10 aryabhatiya iv (kr)

7/23/2019 10 Aryabhatiya IV (KR)

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NPTEL COURSE ON

MATHEMATICS IN INDIA:

FROM VEDIC PERIOD TO MODERN TIMES

LECTURE 10

Aryabhat.ya of Aryabhat.a - Part 4

and

Introduction to Jaina Mathematics

K. Ramasubramanian

IIT Bombay
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OutlineAryabhat.yaof Aryabhat.a Part 4 + Jaina Mathematics

Some algebraic identities

TheKut.t.akaproblem

Meaning of the term + formulation of problem Need for solvingkut.t.akaproblem (Bhaskaras example) Solution to theKut.t.akaproblem Explanation of the verses given by Aryabhat.a Presenting the algorithm in modern notation

Illustrative example

Jaina Mathematics

Introduction Results on Mensuration Notion of infinity Law of indices Typical contents of a Jaina text
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The number of terms in an arithmetical series

Consider an arithmetical series of the form

a+ (a+d) + (a+2d) + (a+3d). . . . . .+ (a+ (n 1)d). (1)

The formula for finding the number termsnin the series, interms of its sumS, the first termaand the common difference dis encoded in the following verse:1

The content of the above verse can be expressed as:

n= 1

2

8Sd+(2a d)22a

d +1

(2)

1Aryabhat.a, Aryabhat. ya,Gan. itapada, verse 20.
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Some algebraic identities

2

Multiply theproductby four, then add the square ofthe difference of thetwo(quantities), and then take the square root. [Set down this square rootin two places]. (In one place) increase it by the difference (of the twoquantities), and (in the other place) decrease it by the same. The resultsthus obtained, when divided by two,give the two factors[of the givenproduct].

The content of the above verse may be expressed as,

If x y= a; and xy= b, then

x =4b+ a2 + a

2

y =

4b+ a2 a

2 .

2Aryabhat.a, Aryabhat. ya,Gan. itapada, verse 24.


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Kut.t.akaalgorithmThe meaning behind the nomenclature kut.t.aka

The problem of solving first order indeterminate equations wasso important to Indian astronomer-mathematicians as there wasacompelling needfor them in choosing a given epoch, withconstraints imposed by various planetary parameters.3

The termKut.t.anain general means pulverisingthe act of

reducing somethingto finer sizesby repeated operation. When applied in the context of mathematics, Kut.t.anarefers to

repeated divisionby which thegiven numbersare made smallerand smaller by mutual division.

This algorithm also plays a key role in finding the solution of themuch more difficult problem namely, second order indeterminateequations (varga-prakr. ti).

3In order to demonstrate its application in a variety of contexts, several

examples have been presented by Bhaskara,

,

,

, and so on.
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Formulation of the kut.t.akaproblem

Suppose there is an integerNwhich when divided by two

integers(a, b)leaves remainders(r1, r2). That is,

N = ax+r1

also = by+r2. (3)

This equation may be written as

by c=ax, where c=r1 r2. (4)

Kut.t.akaproblem: Given (a,b,c) that are integers, we need

to find (x,y)again integersthat will satisfy the above

equation.

Historical error: Usually the equation given above is called

Diophantine equation. Diophantus who lived in 3rd cent

AD was concerned with finding rational solutionsnot

integer solutionsof (2), which is a much simpler problem.
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Kut.t.akaproblem: Bhaskaras example

Statement of the problem

. . .

Consider the equation

byc=ax, where (5)

b no. of man. d. alasof sun in aMahayuga(4320000) a no. ofbhudivasasin aMahayuga(1577917500)

cman. d. alases. a, obtained by observation (86688)
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Solution to the Kut.t.akaproblem

Aryabhat.apresents the solution in two verses Gan. itapada, 3233):

-

remainder (r1)

for which the remainder is large (a)

the dividend (a)

may you divide

the divisor (b)

being mutually divided by the remainder

multiplied bymati(optional multiplier)

added to/subtracted by the diff. of the remainders

(

?

