Sejarah Matematika : Sistem Numerasi Mesir Kuno, Mesopotamia (Babilonia), Yunani Kuno, China Kuno dan Hindu-Arab

@G@ EE ZFA@GOG\GK

Zrbsfs Zrbsfs pfrnfa pfrnfa`gk `gkmgk mgk agtfagt agtfagteng eng deaucg deaucgee sfhgn sfhgn gdgkyg gdgkyg pfrgdg pfrgdg`gk `gk agku agkuse seg. g.

Ngrf Ngrfkg kg

ogky ogkygg

agku agkuse segg

dfkm dfkmgk gk

nfag nfagap apug ugk k

nbau nbauke keng ngse se

dgk dgk

ektfcfmfksekyg ygkm agapu afkleptgngk pfrgdg`gk, heng de`gkdekmngk dfkmgk agnocu agnocun n ygkm ygkm cgek. cgek. \foekm \foekmmg mg agkuse agkusegg agapu agapu afklep afkleptgn tgngk gk `frgdg `frgdg`gk `gk ygkm ygkm dedgsgrngk gtgs nbktgn sbsegc ygkm `frnfa`gkm. Deceogt dgre lgtgtgk sfhgrgo, aukluckyg pfrgdg`gk agkuseg `frgwgc dgre pfrnfa`gkmgk ùdgyg agkuseg ygkm sgkmgt detfktungk bcfo nbkdese mfbmrgies ygkm afkdunukm. Ger denfkgc agkuseg sf`gmge `fkdg ygkm `gkygn afa`frengk pfkmgruo `gme nfoedupgkkyg. @fmetu pucg dfkmgk pusgt pfrgdg`gk pfrtgag agkuseg, ygkm rgtgrgtg `frgsgc dgre dgfrgo ygkm decgcue gcergk sukmge. Degktgrgkyg gdgcg gdgcgo o !fsb !fsbpb pbtga tgaeg, eg, !fse !fser, r,"k "kdeg deg dgk dgk #ekg #ekg.. Zfrg Zfrgdg dg`g `gk k nbtg nbtg !fsb !fsbpb pbtga tgaeg eg afapfkmgruoe pfrgdg`gk sukmge $ec de !fser, sukmge Nukekm de #ekg dgk %fa`go "kdus de "kdeg. Zfrg Zfrgdg dg`g `gk k de kfmg kfmgrg rgkf kfmg mgrg rg tfrsf tfrsfù ùtt afkh afkhgd gdee dgsg dgsgrr aukl aukluc ucky kygg pfkleptggk ngryg agtfagteng. Zgdg &gagk pfrgdg`gk ofcfkesten, agtfagteng @gècbkeg `frpgdu dfkmgk agtfagteng !fser uktun afa`gkmnetngk agtfagteng 'ukgke kgke..

(gsec (gsec

ektf ektfrgn rgnse se

agsy agsygrg grgng ngtt

dfkm dfkmgk gk

pfrg pfrgdg dg`g `gk kpf pfrg rgdg dg`g `gk k

cgek cgek

afapfkmgru afapfkmgruoe oe pfrnfa`gkmg pfrnfa`gkmgk k agtfagteng agtfagteng sfcgkhutkyg. sfcgkhutkyg. $fmgrgkfmgr $fmgrgkfmgrgg tfrsfùt tfrsfùt tfcgo `gkygn afa`frengk sua`gkmgk tfrogdgp pfrnfa`gkmgk agtfagteng sggt eke. eke. @fr` @fr`gmg gmgee ùnt ùntee afku afkukh khun unng ngk k `gow `gowgg stru strunt ntur ur agtfa agtfagte gteng ng ygkm ygkm pgce pgcekm km sfdfrogkg`frupg ècgkmgk tfcgo demukgngk dgre agsg nfagsg aucge dgre `gkmsg @gècbkeg, !fser, 'ukgke, #ekg, dgk "kdeg ygkm nfaudegk desfapurkgngk bcfo ecauwg ecauwgk k Zfrseg Zfrseg Gc Nogwgre Nogwgre&ae &ae.. (gc eke afkukh afkukhunn unngk gk `gowg `gowg pfrnfa pfrnfa`gk `gkmgk mgk agtfagteng agtfagteng dgre pfrgdg`gkp pfrgdg`gkpfrgdg` frgdg`gk gk tfrdgoucu tfrdgoucu afapfkmgru afapfkmgruoe oe pfrnfa`gkmgk pfrnfa`gkmgk agtfagteng pgdg pfrgdg`gkpfrgdg`gk deagsg `frenutkyg.

