Taylor James - Indian Math

8/14/2019 Taylor James - Indian Math

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India's Contribution to Mathematics

©James Taylor 2002

[email protected]

In India, mathematics has its roots in Vedic literature which is nearly 4000 years old. Between

000 B.!. and 000 ".#. various treatises on mathematics were authored $y Indian mathematicians inwhich were set %orth %or the %irst time, the concept o% &ero, the techni'ues o% al(e$ra and al(orithm,

s'uare root and cu$e root. " method o% (raduated calculation was documented in the )ancha*+iddhantia

-ive )rinciples/ in the th !entury, $ut the techni'ue is said to date %rom Vedic times circa 2000 B.!. Ithas now $een (enerally accepted that the techni'ue o% al(e$ra and the concept o% &ero ori(inated in India.

The rudiments o% 1eometry, called eha*1anita in ancient India, were %ormulated and applied in the

dra%tin( o% 3andalas %or architectural purposes. They were also displayed in the (eometric patterns usedin many temple moti%s. 3any moti%s in indu temples and )alaces display a mi5 o% %loral and 1eometric

patterns.

They can also $e credited with the concept o% ne(ative num$ers. They showed that these newnum$ers could $e as use%ul as positive num$ers $y employin( them to represent de$ts. In %act, they

%ormulated the arithmetic operations on ne(ative num$ers with this application in mind.

Algebra- The Other Mathematics?

In ancient India conventional mathematics termed 1anitam was nown $e%ore the development o%al(e$ra. This is $orne out $y the name * Bija(anitam, which was (iven to the al(e$raic %orm o%

computation. Bija(anitam means 6the other mathematics6 -Bija means 6another6 or 6second6 and 1anitam

means mathematics/. The %act that this name was chosen %or this system o% computation implies that itwas reco(nised as a parallel system o% computation, di%%erent %rom the conventional one which was used

since the past and was till then the only one.

"rya$hatta -478 9 0 "#/ has re%erred to Bija(anitam in his treatise on 3athematics, Aryabhattiya. This wor, %inished in 4::, is a summary o% indu mathematics up to that time, written inverse. It covers astronomy, spherical tri(onometry, arithmetic, al(e$ra and plane tri(onometry.

"rya$hata (ives %ormulas %or the areas o% a trian(le and a circle which are correct, $ut the %ormulas %or

the volumes o% a sphere and a pyramid are wron(. Aryabhatiya also contains continued %ractions, 'uadratic e'uations, sums o% power series and a ta$le o%

sines. "rya$hata (ave an accurate appro5imation %or π -e'uivalent to ;.48/ and was one o% the %irst

nown to use al(e$ra. The %irst appearance o% the sine o% an an(le appears in this wor, and he also

introduced the versine - versinθ < 9 cosθ/ into tri(onometry. e (ave ta$les o% hal% chords which now

really are sine ta$les and used jya %or our sine. This same ta$le was reproduced in the wor o%

Brahma(upta -in 82=/ and detailed method %or constructin( a ta$le o% sines %or any an(le were (ive $y

Bhasara in 0. By a tortuous path, the word sine itsel% can $e traced $ac to the "rya$hata. Theindu word jya %or the sine was adopted $y the "ra$s who called the sine jiba , a meanin(less word

with the same sound as jya. >ow jiba $ecame jaib in later "ra$ writin(s and this word does have a

meanin(, namely a 6%old6. ?hen uropean authors translated the "ra$ic mathematical wors into Aatin

they translated jaib into the word sinus meanin( fold in Aatin. In particular i$onacci6s use o% the term sinus rectus arcus soon encoura(ed the universal use o% sine.

"rya$hata also wrote the astronomy te5t Siddhanta which tau(ht that the apparent rotation o% theheavens was due to the a5ial rotation o% the arth. The wor is written in 2 stan&as. It (ives a

remara$ly modern view o% the nature o% the solar system.

mailto:[email protected]

mailto:[email protected]



"rya$hata (ives the radius o% the planetary or$its in terms o% the radius o% the arth+un or$it as

essentially their periods o% rotation around the +un. e $elieves that the 3oon and planets shine $yre%lected sunli(ht, and incredi$ly he $elieves that the or$its o% the planets are ellipses. e correctly

e5plains the causes o% eclipses o% the +un and the 3oon.

is value %or the len(th o% the year at ;8 days 8 hours 2 minutes ;0 seconds is an overestimatesince the true value is less than ;8 days 8 hours

"round the 2th century ".# Bhasaracharya wrote 6+iddhanta*+hiromani6 o% which one section is

entitled Bija(anitam. Thus the techni'ue o% al(e$raic computation was nown and was developed inIndia in earlier times. rom the ;th century onwards, India was su$ject to invasions %rom the "ra$s andother Islamised communities lie the Turs and "%(hans. "lon( with these invaderC came chroniclers and

critics lie "l*$eruni who studied Indian society and polity. It was also the a(e o% the Islamic

enaissance and the "ra$s (enerally improved upon the arts and sciences that they im$i$ed %rom theland they overran durin( their (reat Jehad. )erhaps the most important person mathematically was the

splendidly named "$u "$d*"llah i$n 3usa al6Dhwari&mi -7:0*=40/

"l6Dhwari&mi6s most important wor Hisab al-jabr w'al-muqabala written in =;0 (ives us the word

algebra, %rom the latinised title Ludus algebra at almucgrabalaeque . "l*je$r means Ethe reunionF. This

treatise classi%ies the solution o% 'uadratic e'uations and (ives (eometric methods %or completin( the

s'uare -see $elow/ >o sym$ols are used and no ne(ative or &ero coe%%icients were allowed.

Between the 0th to ;th centuries, the !hristian in(doms o% urope made numerous attempts to

recon'uer the oly Aand %rom its 3ohammedan*"ra$ rulers. These !rusades %ailed in their militaryo$jective, $ut the contacts they created $etween oriental and occidental nations resulted in a massive

e5chan(e o% ideas. The techni'ue o% al(e$ra could have passed on to the west at this time. #urin( the

enaissance in urope, %ollowed $y the industrial revolution, the nowled(e received %rom the east was%urther developed. "l(e$ra as we now it today has lost any characteristics that $etrays its eastern ori(in

save the %act that the term reminds us o% the title 6"l ja$r6 which the "ra$s (ave to Bija(anitam.

Incidentally the term Bija(anit is still use in India to re%er to this su$ject.

