÖÊiÉ½þÉ®úÉ{iÉä¹]õ& ¶Éä¹ÉÉä%º¨ÉÉÊnùilÉÊ¨ÉiÉ®äú¹ÉÉ¨ÉÂ**47

The above stanza means that the denominator of the given sum is combined with

an optionally chosen number and then divided the numerator of the sum so as to

leave no remainder becomes the denominator of the first fraction. The optionally

chosen quantity when divided by the denominator of the first fraction and then by

the denominator of the given sum gives the denominator of the other fraction.

ie., a 1 p

n p n pn.n

a a

where p is the optionally chosen quantity so that 'n +

p' is exactly divisible by 'a'.

b. To decompose a unit fraction as the sum of fractions with given numerators:

±É¤vÉ½þ®ú& |ÉlÉ¨ÉºªÉSUäônù& ºÉº´ÉÉÆ¶ÉEòÉä%ªÉ¨É{É®úºªÉ

|ÉÉEÂò º´É{É®äúhÉ ½þiÉÉä%xiªÉ& º´ÉÉÆ¶ÉäxÉèEòÉÆ¶ÉEäò ªÉÉäMÉä**48

By this sloka if 1

n is the sum of the given fractions and a, b, c, d are the numerators

then

81

1 a b c d

n n n a n a . n a b n a b n a b c d n a b c

a n a b bn c n a bR.H.S.

n n a n a b n a b n a b c

a b 1

n n a b n a b

n a b

n n a b

1L.H.S.

n

So the result is true.

c. Expression for '1' as the sum of 'n' unit fractions

°ü{ÉÉÆ¶ÉEò®úÉ¶ÉÒxÉÉÆ °ü{ÉÉtÉÎºjÉMÉÖÊhÉiÉÉ ½þ®úÉ& Gò¨É¶É&*

ÊuùÊuùjªÉÉÆ¶ÉÉ¦ªÉÉºiÉÉ´ÉÉÊnù¨ÉSÉ®ú¨ÉÉè ¡ò±Éä °ü{Éä**49

As per the sloka if the sum of n unit fractions is one then the denominators of

the fractions are 2x1, 3, 32, 33, ...3n-2, 2x3n-1

ie.,

2 3 2 1

1 1 1 1 1 31

2 3 3 3 3 2.3n n

This rule is true because

82

n 2

2 n 2

n 2

1 1. 1

3 31 1 1

13 3 31

3

1 1 1

2 2 3

so R.H.S. = 2 3 2 1

1 1 1 1 1 3

2 3 3 3 3 2.3

n n

=

n 2

n 2

1 1 1 1 1 1.

2 2 2 3 2 3

= 1 = L. H. S.

d. Expression for '1' as the sum of '2n' unit fractions:

BEòÉÆ¶ÉEò®úÉ¶ÉÒxÉÉÆ tÉtÉ °ü{ÉÉäkÉ®úÉ ¦É´ÉÎxiÉ ½þ®úÉ&*

º´ÉÉºÉzÉ{É®úÉ¦ªÉºiÉÉººÉ´Éæ nùÊ±ÉiÉÉ& ¡ò±Éä °ü{Éä**50

This sloka states that if the sum of different unit fractions is 1 then the

denominators are obtained by multiplying one integer (say n) and the next integer

(n+1) and halving the product. The integer n can take values beginning from 2.

1 1 1 1 11 ....

1 1 1 1 12.3. 3.4. 4.5. 2n 1 2n. 2n.

2 2 2 2 2

This is true because

83

1 1 1 1

R.H.S. 2 ...2.3 3.4 2n 1 2n

1 1 1 1 1 1 1 1 12

2 3 3 4 4 5 2n 1 2n 2n

12. 1

2

e. To express a given fraction as the sum of unit fractions:

In G.S.S. 380 Mahavira51 states that if M

N is the given fraction. Then

M 1 G

N S SN where G is a chosen number so that

N G

M

whole number say

'S' and M<N.

Doing the same procedure on G

SN and repeating it we get the required unit

fractions.

E.g. let 3

5 be the given fraction. Then

3 M

5 N say.

Take G = 1 then 5 1

23

whole no.

3 1 1 1 1

5 2 2.5 2 10

84

2.4.2. Addition and subtraction of fractions

For finding the sum and difference of simple fraction the 1st step given by

Mahavira is to make the denominators equal. The fractions of equal denominators

can be added or subtracted by doing the operation for the numerators and putting

the common denominator.