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G. \estf \estfa a Kuafrg Kuafrgse se Afser Afser Nukb Nukb

@gkmsg

!fser

nukb

tfcgo

afkmfkgc

tucesgk

dgk

sestfa

ècgkmgk.@egsgkyg tucesgk eke detfaungk pgdg sfùgo `gtu eke dgk denfkgc dfkmgk dfkmgk sestfa oefrbmcei. \estfa oefrbmcei oefrbmcei afrupgngk afrupgngk sestfa ècgkmgk ècgkmgk ygkm sfrekm demukgngk demukgngk dgcga nfoedupgk nfoedupgk sfogreogre. )ucesgk oefrbmcei oefrbmcei `fr`fktun `fr`fktun mga`gr mga`gr nflec ygkm afkygtgngk afkygtgngk sfùgo sfùgo ngtg.\estfa ngtg.\estfa oefrbmcei oefrbmcei tfcgo demukgngk bcfo `gkmsg !fser nukb sfhgn sfnetgr tgouk *+- \!. \ea`bc \ea`bc ècgkmgk ècgkmgk gtgu kbtgse agtfagteng agtfagteng !fser Nukb `frseigt dfseagc `fr`gses /-: dgk dedgsgrngk pgdg sea`bcsea`bc oefrbmcei uktun tegp kecge pfrpgkmngtgk /- /, /-, /--, /---, /----, /-----, /------: sgapge dfkmgk sfhutg. sfhutg. )egpt )egptegp egp sea`bc sea`bc eke dgpgt dgpgt detuce detucess sf`gky sf`gkygn gn gpgpuk gpgpuk sfsuge sfsuge dfkmgk dfkmgk ècgkmgk ygkm deekmekngk, sfoekmmg uktun afkucesngk ècgkmgk dfcgpgk pucuo gtgu dfcgpgk rgtus, sea`bc sea`bc /- gtgu /-- detuces sf`gkygn dfcgpgk ngce.Dfkmgk ngce.Dfkmgk eke `frgrte `gowg afrfng afaecene sea`bc tfrpesgo uktun sgtugk, pucuogk, rgtusgk rgtusgk,, reùgk reùgk,, pucuo pucuo reùgk reùgk,, rgtus rgtus reùgk reùgk,, dgk hutggk. hutggk..@f .@fren renut ut eke tg`fc tg`fc `fresengk sea`bcsea`bc ygkm demukgngk `gkmsg !fser nukb desfrtge dfkmgk tfrhfagogk cga`gkm ygkm sudgo netg mukgngk sfngrgkm. 7) \estfa \estfa Zfkucesgk Zfkucesgk Gkmng Gkmng Oefrbmce Oefrbmcei i

7ktun sgtugk gdgcgo sfùgo mgres curus, cfkmnukmgk nf gtgs uktun pucuogk, cfkmnukmgk sftfkmgo cekmngrgk afkygapekm sfpfrte b`gt kyga kygaun un::

uktu uktun n rgtu rgtusg sgk, k, dgk dgk uktu uktun n hutg hutggk gk decg decga a`gkm `gkmng ngk k dfkm dfkmgk gk

sea`bc sfbrgkm cgnecgne ygkm afkgenngk tgkmgk.Zfkucesgk oefrbmcei dgpgt deaucge deaucge dgre ngkgk nf nere, nere nf ngkgk, gtgu dgre gtgs nf `gwgo dgk dgre `gwgo nf gtgs, tftgpe ègsgkyg deaucge dgre ngkgk nf nere sfpfrte dgcga pfkucesgk Grg`, wgcgupuk dgcga pfkucesgk ibragc &gagk sfngrgkm eke afkmmukgngk nere nf ngkgk:.











!esgcky !esgckygg uktun uktun afaùg afaùgtt ècgkm ècgkmgk gk *0<, *0<, gdg ceag ceag `fcgs `fcgs sea`bc sea`bc ygkm ygkm depfrcu depfrcungk ngk4du 4dugg sea`bc sea`bc 6rgtus 6rgtusgk9 gk9,, tuhuo tuhuo sea`bc sea`bc 6pucuo 6pucuogk9 gk9,, dgk fkga sea`bc sea`bc 6sgtugk9. 6sgtugk9. #bktbo #bktbo tucesgk ècgkmgk ècgkmgk *0< dgcga oefrbmcei oefrbmcei tfrceogt pgdg `gtu unergk dgre Ngrkgn, `frgsgc dgre sfnetgr s fnetgr /-- \!, dgk sfngrgkm `frgdg depgafrngk de %bu=rf, Zgres.