Bhasara-4 9 = "#/ represents the pea o% mathematical nowled(e in the 2th !entury

and reached an understandin( o% the num$er systems and solvin( e'uations which was not to $e reached

in urope %or several centuries. Bhasara was head o% the astronomical o$servatory at Gjjain, the leadin(

mathematical centre in India at that time.e understood a$out 0 and ne(ative num$ers, and he new that 5 < : had two solutions. e (ives

the %ormula

a ± β = α + α 2 − b

2±

a − a 2− b

2

Bhasara also studied )ell6s e'uation x2

= 1 + πψ 2

%or p < =, , ;2, 8 and 87. ?hen p < 8 he

%ound the solutions x <778;:04:, y < 228;:0. e studied many #iophantine pro$lems.

Bhasara6s mathematical wors include Ailavati -The Beauti%ul/ and Bija(anita -+eed !ountin(/

while he also wrote on astronomy.httpCwww*history.mcs.st*andrews.ac.uhistory3athematiciansBhasara.htmlIn the year =8, an n(lishman $y the name James Taylor translated Bhasara6s Ailavati into

n(lish. " second n(lish translation appeared in the %ollowin( year -=7/ $y the n(lish astronomer

enry Thomas !ole$rue.. Thus the wors o% this Indian mathematician astronomer were made nownto the western world nearly 700 years a%ter he had penned them, althou(h his ideas had already reached

the west throu(h the "ra$s many centuries earlier. In the words o% the "ustralian Indolo(ist ".A. Basham

-".A. BashamH The ?onder That was India./ ... the world owes most to India in the realm o%mathematics, which was developed in the 1upta period to a sta(e more advanced than that reached $y

any other nation o% anti'uity. The success o% Indian mathematics was mainly due to the %act that Indians



had a clear conception o% the a$stract num$er as distinct %rom the numerical 'uantity o% o$jects or spatial

e5tension. Thus Indians could tae their mathematical concepts to an a$stract plane and with the aid o% asimple numerical notation devise a rudimentary al(e$ra as a(ainst the 1rees or the ancient (yptians

who due to their concern with the immediate measurement o% physical o$jects remained con%ined to

3ensuration and 1eometry.

Geometry And Algorithm

But even in the area o% 1eometry, Indian mathematicians had their contri$ution. There was anarea o% mathematical applications called eha 1anita -Aine !omputation/. The Sulva Sutras, which

literally mean 6ule o% the !hord6 (ive (eometrical methods o% constructin( altars and temples. The

temple layouts were called 3andalas. +ome o% important wors in this %ield are $y "pastam$a,Baudhayana, iranyaesin, 3anava, Varaha and Vadhula.. The Buddhist )a(odas $orrowed their plan o%

construction %rom the (eometric (rid o% the 3andala used %or constructin( temples in India The "ra$

scholar 3ohammed I$n Ju$air al Battani studied Indian use o% ratios %rom eha 1anita and introducedthem amon( the "ra$ scholars lie "l Dhwara&mi, ?ashiya and "$e 3ashar who incorporated the

newly ac'uired nowled(e o% al(e$ra and other $ranches o% Indian mathematics into the "ra$ ideas

a$out the su$ject.

It was "l Dhwara&mis second $oo that $rou(ht the indu num$er system to the "ra$ world,leadin( to the most success%ul num$er system o% all * the indu* "ra$ic num$er system . It is instructive

to realise that it was not an immediate success * merchants thorou(hly %amiliar with the oman system

and %ound the new "ra$ic system very con%usin(, and early e5amples o%ten contain mistaes or mi5 thetwo systemsC

30iv < 04

In an account o% the !ommune o% 1enoa in ;40, the e5penses %or a transaction were 2 07; li$$re,written as as IIAKKIII. The initial II meant two o% what you would e5pect to %ind at the $e(innin( o%

such a $i( num$er, thousands

" $i( impetus was (iven to the system with the pu$lication o% ra6 Auca )acioli6s Summa de

Arithmetica geometria proportioni et proportionalita, pu$lished %irst in 4:4.

>early every num$er in it was written in the new, convenient indu*"ra$ic numerals.

In $ooeepin(, )acioli used oman numerals only %or the year. Gse the ancient letters in main( this

entry,he advised6 i% only %or the sae o% more $eauty.

Incidentally, accordin( to the L5%ord #ictionary, the word al(orithm which we use in the n(lish

lan(ua(e is a corruption o% the name Dhwara&mi which literally means 6-a person/ %rom Dhawari&m6,which was the name o% the town where "l Dhwara&mi lived. Lur de$t to "l Dhwara&mi has $een

acnowled(ed in another wayC there is a Crater Al-Khwarizmi on the moonM

Today, un%ortunately, the ori(inal Indian te5ts that "l Dhwara&mi studied are lost to us, only thetranslations are availa$le . The "ra$s $orrowed so much %rom India in the %ield o% mathematics that even

the su$ject o% mathematics in "ra$ic came to nown as indsa, which means N%rom India and a

mathematician or en(ineer in "ra$ic is called 3uhandis which means 6an e5pert in 3athematics6. The

word 3uhandis possi$ly derived %rom the "ra$ic term mathematics vi&. indsa.

The Concept of Zero

The concept o% &ero also ori(inated in ancient India. This numeral indicated $y a dot and was

termed )ujyam. ven today the Indians use this term %or &ero alon( with the more current term +hunyam

meanin( a $lan. Interestin(ly, the term )ujyam also means holy. )aram*)ujya is a pre%i5 used in writtencommunication with elders. In this case it means respected or esteemed. The reason why the term )ujya *

meanin( $lan * came to $e sancti%ied can only $e (uessed. Indian philosophy has (lori%ied concepts lie

the material world $ein( an illusion -3aya/, the act o% renouncin( the material world -Tya(a/ and the

(oal o% mer(in( into the void o% eternity ->irvana/. erein could lie the reason how the mathematicalconcept o% &ero (ot a philosophical connotation o% reverence. concept o% 6&ero6 or Shunya is derived %rom



Brahma(upta was head o% the astronomical o$servatory at Gjjain which was the %oremost

mathematical centre o% ancient India. e wrote important wors on mathematics and astronomy. ewrote Brahma* sphuta* siddhanta -The Lpenin( o% the Gniverse/, in 2 chapters, at Bhillamala in 82=.

is second wor on mathematics and astronomy is Dhandahadyaa written in 88. Brahma(upta6s

understandin( o% the num$er systems was %ar $eyond others o% the period. e developed some al(e$raicnotation. e (ave remara$le %ormulas %or the area o% a cyclic 'uadrilateral -see pa(e P./ and %or the

len(ths o% the dia(onals in terms o% the sides. Brahma(upta also studied arithmetic pro(ressions,

'uadratic e'uations, theorems on ri(ht*an(led trian(les, sur%aces and volumes. The remainin( chaptersdeal with solar and lunar eclipses, planetary conjunctions and positions o% the planets. Brahma(upta $elieved in a static arth and he (ave the len(th o% the year as ;8 days 8 hours minutes : seconds in

the %irst wor, chan(in( the value to ;8 days 8 hours 2 minutes ;8 seconds in the second $oo.

