El Reino de Los Numeros Isaac Asimov

September 7, 2022 | Author: Anonymous | Category: N/A

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@j _ìad e` Jds Aµ uh`rds uh`rds

Isbbk Bsihdv

µ Iaeik` mà`rbj 2. S`rp`tuiebe S`rp`tuiebe

µ IAEIK@ M@A@_BJ

KbpµĲtujd 2 EµĲmitds y ... eµĲmitds Adhor`s e` jds aµ uh`rds uh`rds @j sàtied e`j aµ u uh`rd h`rd ad `stµb jihitbed b jb rbzb guhbab. Vbrids baihbj`s puèà s`r bei`strbeds pbrb eistiamuir jds aµu uh`rds h`rds àtr` eif`ràt`s dol`tds. Abturbjhàt`, abei` ihbmiab qu` kdaskiàt`hàt` jds kuàtà, p`rd, bj pbr`k`r, r`k` r, puè` puèà a bpr`k bpr`kibr ibr jb eif`r eif`ràkib àkib àtr` aµu uh`rds h`rds pdr jbs eif`ràkibs qu` à jds bsp`ktds dfr`kà jbs eistiatbs kbatiebe`s. Hukgds e` adsdtrds, pdr `l`hpjd, ads muibhds pdr `j bsp`ktd, bj lumbr abip`s, pdkd e`spuµ`s `s e` uab kdrtb fbhijibrizbkiµ dda a kda jds hishds. Sbrb hbydr s`muriebe, kbeb abip` tià` ua auh`rd p`qu`˚a add à jb `squiab sup`ridr izqui`reb, p`rd `j lumbedr prdhèid r`bjhàt` ad jd a`k`sitb. Jds eioulds e` abip`s qu` bpbr`kà b kdatiaubkiµdda a ad tiàà aµ u uh`rd. h`rd. ´J` hdj`stb b `j j`ktdr `sd, d r`kdadk` jds abip`s, e` kubjqui`r hba`rb, kda uab sdjb hirbeb y sia kdatbr su vbjdr4 @j puatd krukibj à jb gistdrib hbt`hµbbtikb tikb e`j gdhor` s` pr`sàtµd kubaed s` a`k`sitbrda hµbbss qu` bsp`ktds> kubaed fu` a`k`sbrid bjmd hµbbss qu` uab hirbeb gbkib jb kbv`rab, pbrb tà`r jb s`muriebe e` qu` bhods ai˚a ads ds s` àkdatrboba à `jjb, d uab hirbeb b su p`rkgb e` gbkgbs, pbrb kdavàk`rs` e` qu` jds kubtrd r`pu`stds e` jbs hishbs `stboba à su jumbr. @a bjmµ u ua a hdhàtd, `j gdhor` àkdatrµd a`k`sbrid kdhuaikbrs` usbaed aµuh`rds. uh`rds. Uuvd qu` ir b kbsb e` ua v`kiad y e`kirj`< ‚Dy`, vi`ld, Ád e`skdjmbst` uab e` his gbkgbs e` pièrb jb µu ujtihb jtihb v`z qu` `stuvist` à hi kbv`rab4‘ @atdak`s, si `j v`kiad j` e`k e`kµµĲb< ‚[batd ki`jd, ´Ruµ` t` gbk` pàsbr `sd4‘, j` kdavàer kdav àerµµĲb `stbr kbpbkitbed pbrb r`spdae`r< ‚Hirb bhimd, yd tà tàµµĲb kubtrd 2

?

(b)

KBS µ IUPJ IUPJD D 2. 2. E µ IMIUD[ Y ... E µ IMIUD[

(o)

(k)

Fimurb 2.2 r`pu`std s bbat`s r`pu`stds at`s e` qu` h` visitbr visitbrbs bs y ssdjbhàt` djbhàt` àkdatrµ` tr` tr`ss e e`spuµ `spuµ`s `s e` qu` t` r`tirbst`‘. Sbrb bor`vibr, `s utij u µ tij tà`r adhor`s pbrb jds eif`ràt`s aµu uh`rds. h`rds. Iaeueboj`hàt`, bj priakipid sdjbhàt` uads kubatds adhor`s fu`rda iavàtbeds> jds suflkiàt`s pbrb àtàe`rs`. Bjmuabs trious prihitivbs, bua à jb bktubjiebe, ad tiàà adhor`s pbrb aµu uh`rds h`rds hbydr`s qu` `j eds y `j tr`s. (@std ad qui`r` e`kir, abturbjhàt`, qu` e`skdadzkba jds aµu uh`rds h`rds hbydr`s> sµddjd jd simaiflkb qu` ad tiàà adhor`s eistiatds pbrb `jjds. Suèà jjbhbr, pdr `l`hpjd, bj auh`rd qu` adsdtrds ‚kubtrd‘,‚tr`s hµbbss uad‘). `hobrmd, àaµ kbsi tdeds jds kbsds, gbaà sied ebeds adhor`s eistiatds b jds[ia prih`rds ei`z uh`rds. uh`rds. @stds adhor`s, `spb˚ adj, adj, sda< uad, eds, tr`s, kubtrd, kiakd, sìs, si`t`, dkgd, au`v` y ei`z. SderµµĲb uad ir iav Sder iavàtbaed àtbaed ijihitbebhàt` adhor`s pbrb jds aµu uh`rds h`rds hbydr`s e` ei`z, p`rd `std jds gbr gbrµµĲb eif eifµµĲkij`s e` hba`lbr. ´Kµddhd hd pd pder erµµĲb uad r`kdrebr quµ` sdaied qu`rr qu`rrµµĲb e`kir ‚kubràtb y tr`s‘ y kuµbbjj ‚s`tàtb y au`v`‘, y bsµ bsµĲ suk suk`sivbh `sivbhàt`4 àt`4 B trbvµ``ss e` ei`z, p pdr dr dtrd jbed, jbs kdsbs fu`rda fµbbkij`s, kij`s, pdrq pdrqu` u` s` tà tàµµĲb b hbad ua sist`hb e` h`hdrib kdhd pbrt` `strukturbj e`j hishd, gbstb qu` jds adhor`s e` jds ei`z aµ u uh`rds h`rds s` fllboba oià à jb hàt`. Kubaed uad e`k e`kµµĲb ‚kubtrd‘, j`v j`vbatbob batbob kubtrd eèds> pbrb sìs, j`v j`vbatbob batbob

; sìs eèds. @atdak`s, si su dyàt` dos`rvbob sus eèds, p pde deµµĲb v`r jd qu` uad qu`rµµĲb e qu`r e`kir, `kir, à kbsd e e`` qu` guoi`rb djviebe djviebedd kuµbbatd, atd, `xbktbhàt`, r`pr`sàtboba jds sdaieds ‚kubtrd‘ y ‚sìs‘. Jb pbjborb jbtiab pbrb ‚eèd‘ `s ‚eµ ‚eµĲmitus‘ y bua v`k`s, à bjmuads ieidhbs, s` jjbhb b jds eèds ‚e ‚eµµĲmitd Ĳmitds‘. s‘. Ad `s bkki bkkieà eàtbj, tbj, pdr jd tba tbatd, td, qu` jds prih`rds ei`z aµ u uh`rds h`rds s`ba tbhoiµ`à a jjbhbe jjbhbeds ds eµĲmitds Ĳmitds.. @a ua priakip priakipid, id, jds eèds y jds aµu uh`rds h`rds fu`rda prµbktikbhàt` bktikbhàt` jb hishb kdsb. SderµµĲb pbr`k`r bj j`ktdr qu` sµĲ tà`hds adhor`s pbrb jds aµu Sder uh`rds h`rds brriob e` ei`z> p`rd `sd `s sµddjd jd à bpbriàkib. Jds kbhoids à `j jàmubl` gba e`fdrhbed tbatd jds adhor`s e` jds aµu uh`rds h`rds qu` g`hds djviebed su simaiflkbed drimiabj. Jb pbjborb ‚dak`‘ ad `s r`bjhàt` ua adhor` s`pbrbed, siad qu` vià` e` jbs pbjborbs prihitivbs e`j tùtµddaikd aikd qu` simaifl simaiflkboba kboba ‚uad sdorba sdorbat`‘. t`‘. @a dtrbs pbjborbs, pde`hds ihbmiabrads b au`strd gdhor` j`vbatbaed sus ei`z eèds y eikiàed ‚Y uad sdorbat`‘. Imubjhàt`, ‚edk`‘ simaiflkbob bj priakipid ‚eds sdorbat`s‘ E` bqu bquµµĲ à be`jbat`, jbs kdsbs sda hµbbss kjbrbs ‚Ur`k`‘ `s, `vieàt`hàt`, uab h`zkjb rµbbpieb pieb e` ‚tr`s y ei`z‘2 , y bs bsµµĲ suk`siv suk`sivbhàt`, bhàt`, kda jds aµu uh`rds h`rds qu` j` simuà. Kubaed jj`mb uad b ‚vìat`‘, tià` yb uab kdrrupkiµdda a e` ‚eds ei`k`s‘, pdr p dr jd qu` ‚vìatitrµ ``s‘ s‘ simaiflkb ‚eds ei`k`s y tr`s‘, ‚trìatb‘, ‚kubràtb‘, ‚kiakuàtb‘ y `j r`std e` jbs e`kàbs s` r`su`jvà à fdrhb sihijbr, y bs bsµµĲ jj`mbhds jj`mbhd s bj ‚advàtb y au`v`‘.

Kjbv`s e` jds eèds S`rd, ´G`hds p`reied jb byueb e` au`strds eèds uab v`z qu` pbsbhds e` ei`z4 ´Kµdhd dhd iaeikbr kda jds eèds, pdr `l`hpjd, ua auh`rd kdhd ‚kiakuàtb y kubtrd‘4 G` vistd bedj`skàt`s qu` borà y ki`rrba rµbbpiebhàt` piebhàt` sus hbads kiakd v`k`s suk`siv suk`sivbhàt`, bhàt`, iaeikbaed bsµ bsµĲ kiakd ei`k`s d kiakuàtb, y ju`md j`vbatba kubtrd eèds. @std `stµb oià, p`rd `j dos`rvbedr tià` qu` `stbr bj`rtb, kdatbaed `j auh`rd e` v`k`s qu` bquµ``jj bor` jbs hbads. Sdr jd 2

A. e`j Urbaskriptdr< `j adhor` e` aµ u uh`rds h`rds à iamj`s b jds qu` s` r`fl`r` sda 2;(tgirt`à), ;(tgr``) y 23 (tà) à `spb˚ adj adj jb sihijitue s` eb b pbrtir e`j 26 (ei`kisµìs) ìs) uab h`zkjb rµbpieb bpieb e` 23(ei`z) y 6(sìs) Jb `tihdjdm `tihdjd mµĲb e` jds aµuh`rds uh`rds à `spb˚ a adj dj sda e` drimà jbtiad, pdr pda`r uads `l`hpjds< ‚trè`kih‘ (2;) uaiµdda a e` tr¯ ``ss (;) y e`k` e`k`h h (23) (23),, ‚qubt ‚qubttudre tudre`kih‘ `kih‘ (25) kdhpu`std kdhpu`std kda qubttudr (5) y e`k`h (23) pdr ujtihd u µjtihd ‚qu¯ ‚qu¯Ĳae`kih‘ (20) kdhpu`std kda qu qu¯¯Ĳaqu` (0) y e`k`h (23)