To make the denominators of fractions equal, Mahavira gives two different

methods.

1

ºÉoù¶É¾þnùSUäônù½þiÉÉè Ê¨ÉlÉÉå%¶É½þÉ®úÉè ºÉ¨ÉÎSUônùÉ´ÉÆ¶ÉÉè*

±ÉÖ{iÉèEò½þ®úÉè ªÉÉäVªÉÉè iªÉÉVªÉÉè ´ÉÉ ¦ÉÉMÉVÉÉÊiÉÊ´ÉvÉÉè ****52

This sloka means that in the operations relating to simple fractions multiply

the numerator and denominator of the first fraction by the quotient obtained when

the denominator of the second fraction is divided by the common factor of the

denominators. Similarly multiply the numerator and denominator of the second

by the quotient obtained when the denominator of the first fraction is divided by

the common factor of the denominators.

Eg:- consider the fractions x y

andab bc

The common factor of the first and the second denominators is b

85

Multiply x and ab by bc

cb

and y and bc by a = ab

b

Then x y x.c y.a xc ya

ab bc ab.c bc.a abc

2. Method by using Niruddha (L.C.M):-

Mahavira introduced the word Niruddha means L.C.M.

UäônùÉ{É´ÉiÉÇEòÉxÉÉÆ ±É¤vÉÉxÉÉÆ SÉÉ½þiÉÉä ÊxÉ¯ûrù ºªÉÉiÉÂ

½þ®ú½þiÉÊxÉ¯ûrùMÉÖÊhÉiÉä ½þÉ®úÉÆ¶ÉMÉÖhÉä ºÉ¨ÉÉä ½þÉ®ú:53

As per the sloka niruddha is obtained by means of the continued

multiplication of all the common factors of the denominators and all their ultimate

quotients. This is modern method to find the L.C.M. Mahavira also says that the

multiplication of the numerators and denominators by the quotient obtained in

division of nirudha by the denominators makes the denominators equal.

Eg:- Consider the fractions x y

and . Herethenirudha abcab bc

Now x x.c xc

ab ab.c abc

and y y.a ya

bc bc.a abc

so x y xc ya xc ya

ab bc abc abc abc

86

In Patiganita v. 37 Sridhara gives a method to add and subtract two simple

fractions.54

Write the fractions one below the other without lines of separation. By the

lower denominator multiply the upper numerator, then by the upper denominator

multiply the lower denominator and then add the product of the numerator and

denominator in the middle to the upper product.

Eg:- To find 2 4

3 5

2 2 5 10 10 12

3 3 3

4 4 4

5 5 3 15 15

2 4 22

3 5 15

This method is applicable only to pairs.

2.4.3. Multiplication of Fractions

For multiplication of fractions Mah¡v¢ra gives the following rule in G.S.S 3-

2.

MÉÖhÉªÉänÆù¶ÉÉxÉÆ¶ÉÉè½þÉÇ®úÉxÉÂ ½þÉ®èúPÉÇ]äõiÉ ªÉÊnù iÉä¹ÉÉ¨ÉÂ*

´ÉXÉÉ{É´ÉiÉÇxÉÊ´ÉÊvÉÌ´ÉvÉÉªÉ iÉÆ Ê¦ÉzÉMÉÖhÉEòÉ®äú ****55

87

This sloka means that in multiplications of fractions numerators are to be

multiplied by the numerators and the denominators by the denominators after

carrying out the process of cross reduction if possible.

In G.S.S. 3.5 an example is given in connection with this : A person gets 3

5

of a pala of long pepper for one pa¸a,. O’ arithmetician mention, after multiplying,

what (he gets) for 9

2 pa¸as.

31 pana of pala

5

9 9 3panas . of a pala)

2 2 5

27of a pala (by the rule)

10

The same rule is given by Sridhara in Patiganita. V. 33(i).

2.4.4. Division of fractions

For the division of fractions, in G.S.S 3-8 Mahavira gives the following rule.

After making the denominator of the divisor its numerator (and vice versa), the

operation to be conducted then is as in the multiplication.56

In G.S.S 3.10 the following example is given.