Brahma(upta6s treatise Brahma*sputa*siddhanta was translated into "ra$ic under the title +ind

ind. or several centuries this translation remained a standard te5t o% re%erence in the "ra$ world. Itwas %rom this translation o% an Indian te5t on 3athematics that the "ra$ mathematicians per%ected the

decimal system and (ave the world its current system o% enumeration which we call the "ra$ numerals.

Ln other matters, we note that• in 800 B!, 00 years $e%ore )ytha(oras, in his $oo !audhayana Sulba Sutra , Baudhayana

states, The area produced $y the dia(onal o% a ri(ht*an(led trian(le is e'ual to the sum o% the

areas produced $uy its two sides.

• in the Surya Siddhanta, dated 400*00 "#, the ancient indu "stronomer Bhasaracharya

states, L$jects %all on the earth due to a %orce o% attraction $y the earth. There%ore, the earth,

planets, constellations, moon and sun are held in or$it due to this %orce.

Jagadguru

I would now lie to discuss the wor o% a most interestin( person, with the %ormida$le name o%is oliness Ja(ad(uru +anaracarya +ri Bharati Drsna Tirthaji 3aharaja. Better nown as NJa(ad(uru

or N1urudeva to his disciples, he was $orn o% learned and hi(hly pious parents in 3arch, ==4. e died

in Bom$ay on ;rd e$ruary, :80, a%ter an e5traordinary li%e devoted to his %aith, and mathematics.

There appears to $e some controversy a$out his assertion that the mathematics he presented as comin(%rom the historic Vedas actually can $e %ound there, $ut there is no dou$t that what he presents is

startlin(, and to western eyes, ori(inal. The %ollowin( e5amples are taen %rom the $oo "edic

#athematics, and represent a summary $y Ja(ad(uru o% the si5teen mathematical %ormulae that hereconstructed %rom the "tharvaveda. In his introduction to this pu$lication, 3anjula Trevedi says C

$bviously these formulae are not to be found in the present recensions of the Atharvaveda%they were actually reconstructed on the basis of intuitive revelation from materials scattered here and there in the Atharvaveda&

Arithmetic

The concept o% the eadhia is central to Ja(ad(urus num$er wor. and he de%ines it as %ollowsC

• I% a num$er ends in :, then the eadhia is one more than the previous num$er. or e5ample, the

eadhias %or :,:,2:,;: are ,2,;,4. This provides the model %or other num$ersC



• or ;,;,2;,;; etc multiply $y ;. The eadhias are then ,4,7,0 etc

c/ or 7,7,27,;7 etc. multiply $y 7. The eadhias are then ,2,:,28 etcd/ or ,,2,; etc. multiply $y :. The eadhias are then ,0,:,2= etc -p27/

>ote that not all num$ers have an eadhia

Application to ec!rring decimals

!onsider the decimal e5pression %or %ractions o% the %orm 119

, 129

, 139

,...

Tae the %irst %raction,1

19.

Ja(ad(uru o$serves that

in every recurring decimal with the standard numerator ie ( it will be observed that ) willinvariably be the last digit of the product of the last digit of the denominator and the last

digit of its recurring decimal equivalent&

or e5ample, the last di(it in the recurrin( pattern %or1

7 must $e 7

ence the last digit o% the recurrin( di(its %or our present pro$lem must $e .)roceed $y multiplyin( the !nits di(it $y the eadhia 2, from right to left , carryin( any tens over as

normal. Ja(ad(uru sets the wor out as %ollowsC

1

19= .

1 0̇51263

111517189

147

1316842 1̇

>ote that the whole = di(its can $e wored out mentally, writin( them down as we proceedM

or1

29, the eadhia o% 2: is ;. epeat the process as a$ove, $ut multiply $y ;C

1

29

= .1 0̇1314248

2217251862

20

26281916155

217

124

11232793 1̇

"nd thats not all. The process can actually $e speeded up, $y noticin( that the di(its in the %irst hal% o%

the e5pansion are the :s complement o% the di(its in the second hal%M

03448275862068

96551724137931

99999999999999

ow do you now when youre hal% way throu(hQ ?hen your multiplyin( (enerates less than the

denominator -2= in this case/.

" %urther e5ample,1

49. +tart with as always, and multiply $y . +top when 4= is reached, and

complete the di(its usin( :s complement.

ere is the calculation o% the last se'uence o% di(its, startin( %rom the ri(ht and Eendin( withF 4=C

48394729

459

41183336172334

4619433837272551

The initial = o% this se'uence is the :s complement o% the %inal . ?e may now write down the %inal

num$erC

.R0204081632653012244 483

947

29

459

411833

36

17

2334

4619

4338

37

27

255 ̇1

emara$leM Its almost a pity the calculator was invented.

Tests for "i#isibity



Ja(ad(uru uses the eadhi a a(ain in a process he calls osculation.

Lsculation consists in multiplyin( the last di(it o% a num$er $y an eadhia, and addin( the result to theremainin( di(its o% the num$er. 5amplesC

eadhia >um$er Lsculation

7 24; 7×; O 24 <4

8;42 ×2 O 8;4 <844

2 78 2×8 O 7 <2:

>ow %or another remara$le resultC

*f the result of osculation is the divisor itself or a repetition of a previous result the given

number is divisible by the divisor& +p,./

5amplesCa/ #etermine whether 2774 is divisi$le $y :.

The eadhia o% : is 2. !ontinual osculation o% 2774 $y 2 (ives 2=, ;= and :.

∴ 2774 is divisi$le $y :

$/ #etermine whether ;= is divisi$le $y 7The eadhia o% 7 is . !ontinual osculation o% ;= $y (ives 40, 4 and :.

∴ ;= is not divisi$le $y 7

c/ #etermine whether 4;727 is divisi$le $y 2:The eadhia o% 2: is ;. !ontinual osculation o% 4;7;2 $y ; (ives 4;7:, 484 and =, 2:.

∴ 4;7;2 is divisi$le $y 2:

+ome num$ers have a lar(e eahdia, and mae this approach impractical. Ja(ad(uru has a method %or

these as well, usin( negative osculators, which we wont (o into here.

"i#ision

Ja(ad(uru has an intri(uin( way o% simpli%yin( certain divisions usin( the eahdia. e constructs what

he calls the auxiliary fraction whose denominator is the eahdia o% the ori(inal %raction.

