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7/24/2019 Indian and Islamic Mathematics
1/101
Indian and
IslamicMathematics
Presented By: Joselito C.Cabije
MAed-Mathematics
7/24/2019 Indian and Islamic Mathematics
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IndianMathematics
7/24/2019 Indian and Islamic Mathematics
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Indus Valley
Civilization
i!hlyso"histicated urban
civilizationthat diedaround #$$$BC
7/24/2019 Indian and Islamic Mathematics
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a re!iono% morethan amillion
s&uare'ilometers
Indus Valley
Civilization
7/24/2019 Indian and Islamic Mathematics
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Indus Valley
Civilization
7/24/2019 Indian and Islamic Mathematics
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Indus Valley
Civilization
7/24/2019 Indian and Islamic Mathematics
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com"risedba'ed claybric'
buildin!s(hi!hlydevelo"ed sea
and river
Indus Valley
Civilization
7/24/2019 Indian and Islamic Mathematics
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)he de!ree o%advancements in
science and technolo!ycan be !au!ed %rom thehi!hly evolve system o%"lumbin!( "ublic baths(
construction o% thecitiesand the
so"histicated sealsandscul"turesthat have
been %ound.
Indus Valley
Civilization
7/24/2019 Indian and Islamic Mathematics
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)he *ecimal +ystem inara""a Period,#$$$-
$$ BC/
standardize 0ei!htsystem based onratios: $.$( $.( $.1($.( ( 1( ( $( 1$($( $$( 1$$( and$$
bronze rod mar'ed in
units o% $.#23 inches
7/24/2019 Indian and Islamic Mathematics
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)he 0ei!hts0eremade in re!ular!eometrical sha"esli'e he5ahedra(barrels( cones( andcylinders( therebydemonstratin!'no0led!e o% basic
!eometry
)he *ecimal +ystem inara""a Period,#$$$-
$$ BC/
7/24/2019 Indian and Islamic Mathematics
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)he *ecimal +ystem inara""a Period,#$$$-
$$ BC/
7/24/2019 Indian and Islamic Mathematics
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a scale 0ith nine "arallele&uidistantmar'in!sthat is
also indicative o% the decimalsystem
)he *ecimal +ystem inara""a Period,#$$$-
$$ BC/
7/24/2019 Indian and Islamic Mathematics
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7/24/2019 Indian and Islamic Mathematics
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Mohenjo-*aro
ruinsin the +indh"rovinceo% Pa'istan
7/24/2019 Indian and Islamic Mathematics
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+eal Arti%acts
7/24/2019 Indian and Islamic Mathematics
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so"histicationo% thecivilization-
"orts(ma!ni6centbric' buildin!sand cities.
Indus Valley
Civilization
7/24/2019 Indian and Islamic Mathematics
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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$$ BC/
7/24/2019 Indian and Islamic Mathematics
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mostly to be %ound inVedic te5tsassociated 0ith ritualactivities
system o% land!rants anda!ricultural ta5
assessments
Activities in the Vedic
Period ,3$8$$ BC/
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convertedrectan!ular "lotsor trian!ular "lotsto s&uares o%
e&uivalent sizes constructed ritual
altars
Activities in the Vedic
Period ,3$8$$ BC/
7/24/2019 Indian and Islamic Mathematics
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)he +ath"atha
Brahmana ,9$$BC/
)his te5trecordsthePytha!orean
theorem $$years be%orePytha!oras.
7/24/2019 Indian and Islamic Mathematics
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)he +ulba +utras
,$$$-1$$BC/
Sulba8 "ieces o%chord or strin!andSutra8 %ormula ora"horism
mathematicaldiscoveriesusin! a"iece o% chord %or
constructions o%
7/24/2019 Indian and Islamic Mathematics
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)he +ulba +utras
,$$$-1$$ BC/
)o 6nd as&uaree&uals to
the sum o%t0o !ivens&uares
7/24/2019 Indian and Islamic Mathematics
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Baudhayana ,;$$
BC/ discovered a "roo%o% the
theorem o% Pytha!oras
7/24/2019 Indian and Islamic Mathematics
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Interestin!
7/24/2019 Indian and Islamic Mathematics
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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$$
BC/
I ti l
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Irrational
?umbers in the+ulba +utras
I ti l
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Irrational
?umbers in the+ulba +utras
7/24/2019 Indian and Islamic Mathematics
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Panini
,1$ BC/ a +ans'rit !rammarian
0ho !ave a
com"rehensive andscienti6c theory o%"honetics( "honolo!y(
and mor"holo!y invented a "er%ect
+ans'rit !rammar
used the conce"t o% zero
7/24/2019 Indian and Islamic Mathematics
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Pin!ala
,1$$ 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 ;$$ears a%ter Pin al
7/24/2019 Indian and Islamic Mathematics
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alayudh
,1$$ 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
7/24/2019 Indian and Islamic Mathematics
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)he Ba'shaliManuscri"t ,#$$A*/
7/24/2019 Indian and Islamic Mathematics
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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
7/24/2019 Indian and Islamic Mathematics
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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
7/24/2019 Indian and Islamic Mathematics
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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
7/24/2019 Indian and Islamic Mathematics
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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
7/24/2019 Indian and Islamic Mathematics
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7/24/2019 Indian and Islamic Mathematics
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7/24/2019 Indian and Islamic Mathematics
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7/24/2019 Indian and Islamic Mathematics
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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
7/24/2019 Indian and Islamic Mathematics
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)ri!onometry inIndia ,#$$ A*on0ards/
)ri!onometry in
7/24/2019 Indian and Islamic Mathematics
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A sidereal year 0as
com"uted as#2.1;; days(0hich is only #minutes and 1$
seconds lon!erthanthe modern value o%#2.12#213 days.