g. Zfku Zfkuce cesg sgk k Gkm Gkmng ng Dfse Dfseagc agc

Dgcga afkucesngk ècgkmgk, susukgk dfseagc tfr`fsgr detuces cfèo dgoucu. @ecgkmgk detuces dgre ngkgk nf nere. @frenut lbktbo pfkucesgk gkmng ?<,*-< 7ktun pfkucesgk tgkdg nbag, ceogt sea`bc oerbmc oerbmcei ei ygkm ygkm tgkdgky tgkdgkygg `fr`gc `fr`gcen en grgo. grgo. !esgcky !esgckygg pgdg pgdg pfkuce pfkucesgk sgk gkmng ?<,*-< sea`bc ygkm `fr`gcen grgo gdgcgo ?---- dgk <---. \fdg \fdgkm kmng ngk k pfku pfkuces cesgk gk sea`b sea`bcc *-*-- dgk dgk < tedg tedgn n `fr` `fr`gce gcen n grgo grgo.. Zfkuce Zfkucesgk sgk tgkdg tgkdg nbag nbag detuce detucess sftfcgo sftfcgo gkmng gkmng ygkm ygkm `fr`gc `fr`gcen en grgo grgo ygnk ygnkee pgdg pgdg gkmn gkmngg ?<.?<.----..

Zflg Zflgogk ogk uktun uktun brgk brgkm m !fser !fser nukb

tfr`gtg tfr`gtgss pgdg pgdg pflgog pflgogk k tukmmg tukmmgcc dfkmg dfkmgk k pfkmfl pfkmflugc ugcegk egk dgre dgre ygkm ygkm sfrek sfrekm m ngce ngce demu demukg kgng ngk k *58 *58 dgk dgk nurgk nurgkm m sfrek sfrekm m demu demukg kgng ngk k 85?: 85?:.. \fù \fùgo go pflg pflgog ogk k tukm tukmmg mgcc gdgcg gdgcgo o `fkt `fktun un /5k /5k deag deagkg kg k gdgcg gdgcgo o ècgkmgk ùcgt dgk eke dewgnece dgcga gkmng oefrbmcei dfkmgk afkfapgtngk sea`bc ygkm afwgnece sfùgo 6aucut9, ygkm `frgrte 6`gmegk9, de gtgs kbabr tfrsfùt. @frenut gdgcgo `f`frgpg lbktbo4











dgcga /5*?> , agng sea`bc 6`gmegk9 detfapgtngk de gtgs 6`gmegk pfrtgag9 ècgkmgk. \ya`bc decftgnngk de gtgs `gmegk pfrtgag ngrfkg ècgkmgk eke de`glg dgre ngkgk nf nere. `. Zfkhu khuacg acggk gk

Zfkhuacgogk sestfa ècgkmgk afser ogaper sfrupg dfkmgk pfkhuacgogk dfkmgk agsg neke ygkm `fr`fdg `fr`fdg ogkygcgo sea`bckyg. ?< 3 *< 1 0*/ 1 0*/

!. Zfrngcegk

!gtfagteng pgpyrus ;oekd gdgcgo sgcekgk dgre sfùgo rgtg rgtg rgtg nrhg nrhg sf`fc sf`fcua uaky kyg, g, !gtfa !gtfagte gteng ng pgpy pgpyru russ roek roekd d desg desgce cek k dgre dgre sfsf sfsfbr brgk gkm m pfku pfkuces ces ygkm ygkm `frk `frkgag gag Goabs oabsff detg detgou ouk k /</<- \!. \!. Deag Deagkg kg pgdg pgdg wgnt wgntu u etu, etu, 2bsfp 2bsfpo o afkh afkhgd gdee mur` mur`fk fkur ur de afser afser.. GcfJgkdfr (fkry ;oekd afapfrbcfokyg de cuJbr, !fser detgouk /++ /++ dgk nfaude nfaudegk gk afa`fce afa`fcekyg kyg deausf deausfua ua ekmmre ekmmress pgdg pgdg tgouk tgouk /+<. +<.!g !gtf tfa agten gtengg afkh afkhgk gkhe heng ngk k