$raction A!%iliary $raction &A'$'(

1

29 0.133

59 0.3621863

49999 2.1863

5

?hat happens now is very curious, $ut e%%ective. ?e do a short division usin( the ".., but prefi% eachquotient with its remainder to %orm the ne5t dividendM

Tae the %raction3

59 !onsider now the short division process %or the "..

0.3

6

8 into ; (oes 0 with remainder ;. ?rite down ) and prefix the 0 with ;8 into ;0 (oes with remainder 0. ?rite down * and pre%i5 the with 0

8 into 0 (oes 0 with remainder . ?rite down ) and pre%i5 the 0 with

8 into 0 (oes = with remainder 2. ?rite down + and pre%i5 the = with 2

The wor is set out as %ollowsC

0.3

6→ 0.30 055 02 84 4 2 73 44 53 716... and so

3

59= 0.0508474576 ...



I% the numerator has more than one di(it, the wor is done in $locs o% di(its. #ivision occurs in the

normal way %or the $loc, and the final remainder is pre%i5ed to the $loc to %orm the ne5t dividend.

!onsider21863

49999

2.1863

5→ 0.34372 26874 45374 49074.... and so

21863

49999= 0.4372687453749074 ....

>ow thats what I call a party tric. I havent the %aintest idea why it wors.

M!ltiplication

Ja(ad(uru has methods %or s'uare roots and cu$e roots, which are o% only passin( interest in this a(e, $ut

his method o% multiplication is intri(uin(. It consists o% %indin( the de%iciency or e5cess %rom a power o%0, and writin( it ne5t to each num$er. or e5ample, let us multiply :2 $y :8C

92 − 8

×

96 − 4

88 / 32

The de%iciency %rom 00 is written to the ri(ht as a ne(ative num$er. The %inal answer is o$tained

$y multiplyin( the di(its on the ri(ht, and com$inin( :8 and 9= across the dia(onal. -Lr, %or that matter,:2 and 94. Ja(ad(uru wrote that Eold historical tradition descri$e this cross*su$traction process as havin(

$een responsi$le %or the acceptance o% the × mar as the si(n o% multiplication./ Below are some %urther

e5amplesC

888 − 112

×

998 − 002

886 / 224

888 −112

×

991 − 009

879/1

008 = 880/ 008

111 +11

×

109 + 9

120/99

1026 + 26

×

997−3

1023 / 0 7 8 = 1022/ 922The method is o$viously very convenient %or num$ers near the power o% ten. 3ethods are (iven

%or dealin( with num$ers %urther away, $ut they are cum$ersome. owever, the a$ove method is worthy

o% note, i% only $ecause you can rattle o%% such thin(s as the s'uare o% ::= in a matter o% secondsM

Algebra

Ja(ad(uru is %ond o% mental e5ercise, and $elieves that we should $e a$le to divide

x3 + 7ξ 2 + 6ξ + 5

ξ − 2

$y si(ht, or $y the Emethod o% ar(umentationF. I a(ree with him. owever, I thin I draw the lineat

12 x 4 + 41ξ 3 + 81ξ 2 + 79 ξ + 42

3ξ 2 + 5ξ + 7

which is one o% his e5amples.

e has some interestin( methods o% %actorisin(. ven today, the methods %or %actorisin( 'uadratice5pressions each have their champions, and Ja(ad(uru (ives (rud(in( approval %or the process o%

splittin( the middle termC



x2

+ 7ξ +10 = ξ 2

+ 2ξ + 5 ξ + 10

= ξ (ξ + 2) + 5( ξ + 2 )

= (ξ + 2)( ξ + 5)

"lthou(h I would not have used this method on such a simple pro$lem, it is certainly the method

I show students i% the leadin( coe%%icient is not C

2 x2

+ 5ξ + 2 = 2ξ 2

+ 4ξ + ξ + 2

= 2ξ (ξ + 2) +1(ξ + 2)

= (ξ + 2)(2ξ + 1)

owever, Ja(ad(uru thins this can cause Ereal harmF i% they are never tau(ht the 'uic method.

To him, this means splittin( the middle coe%%icient so that

the ratio of the first coefficient to the first part is the same as the ratio of the second part tothe last coefficient&

In the case a$ove, this yields 2C4 and C2. >ow this means that - x O 2/ is a %actor, and the second

%actor can $e written down immediately $y looin( at the %irst and last terms.Sou can actually (et 'uite (ood at doin( this, $ut I thin a com$ination o% methods is pre%era$le.

or e5ample, to %actorise 2 x2 O ; x 9 4, %irst %ind two num$ers that multiply to (ive 2 × −4 and add to(ive ;. They are 8 and 9;. Then write

12 x 2 + 13ξ − 4 = 12ξ 2 + 16ξ − 3ξ − 4

= (3ξ + 4)(4ξ − 1)

The last line comes 'uicly $y o$servin(, %rom line , that ; x O 4 is a %actor, and so the second%actor %ollows immediately. In %act, the whole process can $e done mentally.

Ja(ad(uru also maes use o% the ruleC

the product of the sum of the coefficients in the factors is equal to the sum of the coefficientsin the product&

This result is easy enou(h to prove. I%

- x +α )(ξ + β )....(ξ + ζ ) = αξ ν + βξ ν −1

+ ... + C

replacin( x < throu(hout (ives the re'uired result.

Lne application is in %actorisin( cu$ics. I% we now, $y the remainder theorem or otherwise, that

- x O / is a %actor o% x3

+ 6ξ 2

+ 11ξ + 6 , we can write down

x3 + 6ξ 2 + 11ξ + 6 = (ξ +1)( ξ 2 + ...ξ + 6)

$y inspection. ?e could %ind the missin( term $y ar(umentation, considerin( the coe%%icient o% x,

perhaps. owever, usin( the result a$ove, we see that the sum o% the coe%%icients in the second %actor

must $e 2, (ivin( as the re'uired coe%%icient.

eturnin( to 'uadratics, I dont thin many o% us would lie to %actorise

2 x2

+ 6ψ 2 + 3ζ 2 + 7 ξψ + 11ψζ + 7 ζξ

Ja(ad(uru proceeds $y puttin( y 0 separately e'ual to 0 and %actorisin( the remainin( 'uadraticsC



y = 0 : z = 0

2 x2

+ 3 z2

+ 7 zx 2 x2

+ 6 y2

+ 7 xy

( x + 3 z )(2 x + z) ( x + 2 y)( 2 x + 3 y )

!om$inin( the two results we have the %actors - x + 2 y + 3 z)(2 x + 3 y + z ) .

