indian and islamic mathematics

7/24/2019 Indian and Islamic Mathematics

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Indian and

IslamicMathematics

Presented By: Joselito C.Cabije

MAed-Mathematics


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IndianMathematics


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Indus Valley

Civilization

i!hlyso"histicated urban

civilizationthat diedaround #$$$BC


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a re!iono% morethan amillion

s&uare'ilometers

Indus Valley

Civilization


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Indus Valley

Civilization


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Indus Valley

Civilization


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com"risedba'ed claybric'

buildin!s(hi!hlydevelo"ed sea

and river

Indus Valley

Civilization


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)he de!ree o%advancements in

science and technolo!ycan be !au!ed %rom thehi!hly evolve system o%"lumbin!( "ublic baths(

construction o% thecitiesand the

so"histicated sealsandscul"turesthat have

been %ound.

Indus Valley

Civilization


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)he *ecimal +ystem inara""a Period,#$$$-

$$ BC/

standardize 0ei!htsystem based onratios: $.$( $.( $.1($.( ( 1( ( $( 1$($( $$( 1$$( and$$

bronze rod mar'ed in

units o% $.#23 inches


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)he 0ei!hts0eremade in re!ular!eometrical sha"esli'e he5ahedra(barrels( cones( andcylinders( therebydemonstratin!'no0led!e o% basic

!eometry


$$ BC/


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$$ BC/


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a scale 0ith nine "arallele&uidistantmar'in!sthat is

also indicative o% the decimalsystem


$$ BC/


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Mohenjo-*aro

ruinsin the +indh"rovinceo% Pa'istan


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+eal Arti%acts


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so"histicationo% thecivilization-

"orts(ma!ni6centbric' buildin!sand cities.

Indus Valley

Civilization


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7rnaments

objects%ound atboth

Mohenjo-daroandara""a


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evolved around the develo"mentand the "racticeo% the Vedas.

Activities in the Vedic

Period ,3$8$$ BC/


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mostly to be %ound inVedic te5tsassociated 0ith ritualactivities

system o% land!rants anda!ricultural ta5

assessments


Period ,3$8$$ BC/


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convertedrectan!ular "lotsor trian!ular "lotsto s&uares o%

e&uivalent sizes constructed ritual

altars


Period ,3$8$$ BC/


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)he +ath"atha

Brahmana ,9$$BC/

)his te5trecordsthePytha!orean

theorem $$years be%orePytha!oras.


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)he +ulba +utras

,$$$-1$$BC/

Sulba8 "ieces o%chord or strin!andSutra8 %ormula ora"horism

mathematicaldiscoveriesusin! a"iece o% chord %or

constructions o%


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)he +ulba +utras

,$$$-1$$ BC/

)o 6nd as&uaree&uals to

the sum o%t0o !ivens&uares


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Baudhayana ,;$$

BC/ discovered a "roo%o% the

theorem o% Pytha!oras


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Interestin!


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> ,1$$


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>atyayana ,1$$

BC/ authoro% one o% the +ulba +utras


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The ropewhich isstretched alongthe length of the

diagonal of arectangle

producesan areawhich the vertical

and horizontal

sides maketo ether.

>atyayana ,1$$

BC/

I ti l


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Irrational

?umbers in the+ulba +utras

I ti l


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Irrational

?umbers in the+ulba +utras


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Panini

,1$ BC/ a +ans'rit !rammarian

0ho !ave a

com"rehensive andscienti6c theory o%"honetics( "honolo!y(

and mor"holo!y invented a "er%ect

+ans'rit !rammar

used the conce"t o% zero


33/101

Pin!ala

,1$$ BC/ tried to invent ne0

meters %rom the 'no0nVedic meters by varyin!

the syllablesthrou!h"ermutationsandcombinations o% lon! andshort sounds

discovered the MeruPrastara,Pascal@s)rian!le/0hich 0as

discovered by Pascal ;$$ears a%ter Pin al


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alayudh

,1$$ A*/ 0rote a commentary on

Pin!al@s 0or' and in the

"rocess discoveredtheBinomial )heorem$$years be%ore ?e0ton.


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Buddhist

MathematicsBe%ore Christ

)here is evidencethat theJainmathematicians understooddierent orderso% in6nity thereby antici"atin! in many 0ays

the !reat discoveries o% eor! Cantor.