‘In case, a person gets 20

3 of pa¸a for 3

8of a pala of red sandalwood. What

will he get for one pala (of the same wood)’

88

20 3Heregiven that of a pana of a pala

3 8

20

20 8 1603one pala x3 3 3 9

8

The same method is also given by Sridhara in Patiganita.V. 33(ii).

2.4.5. Square, square root, cube and cube-root of fractions

In G.S.S 3-13 Mahavira says that after getting the square, square root, cube

or cube root of the denominator and numerator (of the given fraction) the (new)

numerator is divided by the (new) denominator to get the result of the operation

squaring, cubing, etc.57 The same rule is given by Sridhara in Patiganita.

2.5. Ratio and Proportion - Rule of Three (Trairasika)

The rule of three is found mentioned and used at several places in Jain

mathematical works. The technical terms in connection with the process are phala

(fruit), iccha (requisition) and pramana (argument), the same as found in known

Indian mathematical works. In Patiganita.v. 43 Sridhara has shown the method to

solve problems under this heading.58 Here argument and the requisition which are

of the same denomination has to be set down in the first and last places, the fruit

which is of a different denomination has to be set down in the middle. The middle

quantity multiplied by the last has to be divided by the first.

89

Example, If 11

2 palas of black pepper are obtained for 1

23

pa¸as. Then how

much of that will be obtained for 210

3

pa¸as?

(Pram¡¸a) argument = 12

3pa¸as = 7

3pa¸as

(Phala) fruit = 11

2palas = 3

2palas

(icch¡) requisition = 2

103

pa¸as = 28

3 pa¸as

7 3 28

3 2 3

Applying the rule the required result =

3 28

2 37

3

3 28 36 palas

6 7

But in G.S.S 5.2 Mah¡v¢ra has directly given the formula for problems in

rule of three as phala iccha

pramana

when iccha and pramana are similar.59

In Aryabhatiya 2-26 the rule of three is clearly explained.

90

This is the rule in cases of problems of direct proportion. But for the

problems of inverse proportion the operation is reversed i.e., multiplication and

division are interchanged.

Patiganita 5.45 gives the rule of five, seven and nine.60 Here in these rules 2

sides are referred to as argument side and the requisition side.

The rule is: After transposing the fruit from one side to the other transpose

the denominators in the same manner and multiply the numbers so obtained on

either side, divide the side with larger number of quantities (i.e., numerators) by the

other.

Eg: If 12

2 is the interest of 13

100 for one-third of a month what will be the

interest on 12

50 for 13

6 months.

The two sides referred to are

i) Argument side

1 301100 or

3 3

1

3

1 52 or

2 2

91

ii) Requisition side

1 10150 or

2 2

1 176 or

3 3

These can be written as

(i) (ii)

301 101

3 2

'0' being written in the place of the

desired (unknown) quantity.

1 17

3 3

5

0

2

Transposing the fruit we get

(i) (ii)

301 101

3 2

1 17

92

3 3

0 5

2

Transposing the denominators

(i) (ii)

301 101

2 3

The number of numerators in (ii)

greater than (i). Hence the unknown

quantity

101x3x17x3x5 47121

301x2x1x3x2 1204

1 17

3 3

0

5

2

Lilavati - 82 also explains the rule of five, rule of seven and rule of nine. The

rule says multiplying the 'pramanaphala' by the product of icch rasi and then

dividing by the product of pramana rasis will give the icchaphala.

93

2.6 Permutations and Combinations

The credit of giving the general formula for the number of combinations of

'n' things taken 'r' at a time goes uniquely to Mahavira. It is given in G.S.S. 6-218 as.

BEòÉtäEòÉäkÉ®úiÉ& {Énù¨ÉÚv´ÉäÊvÉªÉÇiÉ& Gò¨ÉÉäiÉÂGò¨É¶É&

ºlÉÉ{ªÉ |ÉÊiÉ±ÉÉä¨ÉPxÉÆ |ÉÊiÉ±ÉÉä¨ÉPxÉäxÉ ¦ÉÉÊVÉiÉÆ ºÉÉ®ú¨ÉÂ

This sloka means that write the numbers starting from 1 upto the given

number in one line. Below that write the same numbers in the reverse order. The

product of 1 2 3 or more numbers in the upper row taken from right to left be

divided by the corresponding product of 1 2 3 or more numbers in the lower row

also taken from right to left is the required combination in each case.