I% two eliminations do not provide su%%icient in%ormation %or the distri$ution o% terms, then the

third elimination is considered. or e5ample, eliminatin( y and 0 %rom the e5pression

x2

+ ξψ − 2ψ 2

+ 2 ξζ − 5ψζ − 3ζ 2

(ives the two pairs - x − ψ )(ξ + 2 y ) and ( x − z)( x + 3 z )

But these are not su%%icient to determine the distri$ution o% terms. liminatin( x, thou(h, (ives a

%urther pair -* y 1 0 /-2 y O ; 0 / and so we have - x − ψ − ζ )(ξ + 2 y + 3 z ) as the re'uired %actorisation.

?ould you lie to %ind the ! o% x3

− 3 ξ 2

− 4 ξ +12 ανδ ξ 3

− 7ξ 2

+ 16ξ − 12 Q

+u$tract themC "dd themC

4 x2

− 20 ξ + 24 2 x3

− 10 ξ 2

+ 12ξ emove any common %actors %rom eachC

x2

− 5ξ + 6 x2

− 5ξ + 6

"nd theres your !. I% the e5treme terms are not as convenient as the ones in the e5ample,multiplyin( $y constants in the normal method %or simultaneous e'uations is used.

,ol#ing .!ations

I% you have a (ood memory %or special cases, Ja(ad(uru has some startlin( results. or e5ample,

the %ollowin( e'uation can $e solved $y si(htC

a + β

ξ + α + β +

β + χ

ξ + β + χ =

2b

x + 2b+

a + c

x + a + c

The answers are x < 0, and x = −α + 2b + c

2. !an you see the patternQ )erhaps a numerical e5ample

may helpC

The solutions %or

2

x + 2+

3

ξ + 3=

4

ξ + 4+

1

ξ + 1

are x < 0 and x < *2..

!onsider now

1

x − 7 +

1

ξ − 9 =

1

ξ − 6 +

1

ξ − 10

"ccordin( to Ja(ad(uru,

the "edic Sutra 2tells us that if other elements being equal the sum total of the

denominators on he LHS and the total on the 3HS be the same then that total is 0ero&

+o the solution to the e'uation is 2 x 9 8 < 0, or x < =.

?e mi(ht have to transpose to o$tain 4 terms with plus si(ns, $ut that can $e done mentally.

ence the solution to1

x + β −

1

ξ + β + δ =

1

ξ + χ − δ −

1

ξ + χ

is x = − 1

2(b + c)

avin( trou$le with - x − 3)3

+ ( x − 9)3

= 2 ( x − 6 )3Q



I(norin( the cu$es, we note the sum o% the %actors is the same. There%ore, $y the Sunyam

Samuccaye %ormula, we may write down immediately that 2 x 9 2 < 0 and hence x < 8.

?hat a$out- x + 3)3

( x + 5)3 =

x + 1

x + 7Q

The same sutra tells us that since 4 1 + ∆1 -within the cu$es/ is the same as 4 2 + ∆2 we may

write 2 x O = < 0 and hence x < *4

" nice variation %or solvin( normal 'uadratic e'uations is that the differential is e'ual to the root

o% the discriminant.5ampleC

+olve ; x2 9 75 9 = < 0

+olutionC6 x − 7 = ± 145

ξ = 7 ± 145

6

It is no 'uicer than usin( the standard %ormula %rom the start, $ut it provides a neat connection.

Coordinate Geometry

uic, nowC what is the e'uation o% the strai(ht line throu(h the two points-=, *2/ and -;, 7/Q

The answer is :5 Oy < 82

The pattern used here is

Υ x * Χy < +

where Υ means Ethe di%%erence o% the y coordinatesF

Χ means Ethe di%%erence o% the x coordinatesF -in the same order as Υ /

+ means Esu$stitute a pointF.

,!mmary

3uch o% Ja(ad(urus wor revolves around special cases, and the student is e5pected to

remem$er a very (reat deal. In the present day, much o% this seems redundant. owever, the startlin(

nature o% some o% these patterns can enliven a lesson, or perhaps stimulate a student to try and determinethe underlyin( mathematical principle.



Exercises

a/ +ee i% you can write down the decimal e5pansion %or1

29 without looin( throu(h the

e5ample.

$/ Try1

39 $y this method and e5plain why it doesnt wor.

c/ ?rite down another %raction o% this type which will %ail.

2. Gsin( the method descri$ed a$ove, determine whether the %irst num$er is a divisor o% the secondC

;: ;748 Uno

2; ;47= Uyes

7 ;208 Uyes

: 2;4 Uno

;. ind the %irst = di(its in the %or the decimal e5pansions o% the %ollowin(, usin( the au5iliary%raction methodC

a/ 5

29 $/ 7.279 c/

9

19d/

3451

5999e/

7183

29999

4. )er%orm the %ollowin( multiplications usin( the e5cessde%icient methodC

88

×

99

898

×

995

109

×

112

994

×

1023

. actorise each o% the %ollowin( $y ; di%%erent methodsC

a/ 6 x2

+ 11ξ − 10 $/ 12 x2

+ 17ξ − 7 c/ 15 x 2

− 14 ξψ − 8ψ 2

8. Gsin( the method o% elimination descri$ed a$ove, %actorise the %ollowin(C

a/ 3 x

2+ ψ 2 − 2ζ 2 − 4ξψ − ψζ − ζξ

$/ 2 x2

+ 2 ψ 2

+ 5 ξψ + 2 ξ − 5ψ − 12

c/ 3 x

2+ ξψ − 2ψ

2+ 19ξζ + 28ζ

2+ 9ξω − 30ω

2− ψζ + 19ωψ + 46ζω

d/ 3 x2 + 7 ξψ + 2 ψ 2 + 11ξζ + 6 ζ 2 + 14ξ + 8ψ + 14ζ + 7ψζ + 8

7. ind the ! o% the %ollowin( pairsC

a/ 4 x3

+ 13ξ 2

+ 19ξ + 4 and 2 x3

+ 5ξ 2

+ 5ξ − 4

$/ 6 x4

− 11ξ 3

+ 16ξ 2

− 22 ξ + 8 and 6 x4

− 11ξ 3

− 8ξ 2

+ 22ξ − 8

c/ x4

+ ξ 3

− 5ξ 2

− 3ξ + 2 and x4

− 3ξ 3

+ ξ 2

+ 3ξ − 2 U52*5*2

=. +olve the e'uations



a/

3

x + 3+

4

ξ + 4=

2

ξ + 2+

5

ξ + 5

$/

1

x − 5+

1

ξ + 7=

1

ξ − 6+

1

ξ + 8

c/

1

x − α −

1

ξ − χ + δ =

1

ξ + β − δ − α −

1

ξ − χ + β

d/ - x + 2 )3

+ ( x − 8)3

= 2( x − 3 )3

e/

- x − 4 )3

( x + 6 )3 =

x + 5

x − 3

:. Veri%y $y e5pandin( the cu$ics that the solution o%

- x − 2a)3

+ ( x − 2b)3

= 2( x − a − b)3

is the solution (iven $y the Sunyam Samuccaye %ormula.

