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Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using positions –Mathematical practices, accounting practices Agathe Keller, CNRS, Université Paris-Diderot Friday, January 6, 2012

Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

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Page 1: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Positional and tabular notations

in Sanskrit mathematical texts (VIIth-Xth century)

SAW seminar January 6, 2011

Using positions –Mathematical practices, accounting practices

Agathe Keller,CNRS, Université Paris-Diderot

Friday, January 6, 2012

Page 2: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Friday, January 6, 2012

Page 3: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

HIGHER SECONDARY – FIRST YEAR

A C C O U N TA N C Y

Untouchability is a Sin

Untouchability is a Crime

Untouchability is Inhuman.

TAMILNADU

TEXTBOOK CORPORATION

College Road, Chennai - 600 006.

CHAIRPERSON

Dr. (Mrs) R. AMUTHA

Reader in Commerce

Justice Basheer Ahmed Sayeed College for Women

Chennai - 600 018.

REVIEWERS

Dr. K. GOVINDARAJAN Dr. M. SHANMUGAM

Reader in Commerce Reader in Commerce

Annamalai University SIVET College

Annamalai Nagar - 608002. Gowrivakkam,Chennai-601302.

Mrs. R. AKTHAR BEGUM

S.G. Lecturer in Commerce

Quaide-Millet Govt. College for Women

Anna Salai, Chennai - 600002.

AUTHORS

Thiru G. RADHAKRISHNAN Thiru S. S. KUMARAN

S.G. Lecturer in Commerce Co-ordinator, Planning Unit

SIVET College (Budget & Accounts)

Gowrivakkam, Chennai - 601302. Education for All Project

College Road, Chennai-600006.

Thiru N. MOORTHY Mrs. N. RAMA

P.G. Asst. (Special Grade) P.G. Assistant

Govt. Higher Secondary School Lady Andal Venkatasubba Rao

Nayakanpettai - 631601 Matriculation Hr. Sec. School

Kancheepuram District. Chetpet, Chennai - 600031.

PREFACE

The book on Accountancy has been written strictly in accordance

with the new syllabus framed by the Government of Tamil Nadu.

As curriculum renewal is a continuous process, Accountancy

curriculum has undergone various types of changes from time to time in

accordance with the changing needs of the society. The present effort

of reframing and updating the curriculum in Accountancy at the Higher

Secondary level is an exercise based on the feed back from the users.

This prescribed text book serves as a foundation for the basic

principles of Accountancy. By introducing the subject at the higher

secondary level, great care has been taken to emphasize on minute

details to enable the students to grasp the concepts with ease. The

vocabulary and terminology used in the text book is in accordance with

the comprehension and maturity level of the students.

This text would serve as a foot stool while they pursue their higher

studies. Since the text carries practical methods of maintaining accounts

the students could use this for their career.

Along with examples relating to the immediate environment of the

students innovative learning methods like charts, diagrams and tables

have been presented to simplify conceptualized learning.

As mentioned earlier, this text serves as a foundation course which

is coupled with sample questions and examples. These questions and

examples serve for a better understanding of the subject. Questions

for examinations need not be restricted to the exercises alone.

Chairperson

© Government of Tamilnadu

First Edition - 2004

Price : Rs.

This book has been prepared by the Directorate of School Education

on behalf of the Govt. of Tamilnadu.

This book has been printed on 60 G.S.M. paper

Printed by Offset at :iii

Friday, January 6, 2012

Page 4: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

!Lall Nigam, B. M. «Bahi-khata: The pre-pacioli indian double entry system

of bookkeeping.» Abacus, 22(2):148–161, 1986.

!!Lall!Nigam, B. M. «Double-entry system of book-keeping». Chartered

Accountant, New Delhi, XXXVI(2), 1987.

!Nobes, C.!W. «The pre-pacioli Indian double-entry system of bookkeeping:

A comment». Abacus, 23(2):182–184, September 1987.

! Scorgie, M.!E. «Indian imitation or invention of cash-book and algebraic

double-entry». Abacus, 26(1):63–70, March 1990.

!Scorgie, Michael E;!Nandy; S. C. « Emerging evidence of early Indian

accounting». Abacus, 28(1):88–97, 1992.

Friday, January 6, 2012

Page 5: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Friday, January 6, 2012

Page 6: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Extracting a Square Root: Âryabhata’s verse

Ab.2.4

Bh!ga" hared avarg!n nitya" dvigu#ena

vargam$lena//

Varg!d varge ßuddhe labdhaµ sth!n!ntare

m$lam//

On should divide, constantly the non-square <place> by twice the square-root /When the square has been subtracted from the square <place>, the quotient is the root in a different place//

Friday, January 6, 2012

Page 7: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Extracting a Square Root: The example of 625

A square place is one that stands for an even power of ten (100, 102, 104, etc.)

uneven even uneven

vi%ama sama vi%ama

102 101 !! ! 100

! square! non-square !square

! varga! a-varga&! varga

6 2 5

Friday, January 6, 2012

Page 8: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Extracting a Square Root: The example of 625

102 101 ! ! 100

!