)h B ' h li


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)he Ba'shaliManuscri"t ,#$$A*/

early manuscri"t( 0ritten onbirch bar' and %oundin the

summer o% ;; near thevilla!e o% Ba'hshali then inIndia and no0 in Pa'istan

commentaryon an earliermathematical 0or'

clear evidenceo% the useo%the decimal system

)h B ' h li


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)he Ba'shaliManuscri"t ,#$$A*/


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volution o%

?umerals

Pierre +imon


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Pierre-+imonDa"lace ,39-;13/

EThe ingenious method of

expressing every possiblenumber using a set of ten

symbols emerged in India. The

idea seems so simple nowadaysthat its signicance andprofound importance is no longerappreciated. It!s simplicity lies in

the wa it facilitated calculation


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volution o%

?umerals Brahmi ?umerals: Around

the )ime o% Christ


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volution o%

?umerals

"rogressionof #rahmi number formsthrough the centuries $column far

left showing forms in use by %&& '()


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volution o%

?umerals *umeral forms found in

#akhshali +anuscriptshowing place value and useof zero

,,&- %/-01&- 20


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+ome Contentso% the Ba'hshaliManuscri"t

+olutiono% linear e&uations 0ith as many as6ve un'no0ns.

Guadratic e&uations 0ith solutions.

Pro!ressions: Both arithmetic and !eometric.

+imultaneous e&uations.


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+&uare root inBa'shaliManuscri"t

)his is stated in the manuscri"t as %ollo0s 8HIn the case of a non3s4uare number- subtract

the nearest s4uare number- divide theremainder by twice this nearest s4uare5 halfthe s4uare of this is divided by the sum of theapproximate root and the fraction. This is

subtracted and will give the corrected root.

+&uare root in


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Ba'shali


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Aryabhata I

,32 A*/ H'ryabhatiya 3

summarizes 6indu

mathematicsup tothat /th 7entury

recordedmany

im"ortantdiscoveriesinmathematicsand

astronomy


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Aryabhata@s

*iscoveries )he earth is round

and revolves around

its a5is. )he sun a""ears !o

around the earth0hen in %act it is the

earth that revolves.

value o% "i correct to%our decimal "laces


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methods o% 6ndin!s&uare roots and cube

roots !eneral solutiono% the

indeterminate e&uationo% 6rst de!ree :by 8

ax9c- wherexand yareunknowns

Indian astronomyon 6rmmathematical %oundation

Aryabhata@s

*iscoveries


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Brahma!u"ta

,9; 8 23$ A*/

made advances innumber systemsincludin!al!orithms%ors&uare roots andthe solution o%&uadratic

e&uations

+ome


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+omeAchievements o%Brahama!u"ta solvedthe e&uation y1?51F

0here 5 and y are un'no0n,Pell@s e&uation o% uler/

a %ormula %or the area o% acyclic &uadrilateral

+ome


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%ormula %ordeterminin! the

dia!onals o% acyclic&uadrilateralin

terms o% itssides

+omeAchievements o%Brahama!u"ta

+ome


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6rst to obtain %ormulas%orthe sum o% s&uares and

cubeso% 6rst nnaturalnumbers

+omeAchievements o%Brahama!u"ta

)ri!onometry in


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)ri!onometry inIndia ,#$$ A*on0ards/

+urya +iddhanta - %ounder

o% modern tri!onometry ma'es distinctive uses o%

the modern tri!onometric

%unctions: +ine ,Jya/(Cosine ,'ojya/( Inverse sine,ot'ram jya/()an!ent(+ecant

)ri!onometry in


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)ri!onometry in


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A sidereal year 0as

com"uted as#2.1;; days(0hich is only #minutes and 1$

seconds lon!erthanthe modern value o%#2.12#213 days.