0. a/ +olve the 'uadratic e'uation x2 9 ;5 9 2 < 0 usin( the %act that the di%%erential is e'ual to

the root o% the discriminant..

$/ #erive this property %rom the 'uadratic %ormula.

. a/ ind the e'uation o% the line passin( throu(h the points -4, ;/ and -*, /

$/ ind the point o% intersection o% the line passin( throu(h the points -, / and -*;, */ withthe line passin( throu(h -*0, *;/ and -=, */

2. )rove Bhasaras %ormulaC

a ± β = α + α 2 − b

2±

a − a 2− b

2



Investigation – The Sri-Yantra

The word Santra means E)ower dia(ramF in +ansrit. The dia(ram may $e descri$ed as a $ilaterally symmetrical %i(ure composed o% nine interwoven isosceles trian(les, usually depicted with

%ive trian(les pointin( downwards and %our pointin( upwards. The %ormer are said to correspond to the

yoni representin( the dynamic %emale principle o% ener(y -Sati/, while the latter correspond to the lingarepresentin( the static male principle o% wisdom -Siva/. The central dot called bindu represents the

ori(inal unity o% the male and %emale principles prior to creation and the parado5ical point %emale principles prior to creation and the parado5ical point %rom which the mani%estation o% the cosmos

emer(es. The interpenetration o% the nine $asic trian(les (ives rise to a num$er o% su$sidiary trian(les -4;includin( the central trian(le enclosin( the bindu/ which %orm the a$odes o% the deities, representin( the

particulari&ation o% the ori(inal creative %orces into more concrete mani%estations. +ometimes the names

o% deities and +ansrit sylla$les are written into these trian(les, or ima(es o% the deities are placed inthem.

The two outer trian(les are constructed so that their points will touch a circle %rom the centre o% thedia(ram. There are = three line crossin( points that must cross e5actly. "lterin( the (eometry o% any one

trian(le will throw the entire coordination o%%. There have $een several attempts to determine the

(eometry used $y the ancient artists. eaders mi(ht lie to try their hand at the dia(ram $e%ore consultin(the method in the appendices.

UThe a$ove %rom alumni.cse.ucsc.eduWmielsriyantra$olton.html



Srinivasa Aiyangar amanu!an

BornC 22 #ec ==7 in rode, Tamil >adu state, India #iedC 28 "pril :20 in Dum$aonam, Tamil >adu state, India

+rinivasa amanujan was one o% India6s (reatest mathematical (eniuses. e made su$stantialcontri$utions to the analytical theory o% num$ers and wored on elliptic %unctions, continued %ractions,

and in%inite series. amanujan was $orn in his (randmother6s house in rode, a small villa(e a$out 400

m southwest o% 3adras. ?hen amanujan was a year old his mother too him to the town o%Dum$aonam, a$out 80 m nearer 3adras. is %ather wored in Dum$aonam as a cler in a cloth

merchant6s shop. In #ecem$er ==: he contracted smallpo5. ?hen he was nearly %ive years old,

amanujan entered the primary school in Dum$aonam althou(h he would attend several di%%erent primary schools $e%ore enterin( the Town i(h +chool in Dum$aonam in January =:=. "t the Town

i(h +chool, amanujan was to do well in all his school su$jects and showed himsel% an a$le all round

scholar. In :00 he $e(an to wor on his own on mathematics summin( (eometric and arithmetic series.

amanujan was shown how to solve cu$ic e'uations in :02 and he went on to %ind his own method to

solve the 'uartic. The %ollowin( year, not nowin( that the 'uintic could not $e solved $y radicals, hetried -and o% course %ailed/ to solve the 'uintic.

It was in the Town i(h +chool that amanujan came across a mathematics $oo $y 1 + !arrcalled Synopsis of elementary 3results in 5ure #athematics. This $oo, with its very concise style,

allowed amanujan to teach himsel% mathematics, $ut the style o% the $oo was to have a rather

un%ortunate e%%ect on the way amanujan was later to write down mathematics since it provided the onlymodel that he had o% written mathematical ar(uments. The $oo contained theorems, %ormulas and short

proo%s. It also contained an inde5 to papers on pure mathematics which had $een pu$lished in the

uropean Journals o% Aearned +ocieties durin( the %irst hal% o% the :th century. The $oo, pu$lished in

=8, was o% course well out o% date $y the time amanujan used it. By :04 amanujan had $e(un toundertae deep research. e investi(ated the series -n/ and calculated uler6s constant to decimal

places. e $e(an to study the Bernoulli num$ers, althou(h this was entirely his own independentdiscovery. amanujan, on the stren(th o% his (ood school wor, was (iven a scholarship to the1overnment !olle(e in Dum$aonam which he entered in :04.

owever, the %ollowin( year his scholarship was not renewed $ecause amanujan devoted more

and more o% his time to mathematics and ne(lected his other su$jects. ?ithout money he was soon indi%%iculties and, without tellin( his parents, he ran away to the town o% Vi&a(apatnam a$out 80 m north

o% 3adras. e continued his mathematical wor, however, and at this time he wored on hyper(eometric

series and investi(ated relations $etween inte(rals and series. e was to discover later that he had $eenstudyin( elliptic %unctions. In :08 amanujan went to 3adras where he entered )achaiyappa6s !olle(e.

is aim was to pass the irst "rts e5amination which would allow him to $e admitted to the Gniversity

o% 3adras. e attended lectures at )achaiyappa6s !olle(e $ut $ecame ill a%ter three months study. e

too the irst "rts e5amination a%ter havin( le%t the course. e passed in mathematics $ut %ailed all hisother su$jects and there%ore %ailed the e5amination. This meant that he could not enter the Gniversity o%

3adras. In the %ollowin( years he wored on mathematics developin( his own ideas without any help

and without any real idea o% the then current research topics other than that provided $y !arr6s $oo.!ontinuin( his mathematical wor amanujan studied continued %ractions and diver(ent series in :0=.