5

KBS µ IUPJ IUPJD D 2. 2. E µ IMIUD[ Y ... E µ IMIUD[

mà`rbj, tià` qu` bs`murbrs` y pr`muatbr< ‚´Kiakuàtb y kubtrd4‘, jd kubj gbk` tdtbjhàt` tdtbjhàt` iaµ utij utij jb `xpdsikiµdda a e` jds eèds. E`se` ju`md, auakb e`sbrrdjjbhds tµ``kaikbs kaikbs be`kubebs e` eèds pbrb jds aµuh`rds uh`rds sup`ridr`s b ei`z pdrqu` bpràe`hds à jb `sku`jb dtrbs y h`ldr`s tr`tbs. [i ad jbs gikiµ``rbhds, rbhds, e`sbrrdjjbr e`sbrrdjjbrµµĲbhds ua r`kursd hµbs bs d hàds bsµ bsµĲ< pd pder erµµĲbhds `stbr bkdre`s à qu` qu`,, kuba kubaed ed tu tuviµ viµ``rbhds rbhds jbs pbjhbs e` jb jbss hb hbads ads gbkib beàtrd, `j aµ u uh`rd h`rd j`vbatbed e` eèds iaeikbrµĲĲbb `j aµu uh`rd h`rd e` ei`k`s y, kubaed jbs pbjhbs e` jbs hbads `stuvi`rba gbkib bfu`rb, `j aµu uh`rd h`rd e` eèds j`vbatbed j`vbatb ed iaei iaeikbrµ kbrµĲb Ĳb `j aµu uh`rd h`rd e` uads. @a `sb fdrhb, p pder derµµĲb uab p p`rsdab `rsdab iaeikbr ‚kiakuàtb y kubtrd‘, tàiàed jds kiakd eèds kda jb pbjhb e` uab hbad gbkib beàtrd y kubtrd eèds kda jb pbjhb e` jb dtrb hbad gbkib bfu`rb. [imuiàed `stb ie`b, eds hdvihiàtds ebrµĲĲba ba kubj kubjqui` qui`rr aµu uh`rd, h`rd, gbstb advàtb y au`v`. @j aµ u uh`rd h`rd ssimuiàt` imuiàt` b advàtb y au`v` `s ‚ei`z ei`k`s‘ y pd pder erµµĲb sim simaiflkbraiflkbrs` pdr ei`z eèds kda jbs pbjhbs e` jbs hbads gbkib beàtrd> p`rd, àtdak`s, ´Ruµ` pd pder erµµĲb gbk`rs` pbrb simaifl simaiflkbr kbr ‚dak` ei`k`s‘4 Kubaed jj`mbhds b ‚ei`z‘ iaikibhds ua au`vd sist`hb e` kdatbr pdr ‚ei`k`s‘, à v`z e` pdr ‚uads‘. Su`std qu` g`hds jj`mbed b ‚ei`z ei`k`s‘, pde`hds `hp`zbr dtrb s`ri` bµu ua, a, y kdatbr pdr ‚ei`z ei`k`s‘. Au`strb pbjborb pbrb ‚ei`z ei`k`s‘ `s ‚kià‘, uab b˚a adsb dsb pbjborb kuyd drimà s` pi`re` à jb batim¨ u uèbe. èbe. BsµµĲ, e`spuµ`s Bs `s e` jj`mbr b ‚kià‘, pde`hds `hp`zbr e` au`v au`vd. d. Pad hµbs bs kià, `s ‚kiàtd uad‘ (´Ruµ` pd pder erµµĲb s`r hµbs bs kjbrd4). Ads àkbhiabhds prdmr`sivbhàt` mr`siv bhàt` gbkib ‚kiàtd vìatitrµ ``s‘, s‘, pbsbhds ‚kiàtd s`tàtb y au`v au`v`‘, `‘, y jj`mbhds b ‚kiàtd advàtb y au`v`‘, s`muied, abturbjhàt`, pdr ‚edskiàtds‘. @a `stb fdrhb pde`hds jj`mbr gbstb ‚adv`kiàtds advàtb y au`v`‘, e`spuµ``ss e` ku kuyb yb kifrb tà`hds ‚ei`z kiàtds‘. Sbrb àtdak`s yb `stbhds gboitubeds b iavàtbr au`vds adhor`s pbrb kubjqui`r aµ uh`rd uh`rd qu` jj`mu` bj puatd ‚ei`z‘. @a `j kbsd e` jds ‚ei`z kiàtds‘ jb au`v au`vbb pbjborb `s ‚hij‘, qu` tbhoiµ`à a `s e` drimà prihitivd. Bp`mµbbaeds` aeds` b `st` priakipid e` adhor`s au`vds pbrb kbeb ei`z e` bjmd, pde`hds kdatiaubr usbaed au`strds eèds. Sde`hds bkdrebr, pdr `l`hpjd, qu` jds eèds bpuatbaed gbkib bobld kda jbs pbjhbs e` jbs hbads gbkib bfu`rb, simaiflquà ‚kiàtds‘. Sdr jd tbatd, si qu`r`hds iaeikbr, pdr hèid e` jds eèds, si`t` hij quiaiàtds vìatikubtrd, pde`hds gbk`rjd à kubtrd hdvihiàtds< si`t` eèds gbkib bobld kda jbs pbjhbs pbrb eàtrd ( 44)> e`sp e`spuµ uµ`s `s kia kiakd kd eèd eèdss tbhoiµ`à a gbkib bobld, kda jb pbjhb pbrb bfu`rb ( 44)> ju`md eds eèds gbkib brriob kda

0

(b) si`t` hij

(o) quiaiàtds

(k) vìat`

(e) kubtrd

Fimurb 2.? jb pbjhb pbrb eàtrd (44) y, flabjhàt`, kubtrd eèds, tbhoiµ`à a gb gbkib kib bbrriob, rriob, kda jb pbjhb pbrb fu`rb(44). @a kdaeikida`s prihitivbs, prµbbktikbhàt` ktikbhàt` auakb `s a`k`sbrid ir hµbbss bjjµb e` jds j ds h hij`s, ij`s, y bsµĲ jjdd e` e`hu`strb hu`strb au`strd sist`hb auhµ``rikd. rikd. Kubaed s` jjj`mb j`mb b ‚ei`z hij‘ ad `xist` adhor` au`vd pbrb `sb kifrb. @s sdjbhàt` ‚ei`z hij‘. y e`spuµ e`sp uµ``ss ‚da ‚dak` k` hij hij‘, ‘, ‚vìat ‚vìatitrµ itrµ``ss hij hij‘‘ y bsµĲ suk` suk`sivbh` sivbhàt`. at`. Jds hbt`hµbbtikds tikds mri`mds sµĲ ie`br ie`brda da ua adho adhor` r` `sp`ki `sp`kibj bj pbrb `j aµu uh`rd h`rd ei`z hij. Jd jjbh jjbhbrda brda ‚hyrib ‚hyribs‘ s‘ (e`j kubj s` e e`rivb `rivb au`strb pbjb pbjborb orb ‚hi ‚hirrµĲbeb‘ ?), p`rd `s` tµ``rhiad rhiad `rb sdjbh` sdjbhàt` at` usbed pdr ua p`qu`˚ a add mrupd `sp`kibjizbed y auakb jj`mµd bj pµuojikd uojikd mà`rbj. @a jb bktubjiebe, tà`hds adhor`s pbrb aµu uh`rds h`rds kdhd ‚hijjµdda‘ a‘ y ‚oijjµdda‘ a‘ p`rd `stds adhor`s fu`rda iavàtbeds à jbs pdst pdstrih`r rih`rµµĲbs e` jjbb @ @ebe ebe Hèib. E` tded td ed `std r`sujtb qu`, pbrb jb hbydr hbydrµµĲb e` jds gdhor`s à jb gistdrib, jbs kubtrd pdsikida`s e` jds eèds gborµbba a sied suflkiàt`s pbrb kbsi tded.

Kifrbs kda pièr`zu`jbs @std ad qui`r` e`kir qu` s`h`lbat` sist`hb e` pdsikida`s e` jds eèds gbyb sied usbed `f`ktivbhàt`. Sbrb kubaed fu` a`k`sbrid utijizbr aµu uh`rds h`rds kdhd kiàtds y hijjbr`s, bjmuià iavàtµd uab kblb e` eèds brtiflkibj`s, b jb kubj jjbhbhds ‚µbbobkd‘ obkd‘ (e`j mri`md ‚µbbobkus‘ obkus‘; ). @j µbbobkd, obkd, à su fdrhb hµbs bs sàkijjb, kdastb e` ua kuberd e` hbe`rb, kda ei`z bjbhor`s pbrbj`jds, à kbeb uad e` jds kubj`s s` àkuàtrba kdjmbeds ei`z eiskds hdvioj`s. (Jds kdastruieds drimiabjhàt` pdr jds mri`mds y jds ? ;

A. e`j Urbaskriptdr< Kbatiebe huy mrbae` ` iae`flaieb. A e`j Urbaskriptdr< Uboj`rd

6

KBS µ IUPJ IUPJD D 2. 2. E µ IMIUD[ Y ... E µ IMIUD[

(b) bobkd

(o) =0?5

Fimurb 2.; rdhbad s kdasis rdhbads kdasisttµĲba à piè pièr`zu`jbs r`zu`jbs r` rèdae`be edae`bebs, bs, kdjd kdjdkbebs kbebs à rb rbaurbs, aurbs, à v`z e` kdjmbebs à bjbhor`s. Jb pbjborb à jbt jbtµµĲa pbrb pièr`zu`jb `s ‚kµbbjkujus‘ jkujus‘ , y jb guhbaiebe gb `stbed usbaed `sbs pièr`zu`jbs kdhd r`pr`sàtbkiµdda a e` aµu uh`rds, h`rds, eurbat` tbatd ti`hpd, qu` bµu ua a e`kihds qu` `stbhds ‚kbjkujbaed‘ kubaed `stbhds trboblbaed kda aµ u uh`rds. h`rds. Y jds eiskds hishds ‛ bua kubaed s` usba pbrb dtrds dol`tds ‛ sda jjbhbeds, à bjmuads ieidhbs, ‚kdatbedr`s‘). Kbeb bjbhor`, kda sus ei`z kdatbedr`s, r`pr`sàtb ua pbr e` hbads kda ei`z eèds. @xist` ua kdrtd `spbkid sdorbat` à kbeb bjbhor` kda dol`td e` qu`, si `hpi`zb uad kda tdeds jds kdatbedr`s b jb izqui`reb, `j hdv`r uad d hµbbss b jb e`r`kgb `quivbjmb b j`vbatbr uad d hµbbss eèds. [upµddambs` ambs` qu` `j bjbhor` d obrrdt` iat`ridr r`pr`sàtb ‚uads‘, `j iahèibtd sup`ridr, ‚ei`k`s‘, `j e` àkihb, ‚kiàtds‘, y `j qu` j` simu` gbkib brriob, ‚hij`s‘ (44). Bgdrb oià, pbrb r`pr`sàtbr si`t` hij quiaiàtds vìatikubtrd, sdjbhàt` `s a`k`sbrid hdv`r ‛si`hpr` gbkib jb e`r`kgb‛ kubtrd kdatbedr`s à `j bjbhor` iaf`ridr, eds à `j s`muaed, kiakd à `j t`rk`rd y tià` à `j kubrtd (44). @std tià` eiv`rsbs vàtblbs sdor` jbs kifrbs kda jds eèds. @a prih`r jumbr, ad tià` uad qu` r`kdrebr si jds eèds vba gbkib brriob d gbkib bobld, d jbs pbjhbs e` jbs hbads pbrb eàtrd d pbrb fu`rb. @sd `vitb `sfu`rzd e` h`hdrib. @a s`muaed, à jbs kifrbs kda jds eèds tià` uad qu` hdstrbr uab kifrb b kdatiaubkiµdda a e` jb dtrb. @j dos`rvbedr eò` r`kdrebr jds si`t` hij`s, hiàtrbs qu` uad pbsb b jds quiaiàtds, y bsµ bsµĲ suk`siv suk`sivbhàt`. bhàt`. @a `j µbbobkd, obkd, à kbhoid, tdebs t debs jbs kbt kbt`mdr `mdrµµĲbs p p`rhba`kà `rhba`kà b jb vi vistb stb si sihujtµ hujtµbba`bhàt` a`bhàt` y puèà p`rhba`k`r p`rhba` k`r bsµ bsµĲ iae`fla iae`flaiebhà iebhàt`. t`. Dtrd `sfu `sfu`rzd `rzd qu` s` `vitb b jb h`hdr h`hdrib. ib. @a t`rk`rd, bmr`mbaed obrrdt`s d bjbhor`s bj µbbobkd, obkd, puè` uad buhàtbr `j aµuh`rd uh`rd e` kifrbs ‛tbatbs kdhd j` pjbzkb‛ sia jb hàdr eiflkujtbe. Fiabjhàt`, `j µbbobkd obkd p`rhit` luatbr eds aµu uh`rds h`rds kda tdeb fbkijiebe y dotà`r à `j bktd

= jb kba kbatiebe tiebe r`pr`s` r`pr`sàtb atbeb eb pdr bhods.

Hbaidorbaed kda jbs pièr`zu`jbs Jb a`k`siebe e` kdhoiabr d ‚suhbr‘ aµu uh`rds h`rds eò` e` gbo`r surmied à jds bjodr`s e` jb gistdrib e` jb guhbaiebe. Sdr `l`hpjd, si uad rdobob jb edtbkiµdda a e` gbkgbs e` su v`kiad (d, pbrb `j kbsd, jbs bequirµĲĲbb g gda`stbh da`stbhàt`), àt`), d si sus odrr odrr`mbs `mbs pbr pbrµµĲba ua kki`rtd i`rtd aµu uh`rd h`rd e` kdre`rds,, e`s kdre`rds e`s`br `brµµĲb, abturbj abturbjhàt`, hàt`, sb sbo`r o`r kuµbbatbs atbs gbkgbs e` pièrb d kuµbbatbs atbs dv`lbs tàib à kdaluatd. Jb fdrhb hµbbss sàkijjb `rb kdatbrjbs. Uàµ àµĲb kiakd, s` bmr`mbrda eds< kuàtb tded `j jdt` y àkuàtrb qu` tià` si`t`.. Sdkd e`spu si`t` e`spuµµ``s, s, y b trb trbvµ vµ`s `s e` uab jbrmb `xp`ri `xp`riàki àkib, b, yb ad tià` qu` kdatbr kiakd y eds. Yb sbo` pdr be`jbatbed qu` sda si`t`. @xist`, pdr p dr supu`std, ua jµĲĲhit` hit` e` bjkbak` e` jb h`hdrib. [i `s a`k`sbrid suhbr vìa vìatitrµ titrµ `s `s y kiakuàtb y kubtrd, uad oià puè` ad sbo`r batikipbebhàt` kubj s`rµb jb r`spu`stb. Y `j kdatbr aµu uh`rds h`rds b `s` aiv`j puè` s`r tèidsd y bua irritbat`. Pa pbstdr prihitivd qu` `stuvi`rb trbtbaed e` kdatbr vìatitrµ `s `s dv` dv`lbs lbs b jbs kubj`s s` guoi`rba bmr`mbed kiakuàtb y kubtrd hµbbss y qu`, pdr s`muaeb v`z guoi`rb p`reied jb kuàtb, ki`rtbhàt` p pder derµµĲb kdav kdav`rtirs` `rtirs` à ua pbstdr àfur`kied e`j qu` kdaviai`rb bpbrtbrs`. @j µbbobkd obkd prdpdrkidab uab sdjukiµdda a b `st` prdoj`hb, pu`std qu` `s ua eispdsitivd h`kµbbaikd aikd qu` gbrµb jbs suhbs kda `j hµĲaihd `l`rkikid e`j iiat`j`ktd. at`j`ktd. Ai siqui`rb a`k`sitbrµb `s` pbstdr p`rhba`k`r k`rkb e` sus irritbat`s dv`lbs, siad qu` pderµb kdatbrjbs sia sbjir e` su kbsb. [i e`s`b suhbr vìa vìatitrµ titrµ``ss y kiakuàtb y kubtrd à `j µbbobkd, obkd, kdjdkbr kdjdkbrµµb prih`rd vìatitrµ ``s, s, hdviàed tr`s kdatbedr`s à `j bjbhor` e` jds ‚uads‘ y eds kdatbedr`s à `j bjbhor` e` jds ‚ei`k`s‘. E`spuµ`s `s bmr`mbrµb jds kiakuàtb y kubtrd, `hpulbaed kubtrd kdatbedr`s hµbbss à `j bjbhor` e` jds ‚uads‘ y kiakd hµbbss à `j bjbhor` e` jds ‚ei`k`s‘. [i kuàtb àtdak`s tdeds jds kdatbedr`s qu` gb hdvied, àkdatrbrµb si`t` kdatbedr`s à jds ‚uads‘ y siàt` à jds ‚ei`k`s‘. Vìatitrµ Vìatitrµ``ss y kiakuàtb y kubtrd sda s`tàtb y si`t`, y ad tuvd qu` kdatbr hµbbss brriob e` ei`z à aiamuab `tbpb e`j prdk`sd. _`bjhàt _`bjh àt`, `, si jd guo guoi`rb i`rb e`s`b e`s`bed, ed, pder pderµµĲb gbo`r bmr`m bmr`mbed bed aµuh`rds uh`rds hµbs bs bjtds, sia eiflkujtbe beikidabj. Sdr `l`hpjd, edskiàtds kiakuàtb y tr`s hij kiàtd edk`, hµbbss ki kiàtd àtd vìatisµ`ìs is h hij ij ddkgdkiàtds kgdkiàtds trìatb y uad, rµbbpiebhàt` piebhàt` jj`mba pdr hèid e`j µbbobkd, obkd, b tr`skiàtds s`tàtb y au`v` hij adv`kiàtds