Eg:- Let the number of things be 5. Then write 1 2 3 4 55 4 3 2 1.

By the rule

15C = 5

51

25C = 5 4

101 2

35C = 5 4 3

101 2 3

45C = 5 4 3 2

51 2 3 4

94

55C = 5 4 3 2 1

11 2 3 4 5

When talking about the different types of fractions in G.S.S 3.138 Mahavira

says that there are 26 types of Bhagamatra fractions (combinations of five simple

varieties of fractions). This number 26 is obtained as follows:-

2 3 4 55C 5C 5C 5C 10 10 5 1 26

Instances can be observed in the Bhagabatis£tra when speculation is made

about different philosophical categories arising out of the combination of 'n'

fundamental objects one at a time (eka-samyoga), two at a time (dvika samyoga), 3

at a time (trika samyoga) or more at a time.61

Silanka the commentator has quoted 2 rules regarding permutations and

combinations. The first one is for determining the total number of transpositions

that can be made out of a number of objects.62

In 'Patiganita' (v. 72, p. 58) Sridhara has given the rule for finding the

number of combinations that can be formed out of a given number of things 1, 2, 3,

.... taken all at a time. The relevant rule has been given as write down the numbers

beginning with 1 and increasing by 1 upto the (given) number of things in the

reverse order. Divide them by the numbers beginning with 1 and increasing by 1 in

the regular order, and then multiply successively by the preceding (quotient), the

succeeding one.

95

Eg:- Let the number of things be 6. The numbers are written down in the reverse

order.

6 5 4 3 2 1

These are divided by 1 2 3 4 5 6 then we get 6

1

5

2

4

3

3

4

2

5

1

6.

Multiplication is made successively as per the above rule.

Thus according to the rule we get

6C1 = 6

1,

6C2 = 6 5

1 2,

6C3 = 6 5 4

1 2 3,

6C4 = 6 5 4 3

1 2 3 4,

6C5 = 6 5 4 3 2

1 2 3 4 5,

6C6 = 6 5 4 3 2 1

1 2 3 4 5 6.

In Dhavala -4 P-124 it is given that by the use of 6 'dravyas' 59' different

combinations can be prepared.63

96

6 dravyas taken 2 at a time = 2

6 56C 15

1 2


6 5 46C 20

1 2 3

.


6 5 4 36C 15

1 2 3 4


6 5 4 3 26C 6

1 2 3 4 5


6 5 4 3 2 16C 1

1 2 3 4 5 6

Total 15 + 20 + 15 + 6 + 1 = 57.

More than these 57 combinations, Jiva and Jiva, Pudgal and Pudgal these 2

combinations are also admissible. So 57 + 2 = 59.64

In Dhavala-1, Dhavala-3 and Dhavala-5 we can see the idea of combination.

The formula for finding the number of combinations of 'n' things taken 2, 3, 4 are

given in Agamas.

But the credit of the formula for permutations and combinations is given to

Herigon (1634).65

97

2.7. Logarithm

Another important concept of jaina mathematics is logarithm with different

bases. The idea of logarithm to the base 2, 3, 4, etc. are clearly given in Dhaval¡.

2.7.1. Arddhaccheda

In Dhavala-3, p. 21 given that ‘ÊVÉiÉxÉÒ ¤ÉÉ®ú EòÉä<Ç ºÉÆJªÉÉ =kÉ®úÉäkÉ®ú +ÉnùÒ - +ÉvÉÒ

EòÒ VÉÉ ºÉEòiÉÒ ½èþ =iÉxÉä =ºÉ ºÉÆJªÉÉ EòÉ +lÉÇSUäônù Eò½äþ VÉÉiÉä ½èþ*’

which means arddhaccheda of a number is equal to the number of times it can be

divided by 2 ie., arddhaccheda of a number which can be put in the power form of

two is the power to which 2 must be raised. ie., if mk 2 then the arddhaccheda of

k is m.

E.g., 32 is divisible by 2 (5 times) or 532 2 .

Hence arddhaccheda of 32 is 5. But according to modern mathematics,

2log 32 5 .

i.e., 2

log x and arddhaccheda of x are same.

The concept of arddhaccheda is also seen in Dhaval¡-3, p-139-141 and p-342-

344.