;oek ;oekd d

pfa` pfa`gl glgg

pgpy pgpyru russ

afcg afcgcu cuee

depfr epfrnf nfkg kgcn cngk gk

ngce ngcea agt

`fre `frenu nut, t,

6

dfkmg fkmgk k Dfkm Dfkmgk gk

afapfcghgre sfaug ogc ygkm `gen, sfaug wgwgsgk gngk tftgp gdg dgk pfkmftgougk dgre rgogseg ygkm tfrsfaùkye, gngk tfrukmngp. Zgdg Zgdg igntgky igntgkyg, g, ogc eke afrupg afrupgngk ngk dfrftgk dfrftgk pfaflgo pfaflgogk gk agsgcgo agsgcgo agtfagteng dgsgr, sfùgo mgres `fsgr obua uktun pfkuces ygkm `frletg letg tekmme, Zfkuces tfrsfùt ogrus dgpgt afkmoetukm dfkmgk pgste `frgpg `gkygn `gtu `gtg `gt g ygkm deùtuongk uktun afa`gkmuk hgcgk dfkmgk nfaerekmgk tfrtfktu dgk `frgpg `gkygn pgpgk rbte ygkm ygkm deù deùtu tuon ongk gk uktu uktun n afa` afa`fre fre agng agngk k ùdg ùdgn n pfnf pfnfrhg rhg dgk dgk sf`gmgekyg. 2guo sf`fcua ngcnucgtbr ngcnucgtbr gtgu `gongk `gongk agtfagteng agtfagteng abdfrk, abdfrk,











ècgkmgk `fsgr dfkmgk lfpgt. @frenut lgrg pfrngcegk kuafrgse !fser4

•

Zfrngcegk Dfkmgk "grg Dbu`cekm

Zfrngcegk dgcga sestfa dbu`cekm denfrhgngk dgre pfkmucgkmgk pfcepgt mgkdggk ècgkmgk dfkmgk uksur pfkmgcekyg nfaudegk afkhuacgongkkyg. #bktbokyg /8 J /* 1  @ugtcgo mgres uktun afaesg afaesgon ongk gk dug dug nbcb nbcba. a. "se "se nbcb nbcba a nf `gwg `gwgo o de sf`fc sf`fcgo go nere nere,, deaucge dfkmgk kbabr /. gkdgngk dgk tuces * de`gwgokyg, cgcu mgkdgngk * etu sfoekmmg afkdgpgtngk gkmng ?, tfrus demgkdgngk sgapge sgapge gkmngk gkmngkyg yg tedgn tedgn afcfèo afcfèoee ygkm ygkm dengcen dengcengk. gk. "secgo "secgo nbcba nbcba ngkgk, ngkgk, tucesn tucesngk gk kbabr kbabr ygkm ygkm ekmek ekmek gkdg gkdg ngceng ngcengk k dgcga dgcga ogc eke, eke, gdgcgo /*:. De`gwgo /*, mgkdgngk dgk tuces *?. gkdgngk cgme *? dgk tuces ?+, )frus sgapge sf`gkygn nbcba nere

\fngrgkm lgre gkmng de nbcba nere ygkm ngcgu detga`gongk gngk afkmogsecngk gkmng pfrtgag ygkm ekmek dengcengk dgcga sbgc eke, /8:. Gkmng / 3 ? 3 +1 /8, cgcu mgres `gwgoe kbabr denbcba ngkgk desf`frgkm kbabr tfrsfùt. !gng ygkm demgres `gwgoe de nbcba ngkgk gdgcgo /* 3 ?+ 3 ><: dgk ngau huacgongk gngk afkdgpgtngk /<, ygkm gdgcgo hgwg`gk tfpgt dgre /8 J /* 1 /< Zfrngcegk #fkmgk "grg Ogc$ekm Zfrngce Zfrngcegk gk dgcga dgcga sestfa sestfa ogc=ek ogc=ekm m `fr`fd `fr`fdgg dfkmgk dfkmgk dbu`ce dbu`cekm km

uktun afapfraudgo cgkmsukm sghg pgdg lbktbo. #bktbokyg /8











dgcga ogc eke /8: de`gme dfkmgk * agng ogseckyg < uktun -, tedgn de tuces:. "secgo nbcba ngkgk, tucescgo kbabr ygkm ekmek gkdg ngcengk dgcga ogc eke gdgcgo /*:. De`gwgo /*, mgkdgngk dgk dgk tuces tuces *?. *?. gkd gkdgn gngk gk cgme cgme *? dgk dgk tuces tuces ?+. ?+. )frus rus sgapge sgapge sf`gkygn nbcba nere /8 < 8 /

/* *? ?+ ><

\fngrgkm lgre gkmng de nbcba nere ygkm mgkhec, ygetu gkmng /8, 8, dgk /. %gcu mgres `gwgoe kbabr de nbcba ngkgk desf`frgkm kbabr kbabr tfrsfù tfrsfùt. t. !gng !gng ygkm ygkm demgres demgres `gwgoe `gwgoe de nbcba nbcba ngkgk ngkgk gdgcgo /* 3 ?+ 3 ><: dgk ngau huacgongk gngk afkdgpgtngk /<, ygkm gdgcgo hgwg`gk tfpgt dgre /8 J /* 1 /<. #. Zfa`g a`gmeg megk