7

(b) Srih`r Sbsd

KBS µ IUPJ IUPJD D 2. 2. E µ IMIUD[ Y ... E µ IMIUD[

(o) [`muaed Sbsd

(k) U`rk`r Sbsd

Fimurb 2.5 kubràtb y tr`s. B p`sbr e` `sd, ad gbor kubràtb gborµµĲb tàie tàiedd qu` kdatbr brriob e` ei`z à aiamuab `tbpb e` jb suhb. S`rd supµdambs` dambs` qu` tuvd a`k`siebe e` suhbr dkgd y si`t`. Buaqu` pbr`zkb `xtrb˚ a ad, d, `std pr`sàtb ua prdoj`hb hbydr qu` jb suhb e` jds kiàtds y hij`s qu` s` bkboba e` hàkidabr. @stb v`z j` gbkà fbjtb hµbbss kdatbedr`s. @hpi`zb hdviàed dkgd kdatbedr`s b jb e`r`kgb (44). [u simui` simuiàt` at` e e`s`d `s`d s`r s`rµµĲb hdv`r si`t` kdatbedr`s hµbbss b jb e`r`kgb, p`rd gboiàed hdvied yb dkgd, sdjbhàt` j` quèba q uèba eds pbrb hdv`r. ´Ruµ` gb gbk`r4 k`r4 Ad dostbat`, jb r`spu`stb `s sàkijjb. Hu`v` uad jds eds, y tià` yb b jb e`r`kgb jds ei`z kdatbedr`s e`j bjbhor` e` jds ‚uads‘. Suè` àtdak`s kbhoibrjds, pdr e`kir bs bsµµĲ, p pdr dr ua kdatbedr à `j bjbhor` e` jds ‚ei`k`s‘, yb qu` ei`z ‚uads‘ `quivbjà b ua ‚ei`z‘. Jj`vb uad gbkib jb izqui`reb sus ei`z kdatbedr`s e` jb gij`rb e` jds ‚uads‘ y, à su jumbr, hu`v` gbkib jb e`r`kgb ua kdatbedr à jb gij`rb e` jds ‚ei`k`s‘ ( 44). B kdatiaubkiµdda, a, kdhpj`tb su hdvihiàtd à jb gij`rb e` jds ‚uads‘. @j dp`rbedr iob b hdv`r si`t` kdatbedr`s, p`rd sdjbhàt` pued hdv`r eds, jd qu` bµ u ua a j` e`lbob kiakd kdatbedr`s pdr hdv`r, jds kubj`s eò` hdv`r ( 44). @j r`sujtbed flabj `s< ua kdatbedr b jb e`r`kgb à jb gij`rb e` jds ‚ei`k`s‘, y kiakd à jb gij`rb e` jds ‚uads‘> p pdr dr jd tbatd dkgd hµbbss si`t` sd sda a quiak`. @stb fdrhb e` kbhoibr ei`z kdatbedr`s pdr uad, dp`rb à tdebs jbs gij`rbs. [i uad a`k`sitb hµbbss e` ei`z ei`k`s, si`hpr` puè` kbhoibr ei`z ei`k`s pdr ua kiàtd> kdhd puè` kbhoibr ei`z kiàtds pdr ua hijjbr, y bsµ bsµĲ suk`siv suk`sivbhàt`. bhàt`. B p`sbr e` tded, auakb `s a`k`sbrid kdatbr hµbbss e` ei`z kdatbedr`s à kubjqui`r dkbsiµdda. a. @a r`bjiebe, ad `s a`k`sbrid kdatbr brriob e` kiakd, pu`std qu` si s` gb `hpulbed hµbbss e` kiakd b jb e`r`kgb, sdjbhàt` s` a`k`sitb kdatbr jds aµ u uh`rds h`rds sdorbat`s b jb izqui`reb (qu` sda si`hpr` hàds e` kiakd) pbrb sbo`r kuµbbatds atds gby b jb e`r`kgb. [i gby µuaikbhàt` uaikbhàt` ua kdatbedr b jb izqui`reb

9 yb s` sbo` qu` gby au`v` b jb e`r`kgb. [ia a`k`siebe e` kdatbr ffµµĲsikbhàt`, puèà bpr`kibrs` b sihpj` vistb kiakd kdatbedr`s d hàds. Sdr `stb rbzµdda, a, b p`sbr e` jb a`k`siebe e` p`rhutbr kdatiaubhàt` ei`z pdr uad, ua gµbboij oij dp`rbedr e` µbbobkd obkd puè` jj`vbr b kbod kdhpjikbebs suhbs y r`stbs (`stbs u ujtihbs µ jtihbs trboblbaed bj kdatrbrid) kda v`jdkiebe hbydr qu` jb qu`, à fdrhb dreiabrib, puè` e`sbrrdjjbr jb hbydr pbrt` e` adsdtrds kda jµbbpiz piz y pbp`j. Pa kbhp`µdda a e` dp`rbkida`s à µbbobkd obkd puè`, à fdrhb sdrpràeàt`, hbatà`r su tµĲtujd kdhpiti kdhpitiàed àed kda kbjkujb kbjkujbedrbs edrbs `jµ`ktrikbs `ktrikbs e` `skritdri `skritdrid. d. Iakieàtbjhàt`, trboblbaed kda `j µbbobkd, obkd, s` puè` e`hdstrbr qu` ad ihpdrtb ihpd rtb kda quµ` aµuh`rd uh`rd e` uab suhb s` `hpi`zb. Oià s`b qu` hu`vb uad si`t` kdatbedr`s prih`rd y e`s e`spuµ puµ``ss ddkgd, kgd, u dkgd prih`rd y e` e`spuµ spuµ``ss ssi`t`> i`t`> e` tdebs hba`rbs tià` quiak` kdhd r`sujtbed. _`kuµ``re`s`, re`s`, kdhd r`mjb mà`rbj, qu` ad ihpdrtb à quµ` dreà s` suh` ua mrup mrupdd e` aµu uh`rds. h`rds.

Kifrbs à j`trbs @j µbbobkd obkd `s ouàd gbstb ki`rtd puatd, p`rd bµu ua a pr`sàtb ua prdoj`hb. ´Kµddhd hd `skrioir aµ uh`rds uh`rds pbrb r`mistrds p`rhbaàt`s4 Jds oboijdaids y jds `mipkids batimuds tuvi`rda kdpidsbs dpdrtuaiebe`s e` `skrioir mrbae`s kifrbs, bj kbjkujbr ihpu`stds y trioutds d bj gbk`r jistbs e` jds brt brtµµĲkujds kdhprbeds pbrb jb kbsb e`j r`y. Sdr supu`std, pder pderµµĲb uad `skrioir aµu uh`rds h`rds kdhd kubjqui`r dtrb pbjborb, y e`kir (kdhd s` `xpus `xpusdd bj priak priakipid ipid e`j kbp kbpµµĲtujd) edskià edskiàtds tds kiak kiakuà uàtb tb y tr`s hij kiàtd edk`, d `j `quivbjàt` à oboijdaid d à `mipkid. @std, sia `hobrmd, s`rµ s`rµĲb huy tèidsd y, pdr jd tbatd, `rb kdav kdavàiàt` àiàt` uab ki`rtb fdrhb tbquimrµbbflkb. flkb. Y tbj fdrhb fu` si`hpr` bedptbeb. Jds bhbauàs`s usboba eiv`rsds simads y sµĲhodjds (fr`kuà (fr`kuàt`hàt` t`hàt` sihpj`s j`trbs e`j bjfbo`td) pbrb iaeikbr jds aµu uh`rds. h`rds. Sdr `l`hpjd, kdasie`r`hds `j sist`hb rdhbad, pdrqu`, bua à jb bktubjiebe, s` usb à jds hdauhàtds y èiflkids pµu uojikds, ojikds, à jds eipjdhbs y à jbs kbrbtujbs e` jds r`jdl`s, y, pdr jd tbatd, ads `s fbhijibr. Sbrb iaeikbr ‚uad‘, jds rdhbads `skrio `skrioµµĲba I, qu` prdoboj`hàt` simaiflkbob ua eèd.Eds, tr`s y kubtrd `rba II, III y IIII, jd kubj `s obstbat` sàkijjd. Sbrb kiakd, `j sµĲhodjd `s V. Abei` sbo sbo`` p pdr dr quµ``.Jb .Jb hµbs bs fbvdr`kieb e` jbs iaeikbkida`s `s qu` r`pr`sàtb uab pbjhb e` jb hbad j`vbatbeb, kda `j pujmbr s`pbrbed e` jds dtrds eèds. B kdatiaubkiµdda a tà`hds VI, VII, VIII y VIIII.

23

KBS µ IUPJ IUPJD D 2. 2. E µ IMIUD[ Y ... E µ IMIUD[

@j ei`z `stµb r`pr`sàtbed pdr W (tbj v`z eds pbjhbs, uab brriob y dtrb bobld). Hµbs bs be`jbat` tà`hds qu` kiakuàtb `s J> kià `s K> quiaiàtds `s E> y hij `s H. Sbrb `skrioir `skrio ir hij adv`kiàtds kiaku` kiakuàtb atb y dkgd d kgd pd pdaer aerµµĲb uad HEKKKKJ HEKKKKJVII VIIII (hij hµbbss quiaiàtds, hµbbss kià, hµbbss kià, hµbbss kià, hµbbss kià, hµbbss kiakuàtb, hµbbss kiakd, hµbbss uad, hµbbss uad, hµbbss uad). Aµddt`s` t`s` qu` à `j sist`hb rdhbad ua sµĲhodjd pbrtikujbr si`hpr` tà tàµµĲb `j hishd vbjdr auhµ``rikd rikd sia ihpdrtbr `j jumbr edae` s` kdjdkbrb. [i à v`z e` gbo`r `skritd HEKKKKJ HEKKKKJVIII VIII guoi`rb `skritd KJKEIIVKHKI, kdatia kdatiaubr ubrµµĲb siàed jb hishb kifrb. Jb u uaikb µaikb rbzµdda a pbrb pda`rjds à dreà e`kr`kiàt` `rb fbkijitbr fbkij itbr bj bhba bhbauàs` uàs` `j pde`r suhbr jds sµĲhodjd Ĳhodjdss rµbbpiebhàt` piebhàt` y dotà`r su simaiflkbed. (@s sihijbr b j`vbatbr uab hbad e` oriem`. Jb hbad tià` `j hishd vbjdr sia ihpdrtbr jb fdrhb à qu` s` brr`mjà jds abip`s, p`rd `j lumbedr jds bkdhdeb à s`ri`s y e` bku`red kda su vbjdr e`kr`kià e`kr`kiàt`, t`, pdr su 5 prdpib kdavàiàkib) . @j g`kgd e` qu` jds aµ uh`rds uh`rds rdhbads ad tàmba ua vbjdr e`t`rhiabed e` bku`red kda `j jumbr à qu` s` kdjdquà, e`struy` `j sist`hb qu` tba sbtisfbktdribhàt` trboblb à `j µbbobkd. obkd. @a µ``st`, st`, kd kdhd hd `s sboied sboied,, `s ihpd ihpdrtbat` rtbat` sbo`r à qu` gij`rb s` gba hdvied jds kdatbedr`s pu`std qu` kbeb uab tià` su prdpid vbjdr. Hµbbss bµ u ua, a, uad pd pder erµµĲb, si jd e`s`b e`s`brb, rb, gb gbk`r k`r suhbs usbaed aµu uh`rds h`rds rdhbads. Sdr `l`hpjd, si quisi`rb suhbr hij adv`kiàtds kiakuàtb y dkgd y eds hij kubtrdkiàtds s`tàtb y eds, pderµĲb Ĳb `skrioir HEKKKKJ HEKKKKJVIII VIII y HHKKKKJWWII kdhd `quivbjàt` e` jds eds aµ u uh`rds, h`rds, y e`spuµ`s `s `skrioir iakdrpdrbaed tdeds jds sµĲhodjds< HHHEKKKKK HHHEKKKKKKKKJJWWVIIII KKKJJWWVIIIII. I. Bgdrb oià, pbrb sihpjiflkbr `std< kiakd I `quivbjà b V, y eds J `quivbjà b K, pdr jd kubj s` puè` `skrioir< HHHEKKKKKKKKKWWVV S`rd kdhd eds V `quivbjà b W y kiakd K `quivbjà b E, jb kifrb `s HHHEEKKKKWWW. [ia `hobrmd, kdhd eds E `quivbjà b uab H, gbk`hds ua u ujtihd µ jtihd kbhoid pbrb `skrioir< HHHHKKKKWWW y tà`hds au`strb r`spu`stb< kubtrd hij kubtrdkiàtds trìatb. 5

@a jb bktubjiebe s` bkdstuhorb pda`r ua sµĲhodjd Ĳhodjd hµbs bs p`qu`˚ ad ad bat`s e` uad mrbae`, kdhd simad e` qu` eò` r`stbrs`, pdr jd kubj IV, à v`z e` IIII, `quivbj` b ‚kubtrd‘, y KH, à v`z e` EKKKK, `quivbj` b ‚adv`kiàtds‘. [ia `hobrmd, `stb fu` uab `vdjukiµda hèi`v hèi`vbj bj pbrb bgdrrbr `spbkid y ad fu` usbeb prihitivbhàt` pdr jds rdhbads