98

2.7.2. Trikaccheda

Trikaccheda of a number is equal to the number of times that it can be

divided by 3 (Dhavala 3-p. 56). ie., trikaccheda of a number which can be put in the

power form of 3 is the power to which 3 must be raised. ie., if mk 3 then

trikaccheda of k is m.

E.g., 81 is divisible by 3, 4 times or 481 3

Hence trikaccheda of 81 is 4

But 3

log 81 4 . So 3

log x and trikaccheda of x are same.

Similarly caturth¡ccheda of a number is the number of times that it can be

divided by 4 (Dhavala - 3. P. 56).

Hence caturthaccheda of 256=4 or 4

log 256 4 .

The following formulae on logarithm are also seen in Dhavala.66

1. 2 2 2

log xy log x log y

2. 2 2 2

xlog log x log y

y

3. y

2 2log x ylog x

4. c

2log 2 c

99

5. 2

a

2 2log a 2alog a

6. 2

aloglog a loga 1 logloga

7. aa

a a alog a a loga

V¢rasena shows that 3 2 2

2 2log .logx x x

In Dhavala we can see 2

log , 3

log , 4

log , 5

log , etc. But as in modern

mathematics loge or

10log are nowhere seen.

Also 2 2log log of a number is given in Dhavala - 3 - p. 21, The name used for

2 2log log x is 'vargasalaka' of x.67

2.8. Mensuration

2.8.1. Measurement of Length

The infinitely minute quantity of matter which is not destroyed by water,

fire and by other such things is called paramanu68.

Then we get the following table

Ananta Paramanu - 1 Anu

8 Anu - 1 Trasarenu

8 Trasarenu - 1 Radharenu

100

8 Radharenu - 1 Uttama balamap

8 Uttama balamap - 1 Madhyama balamap

8 Madhyama balamap - 1 Jaghanya balamap

8 Jaghanya balamap - 1 Karmabhumi balamap

8 Karmabhumi balamap - 1 Liksa

8 Liksa - 1 Thila

8 Thila - 1 Yava

8 Yava - 1 Angula or Vyavaharangula

500 Vyavaharangula - 1 Pramana

9 11

Anu 8 8 8 8 8 8 8 8 8 8 1 Angula

3 3inch 2.54cm

4 4

1.905 cm

1.905Anu =

8 8 8 8 8 8 8 8 8 8

1.757 10 cm 1.757 10 m

101

In Brhatsamhita Varahamihira gives the table of length units as

8 Paramanu - 1 Rajass

8 Rajass - 1 Valagra

8 Valagra - 1 Liksa

8 Liksa - 1 Yuka

8 Yuka - 1 Yava

8 Yava - 1 Angula

1 Paramanu x 8x8x8x8x8x8 = I Angula

= 3

4 inch =

3x 2.54cm

4

= 1.905 cm

1 Paramanu = 7.2 x 10-6 cm = 7.2x10-8 m

It is surprising that not only such small values like valagra but also much

smaller values like 1

10000 of valagra were recognised and utilised in India even

in Upanishadic period. For example, in Svetasvetara upanishad we find,

102

´ÉÉ±ÉÉOÉ¶ÉiÉ¦ÉÉMÉºªÉ

¶ÉiÉvÉÉ EòÎ±{ÉiÉºªÉ SÉ

¦ÉÉMÉÉä VÉÒ´É: ºÉ Ê´ÉYÉäªÉ: 69

which means that Jivanu (Anu) = 1 1

x of valagra100 100

= 1 1 1.905

x x cm100 100 8x8x8x8

= 4.6 x 10-8 cm = 4.6 x10-10m

According to modern mathematics the diameter of the smallest atom (that

of Hydrogen) is 0.92x10-10 m and that of the biggest atom (that of Cecium) is

5.24x10-10m. So the values of G.S.S. and Svetasvetara upanishad lie with in this

range.