Zfa`gm Zfa`gmegk egk dgcga dgcga sestfa sestfa ècgkm ècgkmkg kg afser afser denfrh denfrhgng gngk k dgre dgre pfkmucgkmgk pfcepgt mgkdggk ècgkmgk dfkmgk uksur pfa`gmekyg nfaudegk afkhuacgongkkyg. #bktbokyg >+ 4 0 1  @ugtcgo mgres uktun afaesgongk dug nbcba. "se nbcba nf `gwgo de sf`fcgo nere, deaucge dfkmgk kbabr /. gkdgngk dgk tuces * de`gwgokyg, cgcu mgkdgngk * etu sfoekmmg afkdgpgtngk gkmng ?, tfrus demgkdgngk sgapge gkmngkyg gkmngkyg tedgn afcfèoe ygkm de`gme. de`gme. "secgo nbcba ngkgk, tucescgo kbabr pfa`gmekyg dgcga ogc eke, gdgcgo 0:. De `gwgo 0, mgkd mgkdgn gngk gk dgk dgk tuce tucess /?. /?. gkd gkdgn gngk gk cgme cgme *+ dgk dgk tuce tucess <, <, dgk dgk sftfruskyg. / * ? +

0 /? *+ <

\fngrgkm lgre gkmng denbcba ngkgk ygkm ngcgu detga`gongk













nbcba nere desf`frgkm kbabr eke. !gng ygkm de mgres `gwgoe de nbcba nere gdgcgo * 3 ? 3 +: dgk ngau dgpgt afkdgpgtngk /?, ygkm gdgcgo hgwg`gk tfpgt dgre >+ 4 0 1 /? %) \estf \estfa a Zfkuces Zfkucesgk gk Gkmn Gkmng g Oefrgte Oefrgten n

\fcgag Nfrghggk @gru agsgcgo agtfagtes desfùtngk desfùtngk pgdg Zgpyrus Gkgs Gkgstgs tgsee /, dgk dgk Lec`b ec`bur ur Zgpy Zgpyrus rus dgre dgre wgnt wgntu u ;gaf ;gafssf ssfss """ afklg afklgtgt tgt pfkmunurgk cgogk. Gkmng oefrbmcei gmgn `fr`fdg dgcga pfrebdf ygkm `fr`fdg, kgauk sflgrg uaua afapukyge stycf sfrupg. \estfa ècgkmgk cgek ygkm demukgngk brgkm !fser sftfcgo pfkfaugk tucesgk de pgperus, tfrdere dgre gkmng oefrgtel. Gkmng eke afaukmnekngk ècgkmgk detuces dgcga `fktun ygkm hguo cfèo rgpe dgre sf`fcuakyg sf`fcuakyg sggt afkmmukgngk afkmmukgngk sestfa ygkm afaùtuongk afaùtuongk cfèo `gkygn sea`bc ygkm ogrus deogigc. Gdg sea`bc tfrpesgo uktun4 /, *, 8, ?, , <, 0, +, >, /-, *-, 8-, ?-, -, <-, 0-, +-, >-, /--, *--, 8--, ?--, --, <--, 0--, +--, >--, /---, *---, 8---, ?---, ---, <---, 0---, +---, >---











\estfa ècgkmgk eke dgpgt de`fktun dgre `f`frgpg sea`bc. Gkmng >>>> ogkyg afaecene ? sea`bc oefrgtel sf`gmge pfkmmgkte 8< oefrbmcei. \gcgo sgtu pfr`fdggk pfr`fdggk utgag gktgrg gktgrg gkmng gkmng nfrgagt nfrgagt dgk systfa systfa ècgkm ècgkmgk gk netg netg gdgcgo gdgcgo gkmng gkmng nfrgagt nfrgagt tedgn tedgn afa`fk afa`fktun tun systfa systfa pbsese pbsese sfoekmm sfoekmmgg gkmng gkmng tfrtfktu dgpgt detuces dgcga urutgk gpgpuk. @frenut eke gdgcgo sgcgo sgtu lgrg brgkm !fser afkuces *0< dgcga gkmng oefrgtel

@. \estfa \estfa Kuafrgse Kuafrgse @gècbkeg @gècbkeg &Afsb'bt &Afsb'btgaeg) gaeg)

\estfa \estfa ècgkm ècgkmgk gk `gkmsg `gkmsg \uafre \uafregg afkmmu afkmmukgn kgngk gk sestfa sestfa `gses `gses <- gtgu sestfa sestfa sfJgdf sfJgdfsea seagc+ gc+.. Zfkmmu Zfkmmukgg kggk k sestfa sestfa sfJgdfs sfJgdfseag eagcc agseo agseo netg netg rgsgngk rgsgngk oekmmg sfngrgkm dgcga nfoedupgk sfogreogre, aesgckyg / hga tfrdere dgre