"t this sta(e he $ecame seriously ill a(ain and underwent an operation in "pril :0: a%ter which

he too him some considera$le time to recover. e married on 4 July, :0: when his mother arran(ed%or him to marry a nine year old (irl + Janai "mmal. amanujan did not live with his wi%e, however,

until she was twelve years old. amanujan continued to develop his mathematical ideas and $e(an to

pose pro$lems and solve pro$lems in the Journal o% the Indian 3athematical +ociety. e developed



relations $etween elliptic modular e'uations in :0. "%ter pu$lication o% a $rilliant research paper on

Bernoulli6s num$ers in : in the Journal o% the Indian 3athematical +ociety he (ained reco(nition %orhis wor. #espite his lac o% a university education, he was $ecomin( well nown in the 3adras area as

a mathematical (enius.

In : amanujan approached the %ounder o% the Indian 3athematical +ociety %or advice on a jo$. "%ter this he was appointed to his %irst jo$, a temporary post in the "ccountant 1eneral6s L%%ice in

3adras. It was then su((ested that he approach amachaudra ao who was a !ollector at >ellore.

amachaudra ao was a %ounder mem$er o% the Indian 3athematical +ociety who had helped start themathematics li$rary. e writes in X;0YC*

A short uncouth figure stout unshaven not over clean with one conspicuous feature-

shining eyes- waled in with a frayed noteboo under his arm& He was miserably poor& &&&

He opened his boo and began to explain some of his discoveries& * saw quite at once that

there was something out of the way% but my nowledge did not permit me to judge whetherhe taled sense or nonsense& &&& * ased him what he wanted& He said he wanted a pittance

to live on so that he might pursue his researches&

amachaudra ao told him to return to 3adras and he tried, unsuccess%ully, to arran(e a

scholarship %or amanujan. In :2 amanujan applied %or the post o% cler in the accounts section o%the 3adras )ort Trust. In his letter o% application he wrote X;YC*

* have passed the #atriculation 6xamination and studied up to the 7irst Arts but was

prevented from pursuing my studies further owing to several untoward circumstances& *

have however been devoting all my time to #athematics and developing the subject&

#espite the %act that he had no university education, amanujan was clearly well nown to the

university mathematicians in 3adras %or, with his letter o% application, amanujan included a re%erence

%rom ? 3iddlemast who was the )ro%essor o% 3athematics at The )residency !olle(e in 3adras.

3iddlemast, a (raduate o% +t John6s !olle(e, !am$rid(e, wrote X;YC* I can stron(ly recommend theapplicant. e is a youn( man o% 'uite e5ceptional capacity in mathematics and especially in wor

relatin( to num$ers. e has a natural aptitude %or computation and is very 'uic at %i(ure wor. Ln the

stren(th o% the recommendation amanujan was appointed to the post o% cler and $e(an his duties on

3arch :2. >ow amanujan was 'uite lucy to have a num$er o% people worin( round him with atrainin( in mathematics. In %act the !hie% "ccountant %or the 3adras )ort Trust, + > "iyar, was trained

as a mathematician and pu$lished a paper Ln the distri$ution o% primes in :; on amanujan6s wor.The pro%essor o% civil en(ineerin( at the 3adras n(ineerin( !olle(e ! A T 1ri%%ith was also

interested in amanujan6s a$ilities and, havin( $een educated at Gniversity !olle(e Aondon, new the

pro%essor o% mathematics there, namely 3 J 3 ill. e wrote to ill on 2 >ovem$er :2 sendin(some o% amanujan6s wor and a copy o% his : paper on Bernoulli num$ers. ill replied in a %airly

encoura(in( way $ut showed that he had %ailed to understand amanujan6s results on diver(ent series.

The recommendation to amanujan that he read Bromwich6s Theory o% in%inite series did not please

amanujan much. amanujan wrote to ? o$son and Baer tryin( to interest them in his results $ut neither replied.

In January :; amanujan wrote to 1 ardy havin( seen a copy o% his :0 $oo $rders of *nfinity. In amanujan6s letter to ardy he introduced himsel% and his wor X0YC*

* have had no university education but * have undergone the ordinary school course& Afterleaving school * have been employing the spare time at my disposal to wor at

mathematics& * have not trodden through the conventional regular course which is followed

in a university course but * am striing out a new path for myself& * have made a specialinvestigation of divergent series in general and the results * get are termed by the local

mathematicians as 'startling'&



ardy, to(ether with Aittlewood, studied the lon( list o% unproved theorems which amanujan

enclosed with his letter. Ln = e$ruary he replied to amanujan X;Y, the letter $e(innin(C*

* was exceedingly interested by your letter and by the theorems which you state& 8ou willhowever understand that before * can judge properly of the value of what you have done it

is essential that * should see proofs of some of your assertions& 8our results seem to me to

fall into roughly three classes9 +(/ there are a number of results that are already nown oreasily deducible from nown theorems% +,/ there are results which so far as * now are

new and interesting but interesting rather from their curiosity and apparent difficulty thantheir importance% +:/ there are results which appear to be new and important&&&

amanujan was deli(hted with ardy6s reply and when he wrote a(ain he said X=YC*

* have found a friend in you who views my labours sympathetically& &&& * am already a half starving man& ;o preserve my brains * want food and this is my first consideration& Any

sympathetic letter from you will be helpful to me here to get a scholarship either from the

university of from the government&

Indeed the Gniversity o% 3adras did (ive amanujan a scholarship in 3ay :; %or two yearsand, in :4, ardy $rou(ht amanujan to Trinity !olle(e, !am$rid(e, to $e(in an e5traordinary

colla$oration. +ettin( this up was not an easy matter. amanujan was an orthodo5 Brahmin and so was a

strict ve(etarian. is reli(ion should have prevented him %rom travellin( $ut this di%%iculty wasovercome, partly $y the wor o% >eville who was a collea(ue o% ardy6s at Trinity !olle(e and who

met with amanujan while lecturin( in India.

amanujan sailed %rom India on 7 3arch :4. It was a calm voya(e e5cept %or three days onwhich amanujan was seasic. e arrived in Aondon on 4 "pril :4 and was met $y >eville. "%ter

%our days in Aondon they went to !am$rid(e and amanujan spent a couple o% wees in >eville6s home

$e%ore movin( into rooms in Trinity !olle(e on ;0th "pril. i(ht %rom the $e(innin(, however, he had

pro$lems with his diet. The out$rea o% ?orld ?ar I made o$tainin( special items o% %ood harder and itwas not lon( $e%ore amanujan had health pro$lems. i(ht %rom the start amanujan6s colla$oration

with ardy led to important results. ardy was, however, unsure how to approach the pro$lem o%

amanujan6s lac o% %ormal education. e wrote XYC*

<hat was to be done in the way of teaching him modern mathematics= ;he limitations ofhis nowledge were as startling as its profundity&

Aittlewood was ased to help teach amanujan ri(orous mathematical methods. owever he said

-X;Y/C* ... that it was e5tremely di%%icult $ecause every time some matter, which it was thou(ht that

amanujan needed to now, was mentioned, amanujan6s response was an avalanche o% ori(inal ideaswhich made it almost impossi$le %or Aittlewood to persist in his ori(inal intention.