2.8.2. Measurement of Time

The time in which an atom goes beyond another atom is a samaya70

Asamkhya Samaya - 1 Avali

Asamkhya Avali - 1 Ucchvasa

7 Ucchvasa - 1 Stoka

7 Stoka - 1 Lava

103

12

38 Lava - 1 Ghati

2 Ghati - 1 Muhurta

30 Muhurta - 1 Dina

15 Dina - 1 Paksa

2 Paksa - 1 Masa

2 Masa - 1 Rtu

3 Rtu - 1 Ayana

2 Ayana - 1 Varsa

11 Ucchvasa x 7 x 7 x 38 x 2 x 30 24hrs.

2

24 x 60 x 60Second

24 x 60 x 601 Ucchvasa 0.763second

777 x 7 x x 2 x 30

2

But in Arthasastra the basic unit71 of time is tutam and 1 tutam = 0.06 second

In Tiloyapannatti vv. 4.286 et. seq. the following is given

5 Varsa - 1 Yuga

2 Yuga - 10 Varsa

104

10 x10 Varsa - 1 Satavarsa

10 Satavarsa - 1 Sahasravarsa

10 x 10 Sahasravarsa - 1 Laksa varsa

Continuing like this finally acalatma= 31 90

84 10 yearsisobtained72

2.8.3. Measurement of grain73

4 Sodasika - 1Kudaha

4 Kudaha - 1Prastha

4 Prastha - 1Adhaka

4 Adhaka - 1Drona

4 Drona - 1Mani

4 Mani - 1Khari

5 Khari - 1Pravartika

4Pravartika - 1 Vaha

5Pravartika - 1Kumbha

2.8.4. Measurement of gold74

4 Gandaka - 1 Gumja

105

4Gumja - 1 pana

8 pana - 1 Dharana

2 Dharana - 1Karsa

4 Karsa - 1 Pala

2.8.5. Measurement of silver 75

2Dhanya - 1 Gumja

2Gumja - 1 Masa

16 Masa - 1 Dharana

2 1

2Dharana - 1Karsa or Purana

4 Karsa or Purana - 1 Pala

2.9. Method of Calculation of Interest and Capital

The 6th chapter of G.S.S discusses mixed problems in which a section

contains problems bearing on interest.

Ê´É¦ÉVÉiº´ÉEòÉ±ÉiÉÉÊb÷iÉ¨ÉÚ±ÉºÉ¨ÉÉºÉäxÉ ¡ò±ÉºÉ¨ÉÉºÉ½þiÉ¨ÉÂ

EòÉ±ÉÉ¦ªÉºiÉÆ ¨ÉÚ±É¨É {ÉÞlÉEÂò {ÉÞvÉEÂò SÉÉÊnù¶ÉänÂù´ÉÞnÂùÊvÉ¨ÉÂ 76

106

In this sloka Mahavira explains the method for seperating various interests

for various capitals for various periods of time from the mixed sum of the interests.

Multiply each capital by the corresponding time and the total of the various

interests. This amount is divided by the sum of the products of each capital and

the corresponding time. This will give the interest for the particular capital.

If C1, C2, C3 . . . . . . are the various capitals and t1, t2, t3 . . . . . are the various

periods of time then the corresponding interests are given by

1i = 1 1

1 1 2 2

Ct m

Ct C t ...

2

i = 2 2

1 1 2 2

C t m

Ct C t ...

and so on where m = i1 + i2 + i3 + ..........

Similarly the rule for seperating various capitals from their mixed sum is

also given by Mahavira. As per G.S.S 6.39. If i1, i2, i3 ....... are the interests for

capitals C1, C2, C3 ......... for periods t1, t2, t3 . . . and m = C1 + C2 + C3 .......... then

1C = 1

1 2 3 1

1 2 3

im

i i i t.........

t t t

2C = 2

1 2 3 2

1 2 3

im

i i i t.........

t t t

and so on.

107

References

1 B. B. Datta and A. N. Singh, History of Hindu Mathematics Vol. 1, Bharatiya kala prakashan,

Delhi, 2004, , p. 7.

2 Ibid., p. 9.

3 C.Krishnan Namboodiri op. cit., pp. 86-87

4 Datta and Singh, op cit., p. 10

5 Ibid., p. 11.

6 Ibid., p. 12.

7 Ibid., p. 21

8 Ibid., p. 29.

9 C.Krishnan Namboodiri op cit.,pp. 68-70

10 M. D. Srinivas , Methodology of Indian Mathematics and its contemporary relevance (article),

History of science and technology in india, Vol. II, Sundeep, prakasan, Delhi, 1990., p. 173.

11 Datta and Singh, opcit., pp. 55-60

12 Ibid., pp. 55-104.

13 P. Ramachandramenon, op cit., p. 41.

14 Anupam Jain and N. Shiva Kumar, Arhatvacana Vol. 17, No. 2-3, April-Sept.

pp. 38-40.