\estfa \estfa pbsese pbsese ècgkm ècgkmgk gk @g`yc @g`ycbke bkegg afkmmu afkmmukgn kgngk gk `gses `gses <-. ogc eke ùngk ùngk `frgrte afrfng afaecene <- sea`bc, gngk tftgpe afrfng afkmmukgngk `gses /ducu ducu tfrus tfrus nfteng nfteng sgapge sgapge <- afrfng afrfng afkmuc afkmucgkm gkm cgme cgme sea`bc sea`bckyg kyg.. Dgcga Dgcga @g`y @g`ycb cbke keg, g,

uktu ktun

afkuc fkuces esn ngk

ècgk ecgkmg mgk k

/ />

afkm afkmu ukgng kgngk k

sest sestfa fa

pfkmfcbapbngk sfdfrogkg `fr`gses /-: dgk uktun ècgkmgk <- afkmukgngk sestfa pbsese.

Zbsese Demet / * 8 ?

<-<-/ <-* <-8

7ktun tun

\estfa Zbsese @ecgkmgk @g`ycbkeg $ecge Zbsese Demet Nftfrgkmgk / Zbsese sgtugk s.d. pucuogk s.d. pucuogk
ècg ècgkm kmgk gkky kygg

sfk sfkdere dere

sga sgag

sfp sfpfrte frte

pgdg gdg

sest sestfa fa

pfkmfcbapbngk sfdfrogkg ècgkmgk @g`ycbkeg. )gpe, )gpe, pfrcu depfrogtengk dgcga sestfa eke gdgcgo pfcftgngk dgk pbsese sea`bc tfrogdgp sea`bc cgekkyg.











sfcgek afkukhunngk / humg dgpgt afkukhunngk <-, <- *, <-8, ... <- k. @fmetu humg

sfcgek afkuhunngk /-, humg dgpgt afkukhunngk /-.<-, /-.<- *,

/-.<-8, ... /-.<- k Mcfo ngrfkg etu uktun afkmoedgre nfnfcerugk dgcga afkgiserngk kecge dgre dgre cga`gk cga`gkmc mcga` ga`gkm gkm tfrsf` tfrsfùt ut ègsgky ègsgkygg demukg demukgngk ngk tgkdg tgkdg sfcgkm sfcgkm sf`gmg sf`gmgee pfkgkdg pbsese pucuogk dgk reùgk. Deagkg heng ècgkmgk tgkpg sfcgkm5spgse `frgrte tgkdg sgtugk IIII 1 ?  tgkpg sfcgkm::. 2eng `frsfcgkm5spgse `frsfcgkm5spgs e sgtu `frgrte `frkecge pucuogk I O III

1 <8  pgnge sfcgkm sgtu ::. 2eng

`frsfcgkm5spgse * `frgrte reùgk IO O III 1 <-E 3 8 1 8<-8  pgnge sfcgkm dug ::. Gngk tftgpe, `gme pfkuces ogc eke gngk afkhgde gaèmu heng uktun afa`fdgngk agkg gkmng ygkm `frkecge <- dgk / afkmekmgt afrfng `fcua afkfaungk gkmng - dgcga sestfa ècgkmgkkyg. @frenut eke lbktbo pfkucesgkkyg4











\estfa \estfa gkmng gkmng (ekdu (ekdu egcgo sestfa gkmng gkmng nfdudun nfdudungk gk pfrsfpucuo ygkm de`g de`gkm kmuk uk pgdg pgdg nuru nuruk k nf> nf> bcfo bcfo goc gocee agtf agtfagt agteng eng "kd "kdeg eg,, degd degdgp gptgs tgsee goce goce agtfagten Zgrse GcNogwgre&ae dgcga dgcga ùnu ùnu tfktgk tfktgkm m pfkmer pfkmerggk ggk dfkmgk dfkmgk gkmng (ekdu ygkm detuces sfnetgr +*!: dgk goce agtfagten Grg` GcNekde afkfruse afkfruse ùnukyg ùnukyg tfktgkm pfkmmukgg pfkmmukggk k gkmng "kdeg nfcugrgk nfcugrgk +8-!:, +8-!:, dgk nfau nfaude degk gk tfrsf tfrsf`g `grr nf duke dukegg `grgt `grgt pgdg pgdg Hgagk Zfrtfkm Zfrtfkmgogk gogk.. \fhgrgwgk afcglgn gkmng abdfrk abdfrk de nf`gkygngk `gogsg dfkmgk gkmng gkmng @rgoae ygkm ygkm demukgngk sfnetgr pfrtfkmgogk g`gd nf8 \!. Gkmng @rgoae tfcgo detfaungk dgcga prgsgste de mugmug dgk de nbek de dgfrgo dfngt Zukf, !ua`ge, dgk 7ttgr Zrgdfso. "ke gkmng dfkmgk dfkmgk sfdenet =gregse: ygkm demukgngk demukgngk sfcgag wgntu ygkm lunup cgag oekmmg g`gd nf?. @frenut eke `f`frgpg pfr`fdggk gkmng (ekdu dfkmgk gkmnggkmng cgekkyg.