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Bhas'ara I

continued 0hat Aryabatta le%to

"i as irrational number calculated the sine %unction

0hich 0as 99K accurate

6rst to consider &uadrilateral

0ith all sides une&ualand noneo% the o""osite sides are"arallel


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Bhas'ara II

, 8 ; A*/ Bhas'aracharya 0as

one o% India@s!reatestmathematicians 0homade numerous

im"ortantdiscoveries includin!the discovery o% theCalculus

Contributions o%


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Contributions o%Bhas'ara II

Proo% %ordivision by

zero bein!in6nity

Contributions o%


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6rst to observe that a

"ositive number hast0o s&uare roots

"ro"erties o% +urds

solutions o% Guadratic(

Cubicand Guartice&uations

&uadratic e&uations0ith more than one

un'no0n

Contributions o%

Bhas'ara II

C t ib ti %


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discovery o% the derivative

,tat'ali'a !ati

instantaneous velocity/ sho0ed that the derivative o% the sine

%unction is cosine

discovered Lolles theorem - a s"ecialcase o% the mean value theorem

Contributions o%

Bhas'ara II

Contributions o%


67/101

!ave the cha'ravala,cyclic/ methodto

solve the !eneral%orm o% Pellse&uation

6rst to use symbols

%or un'no0ns inal!ebra

Com"utation o% N,decimal "laces/

Contributions o%

Bhas'ara II

Contributions o%


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discoveredthe

tri!onometric %ormula

Contributions o%

Bhas'ara II

Madhava o%


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Madhava o%+an!ama!ramma #$- 1 A*

made major

discoveries incalculusincludin!im"ortant advances

in in6nite seriese5"ansions %ortri!onometric

%unctions

Credits to


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Credits to

Madhava@s ?ame "o0er series e5"ansion o%

tan-,5/ ,re!ory@s +eries/

Credits to


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com"uted an e5tremelyclose a""ro5imation o% Nas

#.912#9

Credits toMadhava@s ?ame


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+rinivas


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+rinivasLamanujan ;;38 91$

sent a set o% 1$theoremsto Pro%essor

od%rey arold ardyo% Cambrid!e

any bi! number can be0ritten as sum o% notmore than %our "rimenumbers

319 #F 1#

9#F $#


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Islamic

Mathematics

I l i i


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Islamic m"ire

establishedacross Persia( the Middle as

Central Asia( ?orth A%rica( Iberiaand "arto% India%rom the ;th Century on0ards

%usedto!ether the mathematicaldevelo"ments o% both reeceand India.

Islamic


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IslamicMathematics used o% com"le5 !eometric

"atternsto decorate theirbuildin!s

Islamic


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IslamicMathematics

Islamic


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IslamicMathematics

Gu@ranitsel%encoura!ed the

accumulationo% 'no0led!e

9th8 thcentury olden A!e o%Islamic scienceand

mathematics

ouse o%


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ouse o%Oisdom ,9th

Century/ establishedby

Cali"h al-LashidandruledCali"h al-Ma@mun

a libraryand a "lace%or translationand

research translated ree'

and indutreatisesto Arabic

Muhammad Al


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Muhammad Al-

>h0arizmi early *irectoro% the

ouse o% Oisdom in

the 9th Century Hal!orithm is

derived %rom theDatinization o% his

name Eal!ebraE is derived

%rom the Datinizationo% Eal-jabrE

Al->h0arizmi@s


81/101

Al->h0arizmi s

Contributions

+tron! advocacyo%indu numericalsystem

introducedthe

indu conce"t o%decimal "ositionin!notation to the Araband uro"ean 0orlds

Al->h0arizmi@s


82/101

Al >h0arizmi s:itab al3;abrwa


83/101

D)I?)

+GQAL

Al->h0arizmi@s


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Al->h0arizmi s

Contributions develo"eda %ormula %or

systematically solvin!

&uadratic e&uationsbyusin! the methods o%com"letionand balancin!


85/101

)habit ben Gurra


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)habit ben Gurra

,;1289$/ =n the >ustication of

the 'lgebraic "roblems

by ?eometric "roofs studied number theory

"roved theorem about6ndin! "airs o% amicablenumbers

corrected an earliertranslation o% the

@lements

)habit ben Gurra


87/101

develo"eda

!eneral %ormulaby 0hichamicable

numbers couldbe derived

)habit ben Gurra

,;1289$/

)hRbit ibn Gurra


88/101

Gtheorem

0here n is an inte!er andp( 4(and rare "rime numbers( then1nApA4and 1nAr are a "air o%

amicable numbers. )his %ormula

!ives the "airs ,11$( 1;/ %or n1(,3192( ;2/ %or n( and

,9#2#;( 9#3$2/ %or n3( butno other such "airs are 'no0n.