)
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Solution to the Kut.t.akaproblem One step (left as

by Aryabhat.a):

,

,

Having placed the quotients one below the other,below them themati,and underneath the mati the quotient just obtained

Aryabhat.apresents the solution in two verses Gan. itapada, 3233):

. . .

-

- number below multiplied by the one just above it

(

)

added to the last (one just below it)

when divided by the divisor corresponding to the

smaller remainder (b)

the remainder multiplied by the divisorcorresponding the greater remainder (a)

when added to the greater remainder gives

the number corresponding to the two divisors. (agram = sankhya)
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Solution to the Kut.t.akaproblemTranslation of the verses

Below we give the translation of these verses by Datta and

Singh following the interpretation of Bhaskara I:

Divide the divisor corresponding to the greater remainder by the divisor corresponding

to the smaller remainder. The remainder (and the divisor corresponding to the smaller

remainder) being mutually divided, the last residue should be multiplied by such an

optional integer that the product being added (in case the number of quotients of the

mutual division is even) or subtracted (in case the number of quotients is odd) by the

difference of the remainders (will be exactly divisible by the penultimate remainder.

Place the quotients of the mutual division successively one below the other in a

column, below them the optional multiplier and underneath it the quotient just

obtained). Any number below (i.e., the penultimate) is multiplied by the one just above

it and then added to the last (one just below it). Divide the last number (obtained by

doing so repeatedly) by the divisor corresponding to the smaller remainder; then

multiply the remainder by the divisor corresponding the greater remainder and add the

greater remainder. (The result will be) the number corresponding to the two divisors.
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The Kut.t.akaalgorithm

Let the two remainders be such thatr1 >r2, so that(a, b)be thedivisors corresponding to the greater and smaller remainders

respectively. Letc=r1 r2. We write down the procedure, when the number of quotients

(ignoring the first oneq) is even.

b) a (q

bqr1) b (q1

r1q1r2) r1 (q2

r2q2

..

.r2n) r2n1 (q2n

r2nq2nr2n+1


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The Kut.t.akaalgorithmArranging the quotients and thechoice ofmati

The prescription for the choice of the optimal numbert (mati) is:

r2n+1t+cshould be divisible byr2n. (quotients even) r2nt c should be divisible byr2n1 (quotients odd)

Letsbe the quotient

Having foundtands, we have to arrange them in the form of a vall

(column), to generate successive columns.q1 q1 q1 . . . . . . q12n1 + 2n2q2 q2 q2 . . . . . . 2n1. . .

. . .

. . .

q2n1 q2n1 q2n1 1 + t= 2q2n q2nt+ s= 1t t

s

Divideq12n1 + 2n2 byb. The remainder is xandN= ax+ r1.


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The Kut.t.akaalgorithmExample 2: To solve 45x+ 7 = 29y. Here a= 45, b= 29, r1 = 7, r2 = 0.

29) 45 (12916) 29 (1

1613) 16 (1

133) 13 (4

121

Here the number of quotients (omitting the first) is odd. tshould be chosen such that1 t 7 is divisible by 3. Hence tis chosen to be 10. Therefore we have,

1 1 1 92

1 1 51 514 41 4110 101

Now, 92 = 29 3 + 5. Hence,N= 49 5 + 7 = 29 8.Thusx= 5, y= 8.
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Introduction

Jainas seem to have regarded mathematics as an integral

part of their religion. A section of their religious literaturewas namedGan. itanuyoga(system of calculation).

The important Jaina mathematical works include: Jambu-dvpa-praj napti Surya-prajnapti Sthananga-sutra Bhagavati-sutra Uttaradhyayana-sutra Anuyoga-dvara-sutra Trilokasara

Gan. itasarasangraha Our knowledge about the earlier Jaina works is primarily

based on commentaries as many of the original works are

yet to come to light.
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Results on mensuration

One of Umasvatis (c. 219 CE) most important works,

Tattvarthadhigama-sutra-bhas. ya, contains mathematicalformulae and results on mensuration:

1. cirumference of a circle =

10 diameter2. area of a circle = 1

4circumference diameter

3. chord = 4sara(

diameter

sara)

4. sara = 12

[diameter

diameter2 chord2

5. arc of segment less than a semicircle =

6sara2 +chord2

6. diameter = sara2+ 14 chord

2

sara

The term saraemployed above refers to Rversine (RRcos). It has been observed that these results have been taken by

Umasvatifrom other mathematical works extant in his time as hehimself was not known to be a mathematician.