6 2 5- 4 2 2 5

Square root

2

Friday, January 6, 2012

Page 9: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Extracting a Square Root: The example of 625

102 101 ! ! 100

!

2 2 5 2 5 - 2 5 0

Square root

2 5

22/4 = 5 + 2/4

Friday, January 6, 2012

Page 10: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

2 3 5 8

5 8 2 3

4 1 7 6

7 6 4 1

2 3 5 8

5 8 2 3

4 1 7 6

7 6 4 1

2 3 5 8

5 8 2 3

4 1 7 6

7 6 4 1Friday, January 6, 2012

Page 11: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Friday, January 6, 2012

Page 12: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

f. 3r: Elongation of the nodes of the Sun and the Moon for 130 year period

The Kara"akesar#(fl. 1681)

Friday, January 6, 2012

Page 13: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

bh!gaj!ti part class

a1

b1

a2

b2+

LV 30 anyonyah!r!bhihatau har!"%au r!%yor samachedavidh!nam eva"\\

mithas har!bhy!m apavartit!bhy!m yadv! har!"%au sudhiy! atra gu#yau//\\

The numerator and denominator being multiplied reciprocally by the denominators of the

two quantities, they are thus reduced to the same denominators. Or both numerator and

denominator may be multiplied by the intelligent calculator into the reciprocal

denominators abridged by a common measure.

a1

b1

a2

b2

+b2 b1

3

1

1

5+

15

53+

1

3

Friday, January 6, 2012

Page 14: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

prabh$gaj$ti different part classa1

b1x

a2

b2

LV032 lav!llavaghn!% ca har!' haraghn!' bh!gaprabh!ge(u savar#anam sy!t\\

The numerators multiplied by the numerators, and the denominators by the denominators will be same-

coloured when [they are] different parts .

a1

b1

a2

b2

a1

b1

a2

b2x

1

1

1

2x

1280

12

3x

3

4x

1

16x

1

4x

Friday, January 6, 2012

Page 15: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

(sva)bh$g$nubandha%j$ti one’s own part additive class

a2

b2

a1

b1

k$lasa%var"a samecoloured portion

a1

b1

n

LV034 chedaghnar$pe(u lav!' dhana)#am ekasya bh!g!s adhikaunak!% ced//

sv!"%!dhikaunas khalu yatra tatra bh!g!nubandhe ca lav!pav!he/

talasthah!re#a haram nihany!t sv!"%!dhikaunena tu tena bh!g!n//

The integer being multiplied by the denominator, the numerator is made positive or negative, provided

parts of an unit be added or be subtractive. But, if indeed the quantity be increased or diminished by a

part of itself, then, in the addition and subtraction of fractions, multiply the denominator by the

denominator standing underneath, and the numerator by the same augmented or lessened by it own

numerator.

a1(b2 +a2)

b1 b2

a1/b1 + (a1/b1 x a2/b2) n+a1/b1

b1 n+a1

b1

Friday, January 6, 2012

Page 16: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

svabh$g$pavahoj$ti one’s own subtractive part class

bh$g$m$taj$ti mother part class

-a2

a1

b2

b1

-a1

b1

n

LV034 chedaghnar$pe(u lav!' dhana)#am ekasya bh!g!s adhikaunak!% ced//

sv!"%!dhikaunas khalu yatra tatra bh!g!nubandhe ca lav!pav!he/

talasthah!re#a haram nihany!t sv!"%!dhikaunena tu tena bh!g!n//

The integer being multiplied by the denominator, the numerator is made positive or negative, provided

parts of an unit be added or be subtractive. But, if indeed the quantity be increased or diminished by a

part of itself, then, in the addition and subtraction of fractions, multiply the denominator by the

denominator standing underneath, and the numerator by the same augmented or lessened by it own

numerator.

a1 (b2 -a2)

b1 b2

b1 n-a1

b1

a1/b1 - (a1/b1 x a2/b2) n+a1/b1

Friday, January 6, 2012

Page 17: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Rule of three

A measure M produces a fruit F, if I

desire D what is obtained? The fruit

of the desire R

F/M = R/D R= (FxD)/M

Friday, January 6, 2012

Page 18: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Setting number’s down

Bh!skara :

To accomplish intelligently a rule of three

The two same quanitities are <disposed> at the beginning and in the end.

The different quantity is <placed> in the middle.

M! F! D

Friday, January 6, 2012

Page 19: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

(S.R. Sharma) or

M!! ! ! ! M!D

D! ! ! ! ! F

F! ! ! ! ! ! ! !

M D F

Friday, January 6, 2012

Page 20: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Setting down a rule of 2n+1 quantities

N measure quantities (M1, M2,…Mn) produce together a fruit quantity F. We know n desire quantities (D1, D2 … Dn) and we are looking for the fruit of the desire (R).

R = (Fx D1 x D2 x …x Dn) / (M1 x M2 x …x Mn)

Disposition!:

M1& & D1& & .& & .

.& & .

.& & .