)ri!onometry inIndia ,#$$ A*on0ards/
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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
7/24/2019 Indian and Islamic Mathematics
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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%
7/24/2019 Indian and Islamic Mathematics
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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%
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!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 #$- 1 A*
made major
discoveries incalculusincludin!im"ortant advances
in in6nite seriese5"ansions %ortri!onometric
%unctions
Credits to
7/24/2019 Indian and Islamic Mathematics
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Credits to
Madhava@s ?ame "o0er series e5"ansion o%
tan-,5/ ,re!ory@s +eries/
Credits to
7/24/2019 Indian and Islamic Mathematics
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com"uted an e5tremelyclose a""ro5imation o% Nas
#.912#9
Credits toMadhava@s ?ame
7/24/2019 Indian and Islamic Mathematics
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+rinivas
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+rinivasLamanujan ;;38 91$
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 $#
7/24/2019 Indian and Islamic Mathematics
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Islamic
Mathematics
I l i i
7/24/2019 Indian and Islamic Mathematics
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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
7/24/2019 Indian and Islamic Mathematics
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IslamicMathematics used o% com"le5 !eometric
"atternsto decorate theirbuildin!s
Islamic
7/24/2019 Indian and Islamic Mathematics
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IslamicMathematics
Islamic
7/24/2019 Indian and Islamic Mathematics
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IslamicMathematics
Gu@ranitsel%encoura!ed the
accumulationo% 'no0led!e
9th8 thcentury olden A!e o%Islamic scienceand
mathematics
ouse o%
7/24/2019 Indian and Islamic Mathematics
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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
7/24/2019 Indian and Islamic Mathematics
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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
7/24/2019 Indian and Islamic Mathematics
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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
7/24/2019 Indian and Islamic Mathematics
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Al >h0arizmi s:itab al3;abrwa
7/24/2019 Indian and Islamic Mathematics
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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!
7/24/2019 Indian and Islamic Mathematics
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)habit ben Gurra
7/24/2019 Indian and Islamic Mathematics
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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
7/24/2019 Indian and Islamic Mathematics
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develo"eda
!eneral %ormulaby 0hichamicable
numbers couldbe derived
)habit ben Gurra
,;1289$/
)hRbit ibn Gurra
7/24/2019 Indian and Islamic Mathematics
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Gtheorem
0here n is an inte!er andp( 4(and rare "rime numbers( then1nApA4and 1nAr are a "air o%
amicable numbers. )his %ormula
!ives the "airs ,11$( 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-
7/24/2019 Indian and Islamic Mathematics
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Muhammad Al
>araji $thcentury Persian
mathematician
introduced thetheory o% al!ebraiccalculus
6rst to use the
method o% "roo% bymathematicalinduction ,Binomial)heorem/
Ibn Al-aytham
7/24/2019 Indian and Islamic Mathematics
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Ibn Al aytham
,928$$/ author o% numerous
0or's on o"tics(
s"herical !eometry(number theory
discovered Bilson
7/24/2019 Indian and Islamic Mathematics
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7mar >hayyam
$;8#
!eneralized Indian metho
ds %or e5tractin! s&uareand cube roots to include%ourth( 6%thand hi!herroots
studied cubic e&uations
Treatise on(emonstration of
"roblems of 'lgebra
- -)usi
7/24/2019 Indian and Islamic Mathematics
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)usi
#thCentury 6rst to treat
tri!onometryas a
se"aratemathematicaldisci"line
!ave the 6rst
e5tensivee5"osition o%s"hericaltri!onometry
- -)usi
7/24/2019 Indian and Islamic Mathematics
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%ormulated la0 o% sines %or "lanetrian!les
a,sin'/ b,sin #/c,sin 7/ sine la0 %or s"herical trian!lesby
Persians Abul Oa%a Buzjani and Abu ?asrMansur ,$thcentury/
)usi
#thCentury
>amal al-*in al-
7/24/2019 Indian and Islamic Mathematics
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>amal al *in al
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Ian al Banna alMarra'ushi
#thcenturyMoroccanMathematicians
studied !eometry(com"utin! s&uareroots and the
theory o%continued %ractions
binomialcoeWcients
Ian al-Banna al-
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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-
7/24/2019 Indian and Islamic Mathematics
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... 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
7/24/2019 Indian and Islamic Mathematics
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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.html7/24/2019 Indian and Islamic Mathematics
99/101
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-ppt7/24/2019 Indian and Islamic Mathematics
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.html7/24/2019 Indian and Islamic Mathematics
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.htmlRecommended