108

15 L.C. Jain, Exact sciences from jaina sources, Vol. 1. Basic Mathematics, Rajasthan Prakriti

Bharati Samsthan, Jaipur, 1982, p. 25.

16 L.C. Jain, Mathamatical contents in the digambara jain text op.cit, p. 150.

17 Dipak Jadav, Arhat Vacana, Vol.II, July, 1999, pp.54-60

18 Dr. Hiralal jain (ed.), Dhavala3, Hindi translation and notes of Dhavala commentaries of

Virasena (Jivaraj jain Granthamala), Jain Samskrti samraksaka sanga, Sholapur-2, 2002, p.249.

19 RC Gupta, Jaina Cosmography and Perfect Numbers Arhat Vacana, Vol. 4(2,3) April, July 1992,

p. 89-94.

20 Datta and Singh, op cit., p. 8.

21 Ibid., p.124.

22 Ibid., p.124

23 G.S.S op.cit, v.2.1p.22.

24 Sridhara Patiganita, Department of Mathematics and Astronomy Lucknow, 1959, vv18-20.

25 K. V. Sarma (ed.) Lilavati of Bhaskara, Visvesvarant Vedic Research Institute, Hoshiarpur, 1975,

p.10.

26 Patiganita op.cit.,v.22

27 G.S.S , op.cit.pp, v.31, 31-33

28 Lilavati op.cit., v. 20

29 Patiganita op.cit.,v. 22

109

30 Dhavala,3 op.cit., p. 136.

31 G.S.S , op.cit.v.2.36, p.35

32 Ibid. v.1.52,p.16

33 Ibid. v.1.49,p.15

34 Dhavala,3 op.cit., p. 20.

35 L.C. Jain, Exact Sciences from Jaina Sources Vol. 1, Basic mathematics, Rajasthan Prakrit Bharti

Snasthan, Jaipur, 1982, p. 39.

36 Dhavala,3 op.cit., pp. 131-137.

37 Arhad Vachan, Kundakunda jnana pitha, Indore, 14 (2-3) 2002 p. 83.

38 G.S.S. Op cit, v.3.54, p.106.

39 Ibid., v.3.108, p.134.

40 Ibid., v. 3.11, p.83

41 Ibid., v. 3.116. p.138

42 Ibid., v. 3.121, 140.

43 Ibid., v. 3.129, 145.

44 Ibid., v. 3.73, p.114

45 Ibid., v. 3.87, p.125

46 Ibid., v. 3.85, p.124

47 Ibid., v.3.80, p.120

48 Ibid., v.3.78, p.119

110

49 Ibid., v.3.75, p.116

50 Ibid., v.3.77, p.118

51 Ibid., v.3.80, p.120

52 Ibid., v.3.55, p.106

53 Ibid., v.3.56, p.108

54 Sridhara, patiganita, op cit, v.37.

55 G.S.S. op cit, v.3.2, p.81

56 Ibid., v. 3.8, p.83

57 Ibid., 3.13, p.85

58 Patitiganita. op cit.,v 43.

59 G.S.S. op cit., v.5.2, p.194.

60 Patitiganita. op cit., v. 43

61 L. C. Jain Exact Sciences from jaina sources Vol. 1. Basic Mathematics op cit., p.39.

62 Ibid., p. 39.

63 Virasena, Dhavala-4, p.124

64 Pragati Jain and Anupam Jain, Acarya Virasena and his Mathematical contributions, Arhat

Vacana, Kundakunda jnana pitha, Indore, 14 (2-3), 2002.

65 D.E. Smith, History of Mathematics, Vol. 2. p. 527.

66 L.C. Jain, Basic Mathematics op.cit, p.30.

111

67 Dhavala op cit p. 21.

68 G.S.S. op cit., v.1.25-27., p.8.

69 Upanishad Sarasarvasvom Samrat publications, Trissur, 2001, p.106.

70 G.S.S. op cit., v.1.32-35., p.10

71 C.Krishnan Namboodiri Opcit p.55

72 L. C. Jain, Mathematical contents in the Digambara jaina texts of the Karananuyoga group, op

cit., p.148.

73 G.S.S. op cit., v.1.36-38., p.11.

74 Ibid., v.1.39, p.12

75 Ibid., v.1.40, p.12

76 Ibid., v.6.37, p.229

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