?umbers o% the %orm # S 1n

T are'no0n as )habit numbers. In order%or Ibn Gurras %ormula to "roducean amicable "air( t0o consecutive)habit numbers must be "rimeU

this severel restricts the ossible

Muhammad Al-


89/101

Muhammad Al

>araji $thcentury Persian

mathematician

introduced thetheory o% al!ebraiccalculus

6rst to use the

method o% "roo% bymathematicalinduction ,Binomial)heorem/

Ibn Al-aytham


90/101

Ibn Al aytham

,928$$/ author o% numerous

0or's on o"tics(

s"herical !eometry(number theory

discovered Bilson


91/101

7mar >hayyam

$;8#

!eneralized Indian metho

ds %or e5tractin! s&uareand cube roots to include%ourth( 6%thand hi!herroots

studied cubic e&uations

Treatise on(emonstration of

"roblems of 'lgebra

- -)usi


92/101

)usi

#thCentury 6rst to treat

tri!onometryas a

se"aratemathematicaldisci"line

!ave the 6rst

e5tensivee5"osition o%s"hericaltri!onometry

- -)usi


93/101

%ormulated la0 o% sines %or "lanetrian!les

a,sin'/ b,sin #/c,sin 7/ sine la0 %or s"herical trian!lesby

Persians Abul Oa%a Buzjani and Abu ?asrMansur ,$thcentury/

)usi

#thCentury

>amal al-*in al-


94/101

>amal al *in al


95/101

Ian al Banna alMarra'ushi

#thcenturyMoroccanMathematicians

studied !eometry(com"utin! s&uareroots and the

theory o%continued %ractions

binomialcoeWcients

Ian al-Banna al-


96/101

I% 0e denote the binomialcoeWcientpchoose kby p7kthen al-

Banna sho0s that

p71p,p-/X1

p7# p71,p-1/X#

p7k p7k-,p- ,k- / /Xk.

Ian al Banna alMarra'ushi

Ian al-Banna al-


97/101

... the ternary combination is thus obtained

by multiplying the third of the third termpreceding the given number5 and so wealways multiply the combination that

precedes the combination sought by the

number that precedes the given number- andwhose distance to it is e4ual to the numberof combinations sought. From the product-we take the part that names the number of

combinations.

a a a a aMarra'ushi

Le%erences


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Le%erences

http://www.thenagain.info/WebChron/India/Harappa.html

http://www.harappa.com/indus/21.html

http://www.tambourine.in/tmbn_wp/content/sciencepage/and_maths_was_born_episode_2/

http://www.math1!.com/en/mathshistor

mathhistor inindia #a$hshali ba$

Le%erences
http://www.thenagain.info/WebChron/India/Harappa.htmlhttp://www.thenagain.info/WebChron/India/Harappa.htmlhttp://www.harappa.com/indus/21.htmlhttp://www.tambourine.in/tmbn_wp/content/science-page/and_maths_was_born_episode_2/http://www.tambourine.in/tmbn_wp/content/science-page/and_maths_was_born_episode_2/http://www.tambourine.in/tmbn_wp/content/science-page/and_maths_was_born_episode_2/http://www.math10.com/en/maths-history/math-history-in-india/Bakhshali/bakshali.htmlhttp://www.math10.com/en/maths-history/math-history-in-india/Bakhshali/bakshali.htmlhttp://www.math10.com/en/maths-history/math-history-in-india/Bakhshali/bakshali.htmlhttp://www.math10.com/en/maths-history/math-history-in-india/Bakhshali/bakshali.htmlhttp://www.tambourine.in/tmbn_wp/content/science-page/and_maths_was_born_episode_2/http://www.tambourine.in/tmbn_wp/content/science-page/and_maths_was_born_episode_2/http://www.tambourine.in/tmbn_wp/content/science-page/and_maths_was_born_episode_2/http://www.harappa.com/indus/21.htmlhttp://www.thenagain.info/WebChron/India/Harappa.htmlhttp://www.thenagain.info/WebChron/India/Harappa.html