22 [ia eueb `j gµbboij oij bhbauàs` rdhbad, bkdstuhorbed b gbk`r `std, pder pderµµĲb `f`ktubr jb hishb suhb huy rµbpiebhàt`. bpiebhàt`. Ad dostbat`, `xistà dtrds tipds e` hbaipujbkida`s kda kifrbs qu` sda v`rebe`rds rdhp`kbo`zbs usbaed `j sist`hb rdhbad y qu`, sia `hobrmd, sda obstbat` sàkijjbs à `j µbbobkd. obkd. @a r`bjiebe, jb fbjtb e` ua sist`hb be`kubed pbrb `skrioir aµu uh`rds h`rds e`tuvd `j be`jbatd e` jds hbt`hµbbtikds tikds mri`mds, pu`std qu` `j sist`hb mri`md ad `rb hbs sàsi sàsioj` oj` qu` `j rdhbad rdhbad.. [` eik` qu` si Brqu BrquµµĲhè` Ĳhè`s, s, `j hµbs bs adtboj` e` jds hbt`hµbbtikds tikds mri`mds, guoi`rb tàied au`strd sist`hb auhµ ``rikd, rikd, gbor gborµµĲb iavàtbed `j kµbbjkujd jkujd (`j kubj, e` kubjqui`r hba`rb, kbsi iavàtµdd)) y ad gbo gborrµĲb gboied qu` `sp`rbr hij dkgdkiàtds b˚a ads ds pbrb qu` A`wtda jd iavàtbrb. Gbstb `j simjd IW E.L. ad ie`µd `j sist`hb hde`rad ua giaeµu e` adhor` e`skdadkied. @st` e`skuorihiàtd jj`mµd b jds µbbrbo`s, rbo`s, pdr quià`s fu` trbashitied b jds ùrdp`ds, y pdr `sd jjbhbhds ‚brµbboimds‘ oimds‘ b jds aµu uh`rds h`rds hde`rads. @j e`skuorihiàtd à jb Iaeib fu`, sihpj`hàt`, `j hde`jbed e` jds aµu uh`rds h`rds à `j µbbobkd obkd (kdhd s` `xpjikbrµb pdst`ridrhàt`). Y yb qu` `j µbbobkd obkd trboblb tba sbtisfbktdribhàt`, `s e` hbrbvijjbr qu` `j e`skuorihiàtd tbrebrb tbatd à `f`ktubrs`.

2?

KBS µ IUPJ IUPJD D 2. 2. E µ IMIUD[ Y ... E µ IMIUD[

KbpµĲtujd ? Abeb ... y hàds qu` abeb Jds giaeµ u u`s `s `hp `hp`zbrda `zbrda kd kda a au`v` ssµµĲhodjd Ĳhodjdss eif`ràt`s, uad p pdr dr kbeb a aµ u uh`µh`rd e`j uad bj au`v`. @stds gba kbhoibed kda `j ti`hpd, p`rd jj`mbrda b @urdpb à su fdrhb bktubj à `j simjd WVI y bgdrb s` `skrioà 2, ? ,; ,5 ,0 ,6 ,= ,7 y 9. @std, à sµĲ hishd, ad fu` u uaikd. µ aikd. Jds mri`mds y jds gòr`ds, pdr `l`hpjd, usbrda au`v au`v`` sµĲhodjds eif`ràt` eif`ràt`ss pbrb `stds aµu uh`rds. h`rds. @a kbeb kbsd, jds sµĲhodjds `rba jbs prih`rbs au`v` j`trbs e` sus bjfbo`tds. [ia `hobrmd, bhods puòjds kdatiaubrda kda jbs simuiàt`s au`v` j`trbs e` sus bjfbo`tds pbrb e`simabr jds aµu uh`rds h`rds ei`z, viàt`, trìatb y bs bsµµĲ suk`sivbhàt`, suk`sivbhàt`, y kda jbs au`v` j`trbs pdst`ridr`s pbrb jds aµuh`rds uh`rds kià, edskiàtds, tr`skiàtds, `tk. [i `j bjfbo`td `rb suflkiàt`hàt` mrbae` pbrb `j dol`td (s` r`qui`rà vìatidkgd j`trbs pbrb jj`mbr b hij pdr hèid e` `st` sist`hb) s` bmr`mboba j`trbs brkbikbs d j`trbs e` fdrhb `sp`kibj. @j usd e` j`trbs pdr aµ u uh`rds h`rds eid jumbr b kdafusiµda da kda jbs pbjborbs. Sdr `l`hpjd, `j aµ u uh`rd h`rd g` gòr`d or`d ‚qu ‚quiak`‘ iak`‘ gb gbkkµĲb usd e` eds jj`trbs `trbs kd kda a jbs ku kubj`s bj`s `hp`zbob `j adhor` e` Eids (à jàmubl` gòr`d), pdr kuyd hdtivd fu` a`k`sbrid usbr dtrb kdhoiabkiµdda a e` j`trbs. Sdr dtrd jbed, jbs pbjborbs dreiabribs pdeµĲĲba ba s`r kdav`rtiebs à aµu uh`rds, h`rds, suhbaedd `j vbjdr auhµ``rikd suhbae rikd e` jbs j`trbs qu` jbs kdhp kdhpda daµµĲba. @std s` gbk gbkµµĲb `sp`kibjhàt` pbrb jbs pbjborbs y jds adhor`s e` jb Oiojib (ua prdk`sd jjbhbed ‚m`hbtrµĲĲb‘) b‘) y hèibat` `j hishd s` j`µ j`µĲba tdeb kjbs` e` simaifl simaiflkbeds kbeds hµĲstikds y dkujtds. @j `l`hpjd hµbbss fbhdsd y fbhijibr `s `j pbsbl`, à jb _`v`jbki _`v`jbkiµµdda a e` [ba Luba, edae` `j aµu uh`rd h`rd e` ‚o`stibs‘ s` eb kdhd sìskiàtbs s`sàtb y sìs. @std iaeueboj`hàt` simaiflkbob qu` bjmuab kifrb kdat`hpdrµbba`b, a`b, jb kubj `rb p`jimrdsd p`jim rdsd hàk hàkidabr idabr bo boi`rtbhàt` i`rtbhàt` (pr (prdoboj`h doboj`hàt` àt` s` r`f`r r`f`rµµĲb bj `hp `hp`rbedr `rbedr rd rd-2;

25

KBS µ IUPJD IUP JD ?. ABEB ABEB ... Y H@AD[ H@AD[ RP@ ABEB ABEB

hbad A`rµdda) a) tà tàµµĲb ua adho adhor` r` qu` à j`trb j`trbss gòr gòr`bs `bs d mri`m mri`mbs bs suhbob `sb kifrb. [ia `hobrmd, e`se` àtdak`s jb màt` gb `stbed trbtbaed e` kdjdkbr jds adhor`s e` sus à`himds eàtrd e` `sb suhb. Edae` jds giaeµ u u`s `s h`ldrbr `j sist`hb mri`md y `j gòr`d fu` à `j usd e` jbs hishbs au`v` kifrbs pbrb jbs e`kàbs, kàtàbs, hijjbr`s y, à v`rebe, pbrb kubjqui`r obrrdt` d bjbhor` e`j µbbobkd. obkd. E` `sbs au`v` kifrbs e`rivbrda jds giaeµ u u`s `s tdeds jds aµu uh`rds. h`rds. Uded jd qu` s` a`k`sitµd fu` ebr b jbs kifrbs su vbjdr e` pdsikiµdda. a. @j aµ u uh`rd h`rd vìa vìatitrµ titrµ``s, s, pdr `l`hpjd, qu` à `j µbbobkd obkd kdasis kdasisttµĲb à tr`s kdatbedr`s hdvieds b jb e`r`kgb à `j obrrdt` e` jds ‚uads‘ y eds kdatbedr`s à `j obrrdt` e` jds ‚ei`k`s‘, pdeµĲĲbb `skrioirs` ?;, siàed `j a aµ u uh`rd µ h`rd e` jb e`r`kgb `j qu` r`pr`sàtbob bj obrrdt` iaf`ridr e`j µbobkd bobkd y `j e` jb izqui`reb `j kdrr`spdaeiàt` bj obrrdt` iahèibtd sup`ridr. @s dovid qu` trìatb y eds s` `skrioir `skrioirµµĲb ;?, y jds vbjdr vbjdr`s `s e` pdsikiµdda a à jds obrrdt`s bs bsµµĲ jd iaeikbr iaeikbrµµĲba, pu`std qu` ?; y ;? ad sda `j hishd aµu uh`rd. h`rd. Pad `s eds ei`k`s hbs tr`s uads y `j dtrd `s tr`s ei`k`s hµbbss eds uads. @s pdkd prdoboj` qu` jds tbjàtdsds mri`mds ad guoi`rba pàsbed à `std, yb qu` pàsbrda à puatds hukgd hµbbss sutij`s. Jd qu` eò` gbo`rjds e`tàied (y b tdeds gbstb `j eµĲb Ĳb e`j màid giaeµu e`skdadkied) `rb `j eij`hb e`j obrrdt` iatbktd à `j µbbobkd. obkd. [updambhds [upd ambhds qu` uab p`rsd p`rsdab ab qu`r qu`rµµĲb pda` pda`r, r, `à a v`z e` vìatitrµ``s, s, edskià edskià-tds tr`s. @a `j µbbobkd obkd hdv` hdv`rrµĲb eds kdatbedr`s à `j obrrdt` e` jds ‚kiàtds‘ y tr`s à `j obrrdt` e` jds ‚uads‘. @j obrrdt` e` jds ‚ei`k`s‘ p`rhba`k`r p`rhba`k`rµµĲb sia tdkbr. Psbaed `j sist`hb giaeµu pbr`k`rµĲb Ĳb qu` `sb p`rsd p`rsdab ab tà tàµµĲb qu` `skrioir ?;, sdjbhàt` qu` à `stb dkbsiµda da `j eds simaiflkbob eds ‚kiàtds‘ ad eds ‚ei`k`s‘. Sbrb `s` `f`ktd, ´Kµdhd dhd `skrioir `skrioirµµĲb eds hij tr`s, d eds hij trìatb, d eds hij tr`skiàtds4 tr`skiàt ds4 @a kbeb kbsd tàer tàerµµĲb qu` hdv`r eds kdatbedr`s à ua obrrdt` y tr`s à `j dtrd, jd kubj p pder derµµĲb pbr`k`r ?;. Pab sdjukiµdda a pd pder erµµĲb s`r `j u usbr sbr sµĲhodjd Ĳhodjdss eif`ràt`s à kbeb obrrdt`, p`rd pr`kisbhàt` `sd fu` jd qu` giki`rda jds mri`mds, kda r`sujtbeds pdkd sbtisfbktdrids. D pdaer pdaerµµĲb `sb p`rsdab uab `sp`ki` e` sµĲhodjd àkihb e` kbeb kifrb, pbrb iaeikbr `j obrrdt`. Sder SderµµĲb `skrioir ?; kdhd ?˙˙ ;˙ y edskiàtds tr`s kdhd ˙˙ ?˙˙ ;, iaeikbaed qu` à `j s`muaed kbsd `j ? `stbob à `j t`rk`r obrrdt` d s`b à `j e` jds ‚kiàtds‘, à v`z e` à `j s`muaed, d s`b `j kdrr`spdaeiàt` b jds ‚ei ‚ei`k`s `k`s‘. ‘. @std gbrµĲĲbb obs obstbat` tbat` eifµĲĲkij kij jb rµbbpieb pieb j`kturb e` jds aµ u uh`rds, h`rds, bua kubaed à t`dr t`drµµĲb `j sist`hb pd pder erµµĲb fuakidab fuakidabr. r.

20

(b) ?;

(o) ?3;

(e) ?3;3

(k) ?33;

(`) ?;33

Fimurb ?.2

26

KBS µ IUPJD IUP JD ?. ABEB ABEB ... Y H@AD[ H@AD[ RP@ ABEB ABEB

da e` ua sµĲĲhodjd hodjd `sp`kibj pbrb Ad, jb mrba iaadvbkiµdda a giaeµ u fu` jb iavàkiµ uab gij`rb iatbktb e`j µ bobkd. B `st` sµĲho Ĳhodjd djd jds µbbrbo`s rbo`s jd jjbhbrda ‚sifr‘ qu` simaiflkb simaifl kb ‚vbk ‚vbkµµĲd‘, pu`st pu`stdd qu` `j `spbk `spbkid id e`j `xtr `xtr`hd `hd e`r`k e`r`kgd gd e` ua obrrdt obrrdt`` iatbktd e`j µbbobkd obkd `stbob vbk vbkµµĲd. @stb pbjborb gb jj`mbed gbstb adsdtrds kdhd ua ‚kifrb‘ ‚ kifrb‘ d, à fdrhb hbs kdrrdhpieb, kdhd ‚k`rd‘. Au`strd ssµµĲhodjd pbrb ‚k`rd‘‘ `s 3, pdr jd kubj `skrioihd ‚k`rd `skrioihdss v` vìat iatitr itrµµ`s `s kdhd ?; (44), edskiàtds tr`s kdhd ?3; (44), eds hij tr`s kdhd ?33; ( 44), edskiàtds trìatb kdhd ?;3, eds hij trìatb kdhd ?3;3 (44), eds hij tr`skiàtds kdhd ?;33 (44), y bsµĲ suk`sivbhàt`. @a kbeb kbsd, e`hdstrbhds jds obrrdt`s iatbktds e`j µbbobkd obkd usbaed k`rds. (Vìatitr (V ìatitrµµ`s `s pderµĲb Ĳb `skrioirs` kdhd 33?; d 33333333?;, e`pàeiàed e`j tbhb˚ a add e`j µbbobkd, obkd, p`rd `std auakb s` gbk`. [i`hpr` s` supda` qu` tdeds jds obrrdt`s e`j bobkd µbobkd situbeds brriob e`j prih`rd hàkidabed y tdeds jds aµuh`rds uh`rds b jb izqui`reb e`j prih`rd hàkidabed, sda k`rds). Fu` `j k`rd `j qu` gizd tba prµbbktikds ktikds au`strd jjbhbeds aµ u uh`rds h`rds brµbboimds oimds y r`v r`vdjuki djukidaµ daµd `j usd e` jds hishds. (_`bjhàt` `s `xtrb˚a add qu` `j e`skuorihiàtd e` ‚abeb‘ gbyb sied tba huaeibjhàt` ihpdrtbat` y `xkitbat`, td tdeb ebv vµĲb hµbs bs `xtrb˚ a add bµ u ua, a, qu` tbatds hbt`hµbbtikds tikds adtboj`s ad gbyba vistd `sb ‚abeb‘) ‚abeb‘).. @s tbj jb ihpdrtbakib e`j k`rd qu`, à jb bktubjiebe, uab pbjborb pbrb `j hba`ld e` aµuh`rds uh`rds `s ‚kifrbaed‘, y kubaed e`sbrrdjjbhds ua prdoj`hb (buaqu` ad kdhpràeb aµ u uh`rds) h`rds) `stbhds ‚e`skifrbaed.‘ @j g`kgd e` qu` kubjqui`r `skritd s`kr`td, mà`rbjhàt` jjbhbed ua ‚kriptdmrbhb‘, tdmrbh b‘, puèb s`r jjbhbe jjbhbedd tbh tbhoiµ oiµ à à uab ‚kifr ‚kifrb‘, b‘, ads r`ku` r`ku`reb reb `j r`sp`t r`sp`tdd qu` jb màt` tàµĲĲbb pdr jds aµuh`rds, uh`rds, `sp`ki `sp`kibjhàt` bjhàt` bbqu`jjb qu`jjb qu` ad àtàe àtàeµµĲb ssu u fdrhb e` trbobld.