Zfretukmgk

Zgdg aucgkyg `gkmsg "kdeg afkhuacgongk ècgkmgk dgre nere nf ngkgk, tftgpe nfaudegk `fru`go dgre ngkgk nf nere. aesgckyg pfkhuacgogk ècgkmgk *0<













Gtgu dfkmgk prbsfs sf`gmge `frenut 4 2uacgo ècgkmgk sgtugk

4<31

//

2ua 2uacgo cgo ècg ècgkm kmgk gk pucu pucuo ogk

4 0 3 < 1 /8

2uac 2uacgo go ècg ècgkm kmgk gk rgtu rgtusg sgk k

4 * 3 * 1 ?  agng agng ogse ogseck cky yg ?/ ?/

Dfae Dfaene negk gk pucg pucg pfrn pfrngce gcegk gk dug dug ùgo ùgo ècgk ècgkmg mgk, k, aesgc aesgcky kygg /* /* dfkm dfkmgk gk <, prbsfskyg sf`gmge `frenut 4 < J / 1 < detuces degtgs ècgkmgk /*, sfcgkhutkyg < J * 1 /* dfkmgk afkga`gongk ècgkmgk < dfkmgk / < J  1 8- dfkmgk afkga`gongk ècgkmgk * dfkmgk 8 2gde ogseckyg gdgcgo 0-. #grg cgek dgpgt decgnungk sf`gmge `frenut, aesgckyg











%. Ngrgnt Ngrgntfre frest sten en Agtf Agtfagt agteng eng Grg` Grg`

Dgcga sfhgrgo agtfagteng, agtfagteng "scga g`gd pfrtfkmgogk, ègsg desf desfù ùtt agtfa agtfagte gteng ng "scga "scga gtgu gtgu agtfa agtfagte gteng ng Grg`, rg`, afkl afklgn gnup up ngheg nghegk k agtfagteng ygkm decgnungk sfcgag pfrnfa`gkmgk pfrgdg`gk "scga nerg nerg nerg gktg gktgrg rg tgou tgouk k <** <** dgk dgk /<-/<--.. \gek \gekss "scg "scga a dgk dgk agtf agtfag agte teng ng "scg "scga a `frnfa`gkm pfsgt de `gwgo noecgigo "scga ygkm afkmugsge )eaur )fkmgo, )fkmgo, aucge dgre \fafkgkhukm \fafkgkhukm "`freg de `grgt sgapge %fa`go "kdus de teaur dgk Dekgste Gcabrg=ed dgk Nfngesgrgk !gce de sfcgtgk. Dgcga Dgcga ùnu ùnu G (estbry (estbry bi !gtofa !gtofagtel gtels, s, Celtbr eltbr Ngt& Ngt& afkuce afkucess `gowg `gowg \fhgrg \fhgrgo o agtfagt agtfagteng eng "scga "scga g`gd g`gd pfrtfkm pfrtfkmgog gogk k tedgn tedgn dgpgt dgpgt detuce detucess dfkmgk dfkmgk cfkmngp, ngrfkg `gkygn agkusnrep Grg` ygkm `fcua depfcghgre. )ftgp sghg, mgres `fsgrkyg `fsgrkyg sudgo denftgoue. !gtfagtengwgk !gtfagtengwgk "scga afkmfa`gkmn afkmfa`gkmngk gk sestfa sestfa kuafrgc kuafrgceg eg cftgn cftgnkecg kecgee dfseag dfseagcc ygkm ygkm afklgn afklgnup up pflgog pflgogk k dfseagc dfseagc











Zfretukmgk @ecgkmgk #grg cgek aftbdf aftbdf pfrngcegk pfrngcegk ècgkmgk denftgoue denftgoue dgre grg` ygkm dedumg dedumg

lgrg ekededgpgt dgre (ekdu "kdeg: sf`gmge `frenut 4 /*< J /8 1 /<8+











/--- dgk /----. @frenut eke sea`bc ygkm demukgngk dgcga pfkucesgk ècgkmgk sestfa gttel gtgu sestfa glrbpobkel