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Le%erences

http://www.slideshare.net/$rishna$umawat/%edicmathematicsppthttp://pages.intnet.mu/cuebo"/educati

on/maths/histor"/histor"/indiansulbasutras.htm

http://archaeolog"online.net/artifacts/histor"mathematics

http://mathworld wolfram com/C"clic&u

Le%erences
http://www.slideshare.net/krishnakumawat/vedic-mathematics-ppthttp://www.slideshare.net/krishnakumawat/vedic-mathematics-ppthttp://pages.intnet.mu/cueboy/education/maths/history/history/indiansulbasutras.htmhttp://pages.intnet.mu/cueboy/education/maths/history/history/indiansulbasutras.htmhttp://pages.intnet.mu/cueboy/education/maths/history/history/indiansulbasutras.htmhttp://archaeologyonline.net/artifacts/history-mathematicshttp://archaeologyonline.net/artifacts/history-mathematicshttp://mathworld.wolfram.com/CyclicQuadrilateral.htmlhttp://mathworld.wolfram.com/CyclicQuadrilateral.htmlhttp://archaeologyonline.net/artifacts/history-mathematicshttp://archaeologyonline.net/artifacts/history-mathematicshttp://pages.intnet.mu/cueboy/education/maths/history/history/indiansulbasutras.htmhttp://pages.intnet.mu/cueboy/education/maths/history/history/indiansulbasutras.htmhttp://pages.intnet.mu/cueboy/education/maths/history/history/indiansulbasutras.htmhttp://www.slideshare.net/krishnakumawat/vedic-mathematics-ppthttp://www.slideshare.net/krishnakumawat/vedic-mathematics-ppt


100/101

Le%erences

http://mathworld.wolfram.com/'eries()pansion.htmlhttp://mathworld.wolfram.com/*aclaur

in'eries.htmlhttp://www.stor"ofmathematics.com/is

lamic_al$hwari+mi.htmlhttp://www.slideshare.net/guest!,e!!

d islamicmathematics

Le%erences
http://mathworld.wolfram.com/SeriesExpansion.htmlhttp://mathworld.wolfram.com/SeriesExpansion.htmlhttp://mathworld.wolfram.com/MaclaurinSeries.htmlhttp://mathworld.wolfram.com/MaclaurinSeries.htmlhttp://www.storyofmathematics.com/islamic_alkhwarizmi.htmlhttp://www.storyofmathematics.com/islamic_alkhwarizmi.htmlhttp://www.slideshare.net/guest05e00d/islamic-mathematicshttp://www.slideshare.net/guest05e00d/islamic-mathematicshttp://www.slideshare.net/guest05e00d/islamic-mathematicshttp://www.slideshare.net/guest05e00d/islamic-mathematicshttp://www.storyofmathematics.com/islamic_alkhwarizmi.htmlhttp://www.storyofmathematics.com/islamic_alkhwarizmi.htmlhttp://mathworld.wolfram.com/MaclaurinSeries.htmlhttp://mathworld.wolfram.com/MaclaurinSeries.htmlhttp://mathworld.wolfram.com/SeriesExpansion.htmlhttp://mathworld.wolfram.com/SeriesExpansion.html


101/101

Le%erences

http://www.stor"ofmathematics.com/indian.htmlhttp://www.newworldenc"clopedia.org/

entr"/-r"abhatahttp://pages.intnet.mu/cuebo"/educati

on/maths/histor"/histor"/indiansulbas
http://www.storyofmathematics.com/indian.htmlhttp://www.storyofmathematics.com/indian.htmlhttp://www.newworldencyclopedia.org/entry/Aryabhatahttp://www.newworldencyclopedia.org/entry/Aryabhatahttp://pages.intnet.mu/cueboy/education/maths/history/history/indiansulbasutras.htmhttp://pages.intnet.mu/cueboy/education/maths/history/history/indiansulbasutras.htmhttp://pages.intnet.mu/cueboy/education/maths/history/history/indiansulbasutras.htmhttp://pages.intnet.mu/cueboy/education/maths/history/history/indiansulbasutras.htmhttp://www.newworldencyclopedia.org/entry/Aryabhatahttp://www.newworldencyclopedia.org/entry/Aryabhatahttp://www.storyofmathematics.com/indian.htmlhttp://www.storyofmathematics.com/indian.html

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