Hdviàed jds kdatbedr`s à ua pbp`j Bj suhbr kda aµu uh`rds h`rds brµbboimds oimds `s a`k`sbrid, prih`rd, bpràe`r e` h`hdrib jbs suhbs dotàiebs bj kdhoiabr eds aµ u uh`rds h`rds kubj`squi`rb, e`j 3 bj 9. @a prih`r b˚ a ad, d, jds ai˚ a ads, ds, trbobldsbhàt`, bpràeà qu` ? y ; gbkà 0> 5 y 0, 9> 6 y =, 2;, y bs bsµµĲ suk`sivbhàt`. Ubhoiµ`à a bpràeà jd hµbs bs ihpdrtbat` e` tded< qu` 3 y 3 gbkà 3. @a `j µbbobkd, obkd, tbj`s suhbs puèà jj`vbrs` b kbod tàiàed qu` bpràe`r sdjbhàt` jb suhb e` 2 y 2 y siàed a`k`sbrid µu uaikbhàt` aikbhàt` sbo`r kdatbr gbstb `j 23. Ki`rtbhàt`, jb vàtblb qu` tiàà jds aµ u uh`rds h`rds `skritds sdor` `j µbbobkd obkd

2= pbr`k` `stbr oià `skdaeieb à `stb `tbpb. S`rd vbhds b supda`r qu` s` r`qui`r` suhbr mrbae`s aµ u uh`rds, h`rds, p pdr dr `l`h `l`h-pjd, 0795 y ?0=7. Uded jd qu` s` a`k`sitb `s sbo`r kµddhd hd gbk`r jbs suhbs p`qu`˚ a abs. bs. Srih`rd, eò` uad s`pbrbr kbeb aµu uh`rd h`rd pdr hijjbr`s, kiàtds, e`kàbs y uaiebe uaiebe`s. `s. Pab v`z g g`kgd `kgd `std, `j prdoj`h prdoj`hbb quèbrµb bsµ bsµĲ< 0333 y 733 y 93 y 5 hµbs< ?333 y 033 y =3 y 7 gbkà< =333 y 2;33 y 263 y 2? Bgdrb oià, si `j aµuh`rd uh`rd 2;33 s` eivie` à eds kifrbs, 2333 y ;33, `j 263 à 233 y 63, y `j 2? à 23 y ?, `s obstbat` sàkijjd suhbr jds hijjbr`s, jbs kàtàbs, jbs e`kàbs y jbs uaiebe`s pbrb dotà`r< 7333 y 533 y =3 y ? d s`b 75=?. Jb fdrhb à qu` tdeds g`hds sied às`˚ a abeds beds b suhbr usb `st` priakipid,p`rd jd sihpjiflkb bj dhitir jds k`rds y ‚jj`vbaed‘ jds uads, pdr jd kubj `j prdoj`h prd oj`hbb què quèbb bsµĲ< Ĳ< 0795 ?0=7 75=? @a kubjqui`r fdrhb, jd qu` s` gb g`kgd, butdhµbbtikbhàt` tikbhàt` y sia a`k`siebe e` mrba kdakàtrbkiµda, da, `s gbk`r kbhoids e` ei`z pdr uad., @j dp`rbedr `hpulµd ei`z uads eàtrd e` jb kdjuhab e` jds eik`s, ei`z ei`k`s eàtrd e` jb kdjuhab e` jds kiàtds y ei`z kiàtds eàtrd e` jb kdjuhab e` jds hij`s. Jb r`stb `s ua prdkèihiàtd bj kdatrbrid. Sdr `l`hpjd, si `stbhds r`stbaed ?97 e` 0;2, eivieihds jds aµu uh`rds h`rds kdhd simu`< 033 y ;3 y 2 hàds< ?33 y 93 y 7 B prih`rb vistb, pbr`k` kdhd si fu`rb b gbo`r eiflkujtbe pbrb r`stbr 7 e` 2, d 93 e` ;3. Sbrb sihpjiflkbr `j prdoj`hb `skrioihds au`vbhàt` `j aµ u uh`rd h`rd hµbbss mrbae`, mrbae`, piei pieiµµ`àedj` aedj` pr`st pr`stbeds beds 233 b 033, suhbhd suhbhdss `stds kià b jds ;3, pbrb kdav`rtirjds à 2;3, e` edae` tdhbhds 23 pbrb bmr`mbr bj 2. Bgdrb `j prdoj`h prd oj`hbb què quèbb bsµĲ< Ĳ<

27

KBS µ IUPJD IUP JD ?. ABEB ABEB ... Y H@AD[ H@AD[ RP@ ABEB ABEB

533 y 2?3 y 22 hàds< ?33 y 93 y 7 ?33 y ;3 y ; pdr jd kubj jb r`spu`stb `s ?;;. Au`strd hµ``tded tde d usubj e` r`stb ad `s bsµ bsµĲ, yb qu` ads às`˚a aba ba ua hµ`tded `td ed h`kµbaikd baikd qu` dskur`k` `j priakipid> ad dostbat`, `j priakipid `stµb à jb fdrhb e`skritb bat`ridrhà bat`ridrhàt`. t`. Pa gµbboij oij dp`rbedr e` µbbobkd obkd pderµĲb Ĳb r`sdjv`r jds prdoj`hbs hàkidabeds usbaed jds kdatbedr`s à v`z e` jds aµ u uh`rds h`rds kda hukgb hbydr rbpie`z qu` ua `hpj`bed prdhèid eèikbed b trboblbr kda `stb kjbs` e` dp`rbkida`s. [ia `hobrmd, `j µbobkd bobkd r`qui`r` e` gboijiebe hbaubj, kdsb qu` jds aµuh`rds uh`rds ad `xim` `ximà. a. Hµbbss bµ u ua, a, à jb kdhputbkiµdda a auhµ ``rikb rikb td tdebs ebs sus dp`rbkida`s quèba b jb vistb, pdr jd kubj puèà v`riflkbrs` à ouskb e` bjmµu ua a `rrdr, hiàtrbs qu` à `j µbbobkd obkd ad puè` eµddae` ae` ybbsspdr quµ` kdh`tiµ d `jjds `rrdr. Jd qu` `s hµbbs, s, e`j hishd hded uad qu` sbo`r `j µbbobkd obkd `s hµ p`rhba` p`rhbaàt` at` qu` be`hba be`hba`s `s kda jds eèds, jds aµ u uh`rds h`rds sda hµbbss p`rhbaàt`s à ua pbp`j qu` à `j µbbobkd. obkd.

_dhpiàed jb obrr`rb e`j k`rd Pa priakipi p riakipibat` bat` à brithµ``tikb tikb bpràe` rµbbpiebhàt` piebhàt` qu` eds aµ u uh`rds h`rds kubj`squi`rbb puèà suhbrs j`squi`r suhbrs`` y dotà dotà`r `r uab rr`spu`stb `spu`stb rbzdabo rbzdaboj`. j`. Ubhoiµ `à a bprà bprà-e` rµbbpiebhàt` piebhàt` qu` `std ad dkurr` kda jbs r`stbs. [i uad tdhb 0 e` =, quèba ?. [i tdhb = e` =, j` quèba 3. S`rd, ´Suè` tdhbr dkgd e` si`t`4 Jds mri`mds e`kiei`rda qu` ‚¡Ad!‘ kda uab mrba `xkjbhbkiµdda. a. @j r`stbr 7 e` = e`lbr e`lbrµµĲb hàds qu` abeb y ´Kµddhd hd puè` kubjqui`r kdsb s`r hàds qu` abeb, si abeb `s `j hµĲaihd pdsioj`4 @st` rbzdabhiàtd s` simuiµd gbstb `j simjd WVI. Y, sia `hobrmd, si ads e`tà`hds b pàsbr à `j hishd, `s pdsioj` qu` bjmd s`b hàds qu` abeb. [updambhds, [updambh ds, pdr `l`h `l`hpjd, pjd, qu` uab p`rsdab tà tàµµĲb = eµddjbr`s jbr`s y ua bhimd viad b r`kd r`kdrebrj rebrj`` qu` j` eò eòµµĲb. [iàe [iàedd uab p`rsdab gdarb gdarbeb, eb, rµbbpiebhàt` piebhàt` j` eid jds = eµddjbr`s, jbr`s, `xpjikµbbaedj` aedj` qu` `rb tded jd qu` tà tàµµĲb e` hdhà hdhàtd td y prdh`tiµ`àedj` aedj` pbmbr `j edjbr r`stbat` tba prdatd kdhd jd rùai`rb. @sb p`rsdab quèµdd,, pdr jd tbatd, kda hàds qu` su eia`rd, pu`std qu` bµu ua a beùebob ua eµddjbr jbr b su bkr`èdr. @a dtrbs pbjborbs, si uad tdhb 7 e` =, tià` ‚uad hàds e` k`rd‘. ´Ruµ` `s jd qu` tià` e` rbrd d e` iakr` iakr`µµĲoj` `std4

29 D supdambhds qu` uab p`rsdab piàsb kbhiabr gbstb `j puòjd prµddxihd, xihd, qu` `stµb 22 nijµddh`trds h`trds bj sur. @hpi`zb su kbhiabtb à `j puatd 22 nijµddh`trds h`trds bj adrt` e`j puòjd. Bj gbo`r kbhiabed 2 nijµddh`trd, h`trd, `stbrµb 23 nijµddh`trds h`trds bj adrt`> e`spuµ`s `s e` gbo`r kbhiabed ? nijµddh`trds h`trds hµbbs, s, `stbrµb 7 nijµddh`trds h`trds bj adrt`, y bsµĲ gbst gbstbb qu` gbyb kbhiab kbhiabed ed jd jdss 22 nijµdh`trds, dh`trds, bj jj`mbr b kuyd puatd `stbrµ b 3 nijµddh`trds h`trds bj adrt` e`j puòjd> yb jj`mµd. d. S`rd supdambhds qu` `sb p`rsdab `s `xtr`hbebhà `xtr`hbebhàt` t` eistrb eistrbµµĲeb (d `xtr`hbebhàt` t`rkb) y kbhiab 2? nijµddh`trds. h`trds. @sd jb kdjdkb 2 nijµddh`trd h`trd bj sur e`j puòjd. Bgdrb oià, bj kbhiabr jds 22 nijµddh`trds, h`trds, jb eistbakib bj puòjd eishiauyµ d b k`rd. [i s` kbhiab hµbbss e` 22 nijµddh`trds, h`trds , Ád eò`rµĲb Ĳb kdatiaubr eishiauyàed bobld e` k`rd4 [` pderµĲb Ĳb e`kir< ‚Ad. @hpi`zb b buhàtbr au`vbhà au`vbhàt`.‘ t`.‘ S`rd `j buhàtd à eistbakib `s bgdrb gbkib `j sur e`j puòjd, edae` bat`ridrhàt` `stbob `j adrt`. Ád gbk` `sd uab eif`ràkib4 Sbrb v`r si gbk` uab eif`ràkib µu utij, tij, trbk`hds uab jµĲa`b v`rtikbj (qu` s`rµb e` adrt` b sur à `j hbpb usubj). Sdambhds ua puatd sdor` jb hishb qu` r`pr`sàtb jb jdkbjizbkiµdda a e` ua puòjd (d kubjqui`r dtrb kdsb), y jjbh`hds k`rd b `s` puatd. [i hbrkbhds eivisida`s imubj`s e`se` `s` puatd (`std `s, gbkib `j adrt`, kdhd `s kdavàkidabj à au`strds hbpbs) pde`hds pr`tàe`r sda nijµddh`trds h`trds e` s`pbrbkiµdda a y, bj `f`k `f`ktd, td, jds auh auh`rbhd `rbhdss 2, ?, ; y bsµ bsµĲ suk` suk`-sivbhàt`. Sde`hds gbk`r jd hishd pbrb `spbkids d iat`rvbjds imubj`s bobld e`j puatd(bj sur) hbrkµbbaedjds aedjds tbhoiµ `à a 2, ?, ;, `tk. Jbs eivisi eivisida`s da`s gbkib `j adrt` pde`hds jjbhbrjbs aµ u uh`rds h`rds dreiabrids y jbs qu` vba gbkib `j sur jbs jjbhbr`hds jjbhbr `hds a aµ u uh`rds µ h`rds ‚hàdr`s e` k`rd‘. (44) A`k`sitbr`hds bjmµ u ua a ssµµĲhodjd pbrb p pde`r de`r eif`ràki eif`ràkibr br `sds eds kd kdaluatds aluatds e` aµuh`rds. uh`rds. @j sist`hb `f`ktivbhàt` usbed kdhpràe` `j prdk`sd pdr hèid e`j kubj s` dotiàà jds aµ u uh`rds. h`rds. Jds aµ u uh`rds h`rds dreiabrids sda jds u uaikds µaikds qu` s` dotiàà kubaed s` suhba eds aµu uh`rds h`rds dreiabrids. @j sµĲhodjd pbrb jb suhb `s `j simad ‚hµbbs‘ s‘ (+). Sdr jd tbatd jds aµu uh`rds h`rds dreiabrids s` `skrioà +2, +?, +; y bs bsµµĲ suk`siv suk`sivbhàt`. bhàt`. @stds sda jjbhbeds aµu uh`rds h`rds pdsitiv pdsitivds, ds, pbrb ebr jb ihpr`siµda da e` qu` `jjds ‚pdsitivbhàt`‘ `xistà. [da jb ‚kdsb r`bj.‘ Jds aµ u uh`rds h`rds hàdr`s e` k`rd sda dotàieds pdr r`stb kdhd, pdr `l`hpjd, tdhbaed ; e` ? , qu` e`lb ua aµu uh`rd h`rd hàdr e` k`rd. Jb r`stb s` iaeikb pdr hèid e`j simad ‚hàds‘ ( - ), pdr jd kubj jds aµu uh`rds h`rds hàdr`s e` k`rd s` `skrioà< -2, -?, -; y bs bsµµĲ suk`siv suk`sivbhàt`. bhàt`. @stds aµu uh`rds h`rds hàdr`s e` k`rd sda jjbhbeds, hµbbss bprdpibebhàt`, aµ u uh`rds h`rds a`mbtivds, viaiàed e` jb pbjborb ‚a`mbtivd ‚a`mbt ivd‘‘ y µ`stb `stb e` dtrb jbtiab qu` simaiflkb ‚a`mbr‘. B p`sbr e` qu` jds hbt`hµbbtikds tikds fu`rda fdrzbeds, flabjhàt`, b usbr aµ u uh`rds h`rds a`mbtivds, bpbrà-