The war soon too Aittlewood away on war duty $ut ardy remained in !am$rid(e to wor with

amanujan. ven in his %irst winter in n(land, amanujan was ill and he wrote in 3arch : that hehad $een ill due to the winter weather and had not $een a$le to pu$lish anythin( %or %ive months. ?hat he

did pu$lish was the wor he did in n(land, the decision havin( $een made that the results he had

o$tained while in India, many o% which he had communicated to ardy in his letters, would not $e pu$lished until the war had ended.

Ln 8 3arch :8 amanujan (raduated %rom !am$rid(e with a Bachelor o% +cience $y

esearch -the de(ree was called a )h.#. %rom :20/. e had $een allowed to enrol in June :4 despitenot havin( the proper 'uali%ications. amanujan6s dissertation was on Highly >omposite 4umbers and

consisted o% seven o% his papers pu$lished in n(land. amanujan %ell seriously ill in :7 and his

doctors %eared that he would die. e did improve a little $y +eptem$er $ut spent most o% his time in

various nursin( homes. In e$ruary := ardy wrote -see X;Y/C*



!atty Shaw found out what other doctors didn't now that he had undergone an operation

about four years ago& His worst theory was that this had really been for the removal of amalignant growth wrongly diagnosed& *n view of the fact that 3amanujan is no worse than

six months ago he has now abandoned this theory - the other doctors never gave it any

support& ;ubercle has been the provisionally accepted theory apart from this since theoriginal idea of gastric ulcer was given up& &&& Lie all *ndians he is fatalistic and it is

terribly hard to get him to tae care of himself&

Ln = e$ruary := amanujan was elected a %ellow o% the !am$rid(e )hilosophical +ocietyand then three days later, the (reatest honour that he would receive, his name appeared on the list %orelection as a %ellow o% the oyal +ociety o% Aondon. e had $een proposed $y an impressive list o%

mathematicians, namely ardy, 3ac3ahon, 1race, Aarmor, Bromwich, o$son, Baer, Aittlewood,

>icholson, Soun(, ?hittaer, orsyth and ?hitehead. is election as a %ellow o% the oyal +ociety was

con%irmed on 2 3ay :=, then on 0 Lcto$er := he was elected a ellow o% Trinity !olle(e!am$rid(e, the %ellowship to run %or si5 years. The honours which were $estowed on amanujan seemed

to help his health improve a little and he renewed his e%%ors at producin( mathematics. By the end o%

>ovem$er := amanujan6s health had (reatly improved. ardy wrote in a letter X;YC*

* thin we may now hope that he has turned to corner and is on the road to a realrecovery& His temperature has ceased to be irregular and he has gained nearly a stone in

weight& &&& ;here has never been any sign of any diminution in his extraordinary

mathematical talents& He has produced less naturally during his illness but the quality hasbeen the same& &&&& He will return to *ndia with a scientific standing and reputation such as

no *ndian has enjoyed before and * am confident that *ndia will regard him as the treasure

he is& His natural simplicity and modesty has never been affected in the least by success -indeed all that is wanted is to get him to realise that he really is a success&

amanujan sailed to India on 27 e$ruary :: arrivin( on ; 3arch. owever his health was

very poor and, despite medical treatment, he died there the %ollowin( year. The letters amanujan wrote

to ardy in :; had contained many %ascinatin( results. amanujan wored out the iemann series, theelliptic inte(rals, hyper(eometric series and %unctional e'uations o% the &eta %unction. Ln the other hand

he had only a va(ue idea o% what constitutes a mathematical proo%. #espite many $rilliant results, someo% his theorems on prime num$ers were completely wron(. amanujan independently discovered resultso% 1auss, Dummer and others on hyper(eometric series. amanujan6s own wor on partial sums and

products o% hyper(eometric series have led to major development in the topic. )erhaps his most %amous

wor was on the num$er p-n/ o% partitions o% an inte(er n into summands. 3ac3ahon had produced

ta$les o% the value o% p-n/ %or small num$ers n, and amanujan used this numerical data to conjecturesome remara$le properties some o% which he proved usin( elliptic %unctions. Lther were only proved

a%ter amanujan6s death. In a joint paper with ardy, amanujan (ave an asymptotic %ormula %or p-n/. It

had the remara$le property that it appeared to (ive the correct value o% p-n/, and this was later proved $y ademacher. amanujan le%t a num$er o% unpu$lished note$oos %illed with theorems that

mathematicians have continued to study. 1 > ?atson, 3ason )ro%essor o% )ure 3athematics at

Birmin(ham %rom := to : pu$lished 4 papers under the (eneral title ;heorems stated by 3amanujan and in all he pu$lished nearly ;0 papers which were inspired $y amanujan6s wor. ardy

passed on to ?atson the lar(e num$er o% manuscripts o% amanujan that he had, $oth written $e%ore

:4 and some written in amanujan6s last year in India $e%ore his death. The picture a$ove is taen

%rom a stamp issued $y the Indian )ost L%%ice to cele$rate the 7th anniversary o% his $irth.+rinivasa amanujan was elected to the oyal +ociety o% Aondon in :=.

eferences

"edic #athematics, Ja(ad(uru +anaracarya +ri Bharati Drsna Turthaji 3aharaja, +hri Jainendra )ress,

>ew #elhi, ::



;he #an who ?new *nfinity, o$ert Dani(el, +cri$ners, Dent, ::;he #easure of 3eality9 @uantification and <estern Society (,B-(CBB, !am$rid(e Gniversity )ress, ::7, >ew Sor.

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;he !oo of 4umbers, John . !onway and ichard D. 1uy, +prin(er*Verla(, ::8, >ew Sor

A History of #athematics, !arl B. Boyer, >ew Sor, ?iley, ::GAs

httpCwww*history.mcs.st*andrews.ac.uhistory3athematiciansamanujan.html

-"lso used %or Bhasara, alDhwari&mi and amanujanhttpCwww.hindu$oos.or(sudheerZ$irodarindiaZcontri$utionmaths.html

httpCwww.ics.uci.eduWr(uptavedic.html

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ramanujan.html

http://www.hindubooks.org/sudheer_birodkar/india_contribution/maths.html

http://www.ics.uci.edu/~rgupta/vedic.html

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ramanujan.html

http://www.hindubooks.org/sudheer_birodkar/india_contribution/maths.html

http://www.ics.uci.edu/~rgupta/vedic.html

Taylor James - Indian Math

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