Mn& & Dn

F! ! !

Friday, January 6, 2012

Page 21: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Li.80. If the interest of a hundred for a month be five, say

what is the interest of sixteen for a year? (...)

Statement1 12

100 16

5

Friday, January 6, 2012

Page 22: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Li.80. If the interest of a hundred for a month be five, say

what is the interest of sixteen for a year? (...)

Statement1 12 time in months

100 16 capital

5 interest

Friday, January 6, 2012

Page 23: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Li.80. If the interest of a hundred for a month be five, say

what is the interest of sixteen for a year? (...)

Statement1 12

100 16

5

12x16x5=960

The interest obtained is9

3

5

Friday, January 6, 2012

Page 24: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Quantities

R!"i: quantity

Sa!khy!: value

A!ka: digit

[Bhinna, Saccheddha

A while computing

B

When stating an amount

a a

b b°

c c

R"a; Dhana (rules to sum and subtract)

Friday, January 6, 2012

Page 25: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Li.81. If the interest of a hundred for a month and one third

be five and one fifth, say what is the interest of sixty-two

and a half for three months and one fifth?

Statement

4 16

3 5

100 125

1 2

26

5

Friday, January 6, 2012

Page 26: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Li.81. If the interest of a hundred for a month and one third

be five and one fifth, say what is the interest of sixty-two

and a half for three months and one fifth?

4 16

3 5

100 125

1 2

26

5

Friday, January 6, 2012

Page 27: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Li.81. If the interest of a hundred for a month and one third

be five and one fifth, say what is the interest of sixty-two

and a half for three months and one fifth?

4 16

3 5

100 125

1 2

26

5

Friday, January 6, 2012

Page 28: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

Li.81. If the interest of a hundred for a month and one third

be five and one fifth, say what is the interest of sixty-two

and a half for three months and one fifth?

4 16

5 3

100 125

2 1

5 26

The interest is 7

4

5

Friday, January 6, 2012

Page 29: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

BG.E.36. One person has three hundred r$pas and six

horses. Another has ten horses of like price, but he owes a

debt of one hundred r'pas. They are both equally rich. What

is the price of a horse?

300 + 6x=10x-100

4x=400

x=100

y$ 6 r' 300

y$ 10 r' 100°

y!vatt!vat,

avyakta unmanifested quantity

)#a debt, «negative» quantity

dhana wealth, «positive» quantity Friday, January 6, 2012

Page 30: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

BG.E.42. Subtracting from a capital lent at five in the

hundred, the square of the interest, the remainder was lent at

ten in the hundred. The time of both loans was alike, and

the amount of the interest equal. [Say what were the initial

capitals?]

xi two initial capitals; y final interest; z loan time

y= 5x1z/100 = 10x2z/100

x2=x1-y

2

Friday, January 6, 2012

Page 31: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

BG.E.42. Subtracting from a capital lent at five in the hundred, the square of the interest, the remainder

was lent at ten in the hundred. The time of both loans was alike, and the amount of the interest equal.

[Say what were the initial capitals?]

xi two initial capitals; y final interest; z loan time

1 5 time

100 y$ 1 capital

5 interest

2

Solution 1 Assume z=5 Rule of 5

«the interest obtained is y$ 1

4

its square is y$va 1 »

16

y= 5x1z/100 = 10x2z/100

x2=x1-y

y=1/4 x1

y=1/16 x1

2

2

2

Friday, January 6, 2012

Page 32: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

BG.E.42. Subtracting from a capital lent at five in the hundred, the square of the interest, the remainder

was lent at ten in the hundred. The time of both loans was alike, and the amount of the interest equal.

[Say what were the initial capitals?]

xi two initial capitals; y final interest; z loan time

y= 5x1z/100 = 10x2z/100

x2=x1-y 2

1 5 time

100 y$va 1 ° y$ 16 capital

16

10 interest

Rule of 5

x2 is therefore of the form y$va 1 °

16

y$ 16

«the interest obtained is y$va 1° y$ 16 »

32

[-x1+16x1]/162

y= [-x1+16x1]/322

Friday, January 6, 2012

Page 33: Positional and tabular notations in Sanskrit mathematical ... · Positional and tabular notations in Sanskrit mathematical texts (VIIth-Xth century) SAW seminar January 6, 2011 Using

BG.E.42. Subtracting from a capital lent at five in the hundred, the square of the interest, the remainder

was lent at ten in the hundred. The time of both loans was alike, and the amount of the interest equal.

[Say what were the initial capitals?]

xi two initial capitals; y final interest; z loan time

y= 5x1z/100 = 10x2z/100

x2=x1-y 2 y$ 1 ”

4

y$va 1° y$ 16

32

“is equal to

“having reduced by the y$vatt$vat both sides, in

order to equally subtract the two sides are set

down: y$ 1° r' 16

32

y$ 0 r' 1

4

“proceeding as before the measure of the

y$vatt$vat is 8”

[-x1+16x1]/32=x1/42

[-x1+16]/32=0+1/4

x1 = 8

Friday, January 6, 2012