?3

KBS µ IUPJD IUP JD ?. ABEB ABEB ... Y H@AD[ H@AD[ RP@ ABEB ABEB

Fimurb ?.?

?2 t`hàt` tàµĲĲba ba qu` ia iaeikbr eikbr uab ``sp`ki sp`ki`` e` a`mbkiµdda a e` q qu` u` µ``stds std s ‚r ‚r`bjh `bjhàt`‘ àt`‘ `x `xis istµ tµĲĲba. b a. Yb qu` tà` tà`hds hds hbrkb hbrkbeb eb au` au`strb strb jµĲa`b v`r v`rtikbj tikbj (y adtbh adtbhds ds qu` `j 3 ad `s pdsitivd ai a`mbtivd), pde`hds gbk`r suhbs y r`stbs kdor` `jjb. Kdhd jds aµu uh`rds h`rds pdsitivds buhàtba gbkib brriob y jb suhb buhàtb jds aµu uh`rds, h`rds, eimbhds qu` jb suhb simaiflkb hdv`r jb `skbjb gbkib brriob. Kdhd jb r`stb `s `j r`v`rsd e` jb suhb, eò` simaiflkbr hdv`r jb `skbjb gbkib bobld. [updambhds, àtdak`s, qu` qu`r`hds suhbr +? y +0. @std puè` `skrioirs` kdhd (+?) + (+0), siàed usbeds jds pbr pbrµµ`àt`sis at`sis pbrb iaeik iaeikbr br qu` `j simad + eàtrd e` `jjds p`rtà`k` bj aµu uh`rd h`rd y ad `s ua simad e` suhb. [ia `hobrmd, `stbhds tba bkdstuhorbeds b jds aµu uh`rds h`rds pdsitivds sµddjd jd kdhd aµ u uh`rds h`rds qu` `s usubj ad pda`r `j simad ‚hµbbs‘, s‘, b hàds qu` `xistb uab rbzµdda a `sp`kibj pbrb jjbhbr jb btàkiµda da r`sp`ktd b su pdsitiviebe 2. BsµĲ, jb suhb sihpj`hàt` s` `skrio` ?+0. @a +? au`strb `skbjb e` aµ u uh`rds, h`rds, `stdkdatbr simaiflkb qu` brriob eò`hds `hp`zbr à `j puatd y (kdhd `stbhds suhbaed) gbkib kiakd iat`rvbjds. @std ads jj`vbrµb bj puatd +=, e` edae` ?+08=. (Aµddt`s` t`s` qu` si s` guoi`rb `hp`zbed à +0 y kdatbed ? iat`rv iat`rvbjds bjds gb gbkib kib brriob, s` guoi`rb jj`mbed tbhoiµ`à a b +=. Au`v Au`vbhàt` bhàt` gby qu` gbk`r giakbpiµ` à qu` ad ihpdrtb `j dreà à `j kubj suh` uad jds aµu uh`rds< h`rds< ?+080+?). Bgdrb supdambhds qu` jd qu` s` qui`r` `s r`stbr +? e` +0, ua prdoj`hb qu` puè` `skrioirs` (+0) - (+?) d, hµbbss sàkijjbhàt`, 0-?. @a `stb dkbsiµda da `hp`zbhds à `j puatd +0 y kdatbhds eds iat`rvbjds gbkib bobld (pdr qu` `stbhds r`stbaed). @std ads jj`vb bj puatd +;, yb qu` 0-?8;. @j puatd ihp ihpdrtbat` drtbat` bgdrb `s e`skuorir ssµµĲ jds aµu uh`rds h`rds a`mbtivds puèà s`r hba`lbeds pdr hèid e` jbs hishbs tµ``kaikbs kaikbs usbebs pbrb jds aµu uh`rds h`rds pdsitivds. jd puèà àtdak`s, gbkiàed kbsd r`sujtb dhisd e` s`rµĲba Ĳba tba [i u utij`s µ tij`s kdhd s`r, jds aµ uh`rds uh`rds pdsi pdsitivds. tivds. @std r`sujtbr rµĲbsie`‚simaiflkbed‘ iat`rµ`s `s hµbs bs qu` t`µddrikd. rikd. Jds aµ u uh`rds h`rds a`mbtivds ad sdjbhàt` sda àdrh`hàt` µu utij`s tij`s à jb kiàkib y jb iamài`r iamài`rµµĲb, siad qu` tbhoiµ `à a tiàà bpjikbkida`s prµbbktikbs ktikbs à jds a`mdkids eibrids. Sdr `l`hpjd, à kdatboijiebe jds bktivds sda hba`lbeds kdhd aµ u uh`rds h`rds pdsitivds y jds pbsivds kdhd aµ u uh`rds h`rds a`mbtivds. 2

Au`strds simads hµbs bs y hàds ebtba e`j simjd WVI. @j simad hµbs bs surmiµd prdoboj`hàt` e`j gµboitd boitd e` `skrioir tbj kdhd ‚eds y tr`s‘ usbaed `j simad & à bfµba ba e` rbpie`z. @j simad (&) bpbr`k` à jb fdrhb `skritb kdhd , y ? ; rµbbpiebhàt` piebhàt` s` kdavi`rt` à ? ;. @j drimà e`j simad hàds `s hbt`rib e` kdatrdv kdatrdv`rsib. `rsib. Gby auh`rdsbs t`drµ t`drµĲbs, p`rd p `rd aiamuab pbr`k` s`r pbrtikujbrhàt` ouàb

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??

KBS µ IUPJD IUP JD ?. ABEB ABEB ... Y H@AD[ H@AD[ RP@ ABEB ABEB

Hba`lbaed jds a`mbtivds Sbrb `hp`zbr, supdambhds qu` s` qui`r` r`stbr 0 e` ?> `s e`kir, r`sdjv`r `j prdoj`hb ? - 0. Psbaed `j hishd sist`hb bat`ridr, `hp`zbhds à `j puatd +? y kdatbhds 0 iat`rv iat`rvbjds bjds gbkib bobld. @jjd ads jj`v jj`vbb b trbvµ``ss e`j puatd -;> kdas`kuàt`hàt`, ? - 0 8 ;. @std pbr`k` obstbat` kdrr`ktd y ads hu`strb uab kdsb ihpdrtbat`< 0 - ? ad `s  imubj b ? - 0. @a jb r`stb (kdatrbrib b jb suhb) sµĲ ihpd ihpdrtb rtb à quµ` dreà s` tdhba jds aµu uh`rds. h`rds. S`rd, ´Ruµ` pbsb si `hp`zbhds à `j `xtr`hd a`mbtivd e` jb `skbjb4 [updambhds qu` qu`r`hds suhbr -? y +0. (E` bqu bquµµĲ gbstb `j flabj e`j kbp kbpµµĲtujd, vbhds b kdas`rvbr kda s`rvbr jds p pbrµ brµ`àt`sis, at`sis, p pbrb brb ad kdafuaei kdafuaeirr jds aµu uh`rds h`rds a`mbtivds kda r`stb y jds aµ u uh`rds h`rds pdsitivds kda suhb.) @j prdoj`hb bkbobed e` pjbat`br s` `skrio`, pdr jd tbatd, (-?) + (+0). Sbrb gbk`r `std, `hpj`baed `j sist`hb usubj, s` `hpi`zb à `j puatd -? y s` kuàtba gbkib brriob 0 iat`rvbjds. U`rhiabhds à `j puatd +;> e` edae`,(-?) + (+0) 8 +;. ´[` àtiàe` `std4 Oià, si uab p`rsdab eò` ? eµddjbr`s jbr`s y mbab 0, `stµb kdjdkbeb kda ; edjbr`s à aµ u uh`rds h`rds a`mrds, `s e`kir, j` quèba bµu ua a ; eµddjbr`s jbr`s e`spuµ``ss e` gb gbo`r o`r pbmbed su e eùeb. ùeb. Ubho bhoiµ iµ`à a pde`hds hdv hdv`rads `rads gbkib bobld à `j `xtr `xtr`hd `hd a`mbt a`mbtivd ivd e` jb `skbjb. Udh` `j j`ktdr +? e` -?, pdr `l`hpjd> `s e`kir r`su`jvb `j prdoj`hb e` (-?) - (+0). @hpi`k` à `j puatd -? y kuàt` 0 iat`rvbjds gbkib bobld (pu`std qu` `stbhds r`stbaed) jd kubj ads jj`vb bj puatd -=> e` edae`, (-?) - (+0) 8 -=. D à jds hishds tµ``rhiads rhiads prµbktikds bktikds bat`ridr`s, si `hpi`zb uab p`rsdab kda uab eùeb e` ? eµdjbr`s djbr`s y s` v` fdrzbeb b pèir pr`stbeds 0 beikidabj`s, t`rhiab eòiàed = eµddjbr`s. jbr`s. Y, bµ u ua a bsµ bsµĲ, jds aµu uh`rds h`rds a`mbtivds s` `stµbba a kdhpdrtbaed tba adrhbj, e`kàt` y sàsioj`hàt` kdhd jds aµu uh`rds h`rds pdsitivds. @xist`` tdebv @xist tdebvµµĲb uab hbaido hbaidorb rb qu` ad g`hds àsb àsbybe ybed. d. Gbstb bgdrb, sdjbhàt` g`hds `stbed suhbaed y r`stbaed aµ u uh`rds h`rds pdsitivds. ´Ruµ` tbj si àsbybhds suhbr d r`stbr ua aµu uh`rd h`rd a`mbtivd4 @a tµ``rhia rh iads ds pr prµµbbktikds, ktikds, `std sim simaiflkbrµ aiflkbrµĲĲbb `stb `stbrr suhb suhbaed aed d rr`stbaed `stbaed eùebs eùebs.. Bgdrb oià, si fuµ ``rbhds rbhds b kbak kbak`jbr `jbr uab eùeb e` uab p p`rsda `rsdabb pdr 0 eµddjbr`s jbr`s `quivbj `qu ivbjerµ erµĲĲbb b q qu` u` j` eiµ`rbhd `rb hdss 0 eµddjbr`s. jbr`s. [i, à kbhoid, j` trbspbsµbbrbhds rbhds b `sb p`rsdab uab eùeb e` 0 eµddjbr`s jbr`s ( y jb fdrzµbrbhds brbhds à bjmuab fdrhb b qu` jb bk`ptbs`), bk`ptbs `), `s `sbb p p`rsdab `rsdab s` àk àkdatrbr datrbrµµĲb 0 eµddjbr`s jbr`s hµbs bs pdor pdor`> `> `quivbjer `quivbjerµµĲb b qu` j` sbkµbbrbhds rbhds 0 eµddjbr`s jbr`s e` su odjsijjd. Jd kubj simaiflkb, à tµ`rhiads `rhiads brithµ``tikds, tikds, qu` r`stbr -0 `s jd hishd qu` suhbr +0, y qu` suhbr -0 `s jd hishd qu` r`stbr +0.