R l i b h i f di d
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Relation between the circumfrerence, diameter and

area of a circle

4

IfCbe the circumference of the circle,dits diameter, andA the

area, then the formulae given above may be expressed as:

C = 3d (gross)C =

10d2 (subtle)

A = C d4

4Nemicandras Trilokasara.

Dh j b h b dh h
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Dhanur-jya-vyasa-ban. ayoh. sam.bandhah.Relation between the arc, chord, diameter and arrow (Rversine)

5

Ifdbe the diameter of the circle,athe arc length,c the

corresponding chord, andhthe Rversine, then the formulae

given in the above verse may be expressed as:

c2 = 4h(d h)a2 = 6h2 +c2.

5Nemicandras Trilokasara.

N ti f i fi it ( d it diff t t )
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Notion of infinity(and its different types)

Jainas had names for different positionssthanain the numeralsystem: eka, dasa, sata, sahasra, dasa-sahasra,

dasa-sata-sahasra, kot.i, dasa-kot.i, sata-kot.i, etc. It has been stated that very large numbers were used for

measurements of space and time.6

Jainas have classified numbers as

1. enumerables (

)2. unenumerable (

) and3. infinite ( ).

They also talk of different types of infinity:

1. infinite in one direction (

)2. infinite in two directions (

),

3. infinite in area ( )

4. infinite everywhere ( )5. infinite perpetually ( )

6No nation has used such large numbers as theJainasandthe Buddhists.

H t i f i fi it l l b ?
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How to conceive of infinitely large numbers?

It would be difficultand inappropriate tooto find parallelsCantors notions of infinity.

However, it is evident that have conceived of infinity bothspatiallyandtemporallywhich by no means is crude.

Jainas mathematicians have providedpractical examplesthrough which one can conceive of enumerable infinite

Consider a trough whose diameter is of the size of the earth

(100,000yojanas). Fill it up with white mustard seeds

counting them one after another. Similarly fill up with

mustard seeds other troughs of the sizes of the various

lands and seas. Still it is difficult to reach the highest

enumerable number.

They also seem to have developed ( first cent. BC E) afomulation of the law of indices which isquite noteworthyconsidering the fact thatnotations had not been developedintheir full blown form.

Dealing ith la s of indices
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Dealing with laws of indicesExcerpts fromUttaradhyayana-sutraand Anuyoga-dvara-sutra

The textUttaradhyayana-sutra(

third cent. BC E)7 enumerates

powers and roots of numbers: varga-varga((a2)2 =a4) ghana-varga((a3)2 =a6) ghana-varga-varga(((a3)2)2 =a12) varga-mula-ghana((a

12)3 =a

18 )

InAnuyoga-dvara-sutrawe find the statement

the first square root multiplied by the second squareroot, or the cube of the second square root;thesecond square root multiplied by the third square root,

or the cube of the third square root,

which symbolically translates to

a12 a14 = (a14 )3; a14 a18 = (a18 )3.

7CNS, p. 25.

Typical content of Jaina mathematical text
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Typical content of Jaina mathematical textTheSthananga-sutrasupposed to be composed around 300 (BC E), mentions

the following 10 topics:

: The four fundamental operations - subtraction, addition,multiplication, division

: Application of arithmetic to concrete problems

- : Fractions

: Geometry (called Sulvain the Vedic period)

: This may refer to either measurement of grains or it may be

referring to mensuration of plane and solid figures

- : The word of unknown quantity x, using the algebraicsymbolya

: Permutations and combinations, discussed in the next

section

: Squaring

: Cubing

- : This and the previous need not necessarily mean square,cube and square-square. They may also refer to higher

powers and roots.

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10 aryabhatiya iv (kr)

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