7ktun ècgkmgk sgtu desea`bcngk dfkmgk tbkmngt 6 E9 ygkm ùngk afrupgngk ourui gwgc kgag ècgkmgk. \fcgkhutkyg, ècgkmgk ygkm cgek detuces sf`gmge nba` nbaèk ekgs gsee dfkm dfkmgk gk sea`b sea`bcs csea ea`b `bcc ygkm ygkm cgek. cgek. @fren @frenut ut eke eke sea`b sea`bcc ygkm ygkm demukgngk uktun afkuces gkmng / sgapge /- dgcga sestfa glrbpobkel. glrbpobkel .

\fcgkhutkyg, `gkmsg 'ukgke afkmfa`gkmngk ècgkmgk -, --, ---, dgk ---- ygkm depfrbcfo dgre pfkmmgùkmgk sea`bc  dfkmgk sea`bcsea`bc uktu uktun n pucu pucuog ogk, k, rgtu rgtusg sgk, k, reù reùgk gk dgk dgk pucu pucuog ogk k reù reù.. @fre @frenu nutt eke eke ogse ogsecc











Zgdg tgouk -- \!, sestfa ècgkmgk gttel demgkte dfkmgk sestfa gcpog`ftel. )frdgpgt )frdgpgt *0 ourui dgcga dgcg a sestfa gcpog`ft 'u 'ukgke ncgsen, gngk tftgpe tfrdgpgt 8 ourui ygkm oecgkm dgre pfkucesgk. @frenut eke ourui gcpog`ftel 'ukgke `fsfrtg ourui ngpetgc dgk ourui nflec.













@frd @frdgsg gsgrn rngk gk ouru ouruio iour urui ui gcpo gcpog` g`ft ftel el degtgs, degtgs, `gkmsg `gkmsg 'ukgke kgke afkuru afkurukng kngk k afkhgde sestfa ècgkmgk `gru ygetu sestfa gcpog`ftel. !fsnepuk sea`bc ourui uktu uktun n demgaag, demgaag, nbppg, nbppg, dgk sgape pgdg tg`fc degtgs tedgn gdg, `frenut eke gngk detukhunngk sea`bc `gru dgcga sestfa ècgkmgk gcpog`ftel. \ea`bc ècgkmgk sgtugk dgcga sestfa gcpog`ftel

\ea`bc ècgkmgk pucuogk dgcga sestfa gcpog`ftel











#bktbo pfkucesgk cga`gkm ècgkmgk pflgogk

Mpfrgse ècgkmgk gtgu `froetukm 'ukgke 'ukgke Nukb











`gaù

nflec

ygkm

desusuk

uktun

afwgnece

gkmng

sgtu

sgapge

sfaècgk.\estfa ècgkmgk eke dekgagngk ècgkmgk \uzobu dgcga estecgo #ekg. #ekg. Dgcg Dgcga a sest sestfa fa rbd rbd kuafrgc kuafrgc , `gtgkm `gtgkm `gaù `gaù nflec nflec desusu desusuk k uktun uktun afwgne afwgnece ce gkmng dgre / sgapge ><

)edgn )edgn dgpgt denftgoue dfkmgk dfkmgk pgste `frgpg useg rbd kuafrgc , gngk tftgpe sestfa ècgkmgk eke sudgo demukgngk `f`frgpg rgtus tgouk ygkm cgcu hguo h guo sf`fcua kbtgse











#bktbo Zfkucesgk @ecgkmgk











uktun afaùgt gkmng kfmgtei, dgk afkmurgkmngk gkmng kfmgtei dgre kbc uktun afaùgt gkmng pbsetei .













`gtgkm aukmnek tedgn pbsesebkgc, tftgpe pgdg pgpgk pfkmoetukmgk, afrfng pbsesebkgc. gdgcgo mga`gr ygkm tfpgt dgre huacgo pfkmoetukmgk `gtgkm ?- de gtgs afhg gtgu cgktge.











De 2fpgk 2fpgkm, m, agtfa agtfagte gteng ngwg wgk k afkf afkfap apgt gtng ngk k afkm afkmoe oetu tukm km `gtgk `gtgkm m pgdg pgdg pgpg pgpgk k pfkmoetukmgk, sfcfa`gr ngek dfkmgk mred, dgk demukgngk ogkyg `fktun =frtengc

Sejarah Matematika : Sistem Numerasi Mesir Kuno, Mesopotamia (Babilonia), Yunani Kuno, China Kuno dan Hindu-Arab

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