?; @std ads eb jb dpdrtuaiebe e` suprihir tdtbjhàt` jb r`stb, yb qu` kubjqui`r prdoj`hb, tbj kdhd 0 - ?, `s r`bjhàt` (+0) - (+?) y puè` s`r `skritd, e` bku`red kda jb r`mjb bkbobeb e` hbaif`stbr, kdhd (+0) + (-?). Bhobs v`rsida`s eba jb hishb r`spu`stb. @a jds eds kbsds `hp`zbhds kda `j puatd +0 y hdv`hds ? iat`rvbjds gbkib bobld. @a `j kbsd e` (+0) - (+?), `s `std dovid> dov id> r`st r`stbr br simaifl simaiflkb kb ‚hdv ‚hdv`r `r gbkib bob bobld‘. ld‘. @a `j kbsd e` (+0) + (-?), `s hàds dovid. @j simad ‚hµbbs‘ s‘ àtr` jds aµu uh`rds h`rds `s suhb y simaiflkb ‚ir brriob‘. S`rd s` `stµb suhbaed ua aµu uh`rd h`rd a`mbtivd, jd kubj iavi`rt` jbs kdsbs, y bs bsµµĲ, jds eds g`kgds luatds simaiflkba ‚ir gbkib brriob iavirtiàed‘ y `sd `quivbj` b ‚ir gbkib bobld‘. @a kubjqui`r kbsd, jb r`spu`stb `s +;. S`rd, ´Sdr quµ` e` e`s`br s`brµµĲb kubjqui kubjqui`rb `rb kbhoibr uab r`stb b uab suhb kubaed sµddjd jd simaiflkb e`svibrs` y tà`r qu` e`kir ‚yàed gbkib brriob iavirtiàed‘ à v`z e` sihpj` y eir`ktbhàt` ‚ir gbkib bobld‘ ¡Bg! , `s qu` jb suhb tià` à mà`rbj `stb vàtblb sdor` jb r`stb< à jb suhb ad ihpdrtb `j dreà e` jds aµu uh`rds, h`rds, p p`rd `rd µ``st` st` si ihpdrtb à jb r`stb. Jb `xpr`siµdda a (+0) - (+?) ad `s imubj qu` (+?) - (+0) kdhd yb s` gb g`kgd adtbr. Jb r`spu`stb b jb prih`rb `s +;, y b jb µu ujtihb, jtihb, -;. Sdr dtrd jbed, jb `xpr`siµdda a `quivbjàt` (+0) + (-?) `s imubj b (-?) + (+0). @a bhods kbsds jb r`spu`stb `s +;. @vieàt`hàt`, àtdak`s, bj kbhoibr b suhb kubatbs v`k`s s`b pdsioj` `jihiab jb `vàtubjiebe e` ua tipd e` `rrdr< `j kdjdkbr bkkieàtbjhàt` jds aµ u uh`rds h`rds à dreà iakdrr`ktd. Jb situb situbkiµ kiµ dda a `s s`h`lbat` bj r`stbr ua aµu uh`rd h`rd a`mbtivd. @j prdoj`hb (+0) - (-?) puè` `skrioirs`, si jd e`s`bhds, (+0) + (+?). @a `j prih`r kbsd, `hp`zbhds à `j puatd +0 e` jb `skbjb y ‚vbhds gbkib bobld iavirtiàed‘, ? iat`rvbjds. @j ‚ir gbkib bobld iavirtiàed‘ `s ‚ir gbkib brriob‘ y, `j ir ? iat`rvbjds gbkib brriob ads jj`vb b +=. Y, abturbjhàt`, `sd `s `xbktbhàt` jd qu` sukè` si r`sdjv`hds (+0) + (+?). @st` bsuatd e` `jihiabr jb r`stb y trboblbr b obs` e` suhbs, mà`rbjhàt` `s jd prih`rd qu` àkuàtrba jds `stueibat`s bj iaikibr sus `stueids à µbbjmòrb. jmòrb. Sdr `stb rbzµdda, a, jb suhb e` aµu uh`rds h`rds a`mbtivds (à v`z e` r`stbr jds pdsitivds) `s jjbhbeb fr`kuàt`hàt` suhb bjmòrbikb. @a r`bjiebe, ad `xist` rbzµdda a pbrb eikgd adhor`. @s uab p`rf`kt p`rf`ktbb dp`rb dp`rbkiµ kiµdda a brithµ``tikb tikb à jb qu` `j µbbjmòrb jmòrb ad tià` abeb qu` v`r.

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KBS µ IUPJD IUP JD ?. ABEB ABEB ... Y H@AD[ H@AD[ RP@ ABEB ABEB

KbpµĲtujd ; Sbsbaed pdr bjtd jb suhb Hµ bs y hµ bs y hµ bs y hµ bs [updambhds qu` `j j`ktdr trbzµd ua kuberbed qu` tàib 2 kàt kàtµµĲh`trd pdr jbed. Sder SderµµĲb jjbhbrjd ua ‚kàt ‚kàtµµĲh`trd kuberbed‘ y usbrjd kdhd uaiebe e` sup`rflki` sup`rflk i` (44). Kdasie`r`hds b kdatiaubkiµdda a ua s`muaed kuberbed, qu` hie` ? kàt kàtµµĲh`trds pdr jbed. [i s` eivie eivie`` `st` kuberbe kuberbedd e` ? kà kàttµĲh`t Ĳh`trds rds à hitbe`s b jd jbrmd y b jd bakgd, t`rhiab uad tàiàed kubtrd kuberbeds hµbbss p`qu`˚ a ads, ds, kbeb uad e` jds kubj`s hie` 2 kàt kàtµµĲh`trd kuberbed (44). Dtrd kuberbed qu` s`b e` ; kàt kàtµµĲh`trds e` jbrmd y ; kàt kàtµµĲh`trds e` bakgd, puè` kdrtbrs` à s`kkida`s e` 2 k` kàt atµµĲh`trd kuberbe kuberbed, d, eivieiµ`àedjd aedjd à tr`s pbrt`s imubj`s tbatd b jd bak bakgd gd kdhd b jd jbrmd. @j r`suj r`sujtbed tbed `s 9 kà kàttµĲh`tr Ĳh`trds ds kuber kuberbeds beds eà eàtrd trd e` au`strd kuberbe kuberbedd e` ; kkàt àtµµĲh`trds (44). Fiabjhàt`, ua trbtbhiàtd s`h`lbat` b ua kuberijdamd e` 6 kàt kàtµµĲh`trds kuberbeds. Uded `std s` e`hu`strb à jds eibmrbhbs simuiàt`s. @a kbeb uad e` `stds kbsds, jbs pdrkida pdrkida`s `s e` ua kà kàttµĲh`tr Ĳh`trdd kuber kuberbed bed `stµba ba brr`mjbebs pdr fljbs (gij`rbs gdrizdatbj`s) y kdjuhabs (gij`rbs v`rtikbj`s). Kbeb fljb kdatià` tbatds kàt kàtµµĲh`trds kuberbeds kdhd `j kuberbed d `j kuberijdamd kuber ijdamd tià` e` jbrmd (à kà kàttµĲh`tr Ĳh`trds) ds) y gby tba tbatbs tbs fljbs à kdalu kdaluatd atd kdhd `j kuberbed d `j kuberijdamd tià` e` bakgd (à kàt kàtµµĲh`trds). @a `j kuber kuberbed bed e` ? kà kàttµĲh`tr Ĳh`trds ds `xist `xistà à eds kuber kuberbeds beds à kbeb uab e` jbs eds fljbs, e` edae` edae`,, ? + ? 8 5. @a `j kuberbed e` ; kàt kàtµĲh`t Ĳh`trds, rds, gby ? kuberb kub erbeds eds à kbe kbebb uab e` jbs tr`s fljbs fljbs,, e` edae`, edae`, ; + ; + ; 8 9. Fiabjh Fiabjhà àt`, t`, à `j kuberijdamd e` 6 pdr 9 kàt kàtµµĲh`trds ( 44), `xistà 6 s`kkida`s kuberbebs ?0

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KBS µ IUPJD ;. SB[BAED B[BAED SD_ BJUD BJUD JB [PHB

(b) Kuberbed e` 2 K` Kàtµ atµĲh`Ĳh `trd

(o) (o) Kube Kuberb rbed ed e` ? K` Kàtµ atµĲh`t Ĳh `trd rdss

(k) Kuber K uberbed bed e` e ` ; Kàt K` atµµĲh`trds Ĳh`t rds

Fimurb ;.2 à kbeb uab e` jbs 9 fljbs d (si s` dos`rvb jbt`rbjhàt`) 9 s`kkida`s kuberbebs à kbeb uab e` jbs 6 kdjuhabs. @a `j prih`r kbsd `j aµu uh`rd h` rd e` kàtµĲĲh`trds h`t rds kuberbeds `s< 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 y à `j s`muaed kbsd `s< 9 + 9 + 9 + 9 + 9 + 9. Ad ihpdrtb kuµbj s`b `j kbsd qu` `jilb pdrqu` bhods suhba 05. @a mà`rbj, jb dotàkiµdda a e` sup`rflki`s e` pjbads d flmurbs kdhpràe` `stb `sp`ki` e` suhb r`p`tieb. @a jds kuberbeds y jds kuberijdamds `j bsuatd `s obstbat` sàkijjd. S`rd s` gbk` ua pdkd hµbbss kdhpjikbed à jds triµbbamujds amujd s y jds j ds kµĲrkujds Ĳrkujds,, y h hµµbbss kdhpjikbed bµ u ua a à jbs sup`rflki`s e` fdrhb irr`mujbr. [ia `hobrmd, `stb kjbs` e` kµbbjkujds jkujds gb tàied qu` gbk`rs` kdatiaubhàt`, bua à jbs prihitivbs kivijizbkida`s bmr bmrµµĲkdjbs (hbs bµu ua a à jb au`strb bjtbhàt` t`kadjµddmikb). mikb). Jbs mrbalbs d gbkiàebs tà tàµµĲba qu` s`r hèiebs, kbjkujbaed jbs sup`rflki`s, bua kubaed sµddjd jd fu`rb b `f`ktd e` fllbr jds ihpu`stds. (Bjmuabs v`k`s piàsd qu` fu`rda jds ihpu`stds, hµbbss qu` kubjqui`r dtrb kdsb, jd qu` bjàtµd `j e`sbrrdjjd e` jb brit br ithµ hµ``tik t ikb. b.)) [ia `hobrmd, bsµĲ kdhd jb a`k`siebe e` kdatbr r`p`tiebhàt` fdrzµ d `j e`sbrrdjjd e` ua sist`hb e` suhbs, jb a`k`siebe e` suhbs r`p`tiebs fdrzµd `j e`sbrrdjjd e` ua au`vd tipd e` hbaipujbkiµdda a e` jds aµ u uh`rds. h`rds. Sbrb `hp`zbr, bedpt`hds uab au`vb badtbkiµdda a y `skriobhds 6 9 (‚sìs v`k`s au`v`‘, siàed `j ‚simad e` hujtipjikbkiµdda‘ a‘ kubaed qu`rbhds simaiflkbr au`v` sìs`s suhbeds kdaluatbhàt`, d sìs au`v`s à imubj fdrhb).

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Fimurb ;.?

?7

KBS µ IUPJD ;. SB[BAED B[BAED SD_ BJUD BJUD JB [PHB

Kdhd g` e`hdstrbed à `st` kbsd pbrtikujbr (y `j j`ktdr pderµb v`riflkbr à dtrds `l`hpjds) ad ihpdrtb kuµbbjj s`b `j kbsd< 6 9 8 9 6. Sde`hds usbr `stb au`vb badtbkiµdda a hbaif`stbaed, kdhd r`mjb mà`rbj, `j g`kgd e` qu` jb sup`rflki` e` kubjqui`r kuberbed d kuberijdamd `s imubj b su jbrmd hujtipjikbed pdr su bakgd. @j simuiàt` pbsd `s àkdatrbr uab fdrhb sàkijjb pbrb `j hba`ld e` jb hujtipjikbkiµdda. a. Sdr supu`std, si`hpr` tàer`hds jb bjt`rabtivb e` suhbr r`p`tiebhàt`, kdhd à `j kbsd e` 6 9, p`rd `std pd pder erµµĲb r`sujtb r`sujtbrr kdhpjik kdhpjikbed. bed. [i `stuviµ``rbhds rbhds iat`r`sbeds à dotà`r jb sup`rflki` e` ua jdt` e` fdrhb kuberijdamb kuberij damb qu qu`` tuvi`rb 22?9 ?9 h`trd h`trdss e` jbrm jbrmdd y 05 h`t h`trds rds e` ba bakgd, kgd, tàerµĲĲbhds bhds qu` hujtipjikbr 2?9 pdr 05 pbrb kdadk`r jb r`spu`stb (à h`trds kuberbeds, abturbjhàt`, abturbj hàt`, ad à kkàt àtµµĲh`trds kuberbe kuberbeds). ds). @std simaifl simaiflkbr kbrµµĲb suhbr 2?9 kiakuàtb y kubtrd v`k`s d 05 kiàtd vìatia vìatiau`v` u`v` v`k`s. @a kubjqui`r kbsd, s`r s`rµµĲb uab tbr`b bmdtbedrb bmdtbedrb.. D, si à uab trbasbkkiµdda a kdh`rkibj, fu`rba kdhprbebs ?05 edkàbs e` brtµµĲkujds b ==?? kàtbvds jb edkà brt edkàb, b, `j pbmd tdtbj s`r s`rµµĲb (à kàtbvds) ?0 ?055 =?. Gbkiàed `stb dp`rbkiµdda a pdr suhbs, tàer tàerµµĲbhds qu` suhbr ?05 s`tàtb y eds v`k`s, d =? edskiàtds kiakuàtb y kubtrd v`k`s. @stb kjbs` e` prdoj`hbs s` pr`sàtba kdastbat`hàt` kdastbat`hàt` à jb vieb eibrib y, pdr jd tbatd, vbjer vbjerµµĲb jb pàb ie`br ua hded bor`vibed qu` `jihiabrb jb r`p`tikiµdda a e` suhbs.

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Udeds pdr tdeds Au`vbhàt` `j s`kr`td rbeikb à `j h`hdrishd. @s a`k`sbrid bpràe`r e` h`hdrib jb ‚tbojb e` hujtipjikbr‘, qu` kdastb e` tdebs jbs kdhoiabkida`s pdsioj`s e` eds eµĲmitds gbstb 9 9. @j ai˚ a add tià` qu` r`kitbr à jb `sku`jb

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0 ? 8 23, y 7 = 8 06, gbstb qu` pbr`kà `stbr sbjiàed e` sus dµĲeds. [ia `hobrmd, uab v`z qu` s` gba mrbobed eikgbs tbojbs à su hàt` àkuàtrb qu` `s tded jd qu` a`k`sitbrµb pbrb hujtipjikbr eds aµu uh`rds h`rds kubj`squi`rb, sia ihpdrtbr jd mrbae`s qu` s`ba. Pab kdsb qu` `s pbrtikujbrhàt` ihpdrtbat` r`kdrebr `s qu` jb hujtipjikbkiµdda a pdr k`r k`rdd eb uab r`s r`spu` pu`stb stb qu` si` si`hpr hpr`` `s k` k`rd. rd. Bs BsµµĲ, 0 3 8 3> 200 3 8 3> 257?=;690 3 8 3> y, abturbjhàt` 3 3 8 3. Suè` v`rs` qu` tdeds `stds `l`hpjds r`pr`sàtba jb suhb e` 0 k`rds, 200 k`rds, y bs bsµµĲ suk`si suk`sivbhàt`. vbhàt`. Abturbjh Abturbjhàt`, àt`, ad ih ihpdrt pdrtbb kubatds k`rds su suh` h` uad à kdaluatd> `j r`sujtbed si`hpr` `s k`rd. @std simaiflkb qu` gboiàed bpràeied qu` ; = 8 ?2, `s `vieàt` qu`

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?9

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;3 = 8 ?23 y ; =3 8 ?23. @j k`rd à aiamµua u a kbsd `s bf`ktbed pdr jb hujtipjikbkiµ dda. a. [i hujtipjikb uad ;3W =3, =3, `j r`sujtbed `s ?233, bhods k`rds p`rhba`kà iabf`ktbeds pdr `j prdkèihiàtd. @a s`muieb, si s` qui`r` hujtipjikbr ua aµ u uh`rd h`rd qu` kdatàmb eiv`rsds eµĲmitds qu` ad s`ba k`rd k`rds, s, sdjbh` sdjbhàt` at` `s a`k` a`k`sbrid sbrid frbkk frbkkidabrj idabrjdd à jb fdrhb usubj. Sdr `l`hpjd, ´Kuµbj bj `s `j prdeuktd e` ;960 =4 (@j r`sujtbed e` uab hujtipjikbkiµ dda a s` jjbhb ‚prdeuktd‘, bs bsµµĲ, kdhd `j r`sujtbed e` uab beikiµdda a s` jjbhb ‚suhb‘). Jb pdsikiµdda a e` jb badtbkiµda da ads p`rhit` frbkkidabr ;960 à ;333 y 933 y 63 y 0. @std ads eb aµ u uh`rds h`rds qu` kdastba e` ua sdjd eµĲmitd y vbrids vbr ids k`rds> bsµĲ pu`s td tded ed jd qu` a`k`sitbhds `s au`strb tbojb e` hujtipjikbr. Kbeb aµ uh`rd uh`rd s` hujtipjikb pdr => jds prdeuktds pbrkibj`s s` suhba y s` dotià` `j prdeuktd kdhpj`td, kdhd simu`<

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;333

y 933

y 63

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y

0 = ;0 5?3 6;33 ?2333 ?==00

Ad ihpdrtb `j dreà à `j kubj jj`v`hds b kbod jbs hujtipjikbkida`s< e` izqui`reb izqui `reb b e`r` e`r`kgb kgb,, e` e`r` e`r`kgb kgb b izqui izqui`reb, `reb, d sbjtba sbjtbaed ed bjrè bjrèèdr. èdr. Bqu BquµµĲ s` gb g`kgd e` e`r`kgb b izqui`reb, pdr s`r µ``stb stb jb fdrhb qu` mà`rbjhàt` s` às`˚ a abb à jds kursds `j`hàtbj`s. @s hµbbs, s, s` ads às`˚ a abb b e`lbr `j aµu uh`rd h`rd sia frbkkidabr y b imadrbr jds k`rds. Sdr jd tbatd, `j prdoj`hb, bj s`r r`su`jtd, bpbr`k` kdhd simu`< ;960 x = ;0 5? 6; ?2 ?==00 [` ads às`˚ a ab, b, be`hµbbs, s, b jj`vbr hàtbjhàt` uads aµ u uh`rds, h`rds, pbrb bmr`mbrjds b jds prdeuktds pbrkibj`s à kbeb dkbsiµdda. a. Uded `std `s ua bsuatd e` h`kµbaikbs baikbs qu` ad bf`ktb bj priakipid e` jb hujtipjikbkiµdda. a. Jb v` vàtblb atblb e` `stbs

;3

KBS µ IUPJD ;. SB[BAED B[BAED SD_ BJUD BJUD JB [PHB

h`kµbaikbs baikbs rbeikb à qu` `j ldvà bpràe` b hujtipjikbr rµbbpiebhàt` piebhàt` y kda ua hµĲai Ĳaihd hd e` r`﬏` r`﬏`xiµ xiµdda. a. Jb e`svàtblb `s qu`, bjmuabs v`k`s, auakb àkuàtrb jb rbzµdda a pdr jb kubj gbk` jd qu` j` às`˚a abrda> brda> pdr `l`hpjd, pdr quµ` bpjikb bpj ikb jd jdss prdeuktds pbrkibj`s. Kubaed à ua prdoj`hb e` hujtipjikbkiµdda, a, bhods aµu uh`rds h`rds sda hbydr`s e` 23, `xist` uab jim`rb kdhpjikbkiµdda a beikidabj. Jds eds aµu uh`rds h`rds puèà s`r frbkkidabeds, y àtdak`s, kbeb uab e` jbs pbrt`s e`j aµu uh`rd h`rd sup`ridr s` hujtipjikb pdr kbeb uab e` jbs pbrt`s e`j aµu uh`rd h`rd iaf`ridr, suhµbbaeds` aeds` pdst`ridrhàt` jds p prdeu rdeuktds ktds pbrkibj pbrkibj`s. `s. B BssµĲ, ;0 ?7 s` frbkkidab kdhd simu` (44 y `s jb pr`sàkib e` jbs jµĲĲa`bs a`bs kruzbebs iaeikbaed jb fdrhb e` hujtipjikbkiµ dda a jb qu` puè` gbo`r drimiabed jb kdhd simad e` jb hujtipjikbkiµdda)< a)<

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B p`sbr e` tded, jds aµ u uh`rds h`rds hbydr`s sda trbtbeds à imubj fdrhb. @j sist`hb e`bed, hujtipjikbkiµ da da eùsbhds tded jd`je` brriob pdrbbaikd tdedàs`˚ jd e` bobld gbk` kdhpjik kdhpjikbed, y bua kubaed hµ``tded tded h`kµ aikd a abed bed à jbs``sku`jb, `s pdsioj` p`re`rs`, kdhd pderµb kubjqui`rb dos`rvbr si trbtb e` s`muir jb simuiàt` hujtipjikbkiµdda< a< ;960 x ?29= ?==00 ;0670 ;960 =9;3 7=22230

;2 [ia `hobrmd, gborµb qu` behitir qu` `std `s ua be`jbatd sdor` jb a`k`siebe e` suhbr ;960 + ;960 + ;960 + ;960 + ;960 ... ?29= v`k`s.

Hujtipjikbkiµ da da iav`rtieb BsµĲ kd Bsµ kdhd hd jb suhb tià` su kdatrbrid à jb r`stb, tbhoiµ`à a jjbb hujtipjikb hujtipjikbkiµ kiµda da jd tià` à ua tipd e` hbaipu hbaipujbkiµ jbkiµ dda a e` aµ u uh`rds h`rds jjbhbed ‚eivisiµdda‘. a‘. Hiàtrbs qu` jb hujtipjikbkiµdda a `s uab `sp`ki` e` suhb r`p`tieb, jb eivisiµdda a `s uab `sp`ki` e` r`stb r`p`tieb. Sdr `l`hpjd, supdambhds qu` qui`r` uad eivieir 20 àtr` ;, ua prdkèihiàtd qu` puè` s`r sihodjizbed kdhd 20 ;, siàed `j simad e` ‚eivisiµdda‘ a‘ (e` drimà iaki` iaki`rtd) rtd).. Pab fdrhb e` gbk`r gbk`rjd jd `s r`st r`stbr br ; uab v` v`zz y dtrb. Bs BsµµĲ, 20 ; 8 2?> 2? ; 8 9> 9 ; 8 6> 6 ; 8 ; y ;‛; 8 3. D s`b, kdaeàsbaed, 20 ; ; ; ; ; 8 3. Dosµ``rv`s` rv`s` qu` s` gba g`kgd kiakd r`stbs> p pdr dr jd tbatd, 20 ; 8 0. [ia `hobrmd, auakb s` gbk` à `stb fdrhb. @a jumbr e` `sd, s` gbk` usd e`j bvbazbed kdadkihiàtd e` jds r`sujtbeds e` jb hujtipjikbkiµdda. a . E`s E`spuµ puµ`s `s e` tded, si 20 `s eivieied àtr` ; pbrb ebr ua aµ u uh`rd, h`rd, `s` aµ u uh`rd h`rd hujtipjikbed pdr ; eò` ebr au`vbhàt` 20. [` gb ied gbkib be`jbat`, y ju`md gbkib btrµbbs, s, y eò` t`rhiabrs` à jumbr drimiabj. (@stb `s uab situbkiµdda a s`h`lbat` b jb e` jb suhb y jb r`stb< tbhoiµ`à a iavi`rt` `j prd prdkèihiàtd. kèihiàtd. [i = 5 8 ;, àtdak`s ; + 5 8 =). Kdas`kuàt`hàt`, kubaed kdafrdatb uad ua prdoj`hb kdhd 20 ;, jd qu` j` fu` às`˚a abed bed eò` pr`muatbrs` `s< ´Ruµ` aµu uh`rd h`rd hujtipjikbed pdr ; ebrµb 204 Sdr jb tbojb e` hujtipjikbr (qu` s` bpràeiµd e` h`hdrib b fu`rzb e` hbkgbkbr) sbo` uad qu` `j aµ u uh`rd h`rd `s 0 y µ``sb sb `s su r`spu`stb. @a r`bjiebe, s` sbo` uad tba oià sus tbojbs e` hujtipjikbr qu` butdhµbbtikbhàt` tikbhàt` eik` ‚kiakd‘ à r`spu`stb b `st` prdoj`hb, sia ebr bj hishd hbydr kdasie`rbkiµdda. a. Kdhd 0 ; `s imubj b ; 0 y, à bhods kbsds, `j r`sujtbed `s 20, s` eèuk` qu` 20 0 8 ; y 20 ; 8 0. @j hishd aµu uh`rd h`rd à jb tbojb e` hujtipjikbr sirv` pbrb ebr jb r`spu`stb b bhods prdoj`hbs e` eivisiµda da 2

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Dosµ`rv`s` `rv`s` qu` à jb eivisi eivisiµµdda, a, kdhd à jb r`stb, `s ihpdrtbat` `j dreà e` jds aµu uh`rds. h`rds. Bua kubaed 0 ; `s imubj b ; 0, 20 0 ad `s, pdr aiamµ u ua a hdtivd, imubj b 0 20. @j aµu uh`rd h`rd b jb izqui`reb e`j simad e` jb eivisiµda da `s `j eivieàed (e` uab pbjborb jbtiab qu` simaiflkb ‚`sd qu` `s eivieied‘), hiàtrbs qu` `j aµuh`rd uh`rd b jb e`r`kgb `s `j ‚ eivisdr ‘ (‚`sd qu` `stµb eivieiàed‘). @j r`sujtbed e` uab eivisiµda da ‛ `s e`kir, jb r`spu`stb b jb pr`muatb ‚´Kuµ bbatbs atbs v`k`s kbo` ua aµuh`rd uh`rd eàtrd e` dtrd4‘ ‛ `s `j kdkiàt`  (e` uab pbjborb jbtiab

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KBS µ IUPJD ;. SB[BAED B[BAED SD_ BJUD BJUD JB [PHB

[ia `hobrmd, `j g`kgd e` qu` jb eivisiµdda a trbobl` gbkib btrµbbss jb gbk` hµbs bs eifµ eifµĲkij qu` jbs dtrbs dp`rb dp`rbkida`s kida`s brithµ``tikbs> tikbs> µ``stb stb `s uab r`bjieb r`bjiebe e qu` `j ldvà `skdjbr e` mrbed `j`hàtbj kdadk` huy oià. [updambhds, pdr `l`hpjd, qu` fu`rb a`k`sbrid eivieir ==20 àtr` 0. @j kdatbr `j aµ u uh`rd h`rd e` v`k v`k`s `s qu` tàer tàerµµĲb uad qu` r`st r`stbr br 0 e` ==20 pbrb jj`mbr b 3 s`r s`rµµĲb tèid tèidsd sd y jb tbojb e` hujt hujtipjik ipjikbr br ad j` eik` qu` aµu uh`rd h`rd eò` s`r hujtipjikbed pdr 0 pbrb ebrj` ==20. S`rd bqu b quµµĲ `s ` s edae` jb hujtipjikb hujtipjikbkiµ kiµda da pdr 3 `s au`vbhàt` u utij. µ tij. [` sbo` qu` 2 0 8 0> pdr jd tbatd, 2333 0 8 0333. @stb `s jb kifrb hbs k`rkbab b ==20 b qu` puè` uad jj`mbr usbaed sdjbhàt` eµĲmitds y k`rds> p p`rd `rd bbµ u ua µ a j` quèba ?=20, kbatiebe qu` s` eivie` àtr` 0. Jb tbojb e` hujtipjikbr eik` 0 0 sda ?0 pdr jd qu` 033 0 8 ?033, y sµddjd jd quèba yb ?20. Sruµ`ò`s` o`s` `stb u ujtihb µjtihb kifrb. Kdhd 5 0 8 ?3, àtdak`s 53 0 8 ?33. Bs BsµµĲ, quèba u uaikbhàt` µaikbhàt` 20, y eivieir `std àtr` 0 `s fµbkij. bkij. [bo`hds qu` `j r`sujtbed `s ;. @a tdtbj, 0 gb sied hujtipjikbed, prih`rd pdr 2333, ju`md pdr 033, e`spuµ`s `s pdr 53, y flabjhàt` pdr ;. [µuh`s` uh`s` `std y s` àkdatrbrµb qu`, à tdtbj, `j 0 gb sied hujtipjikbed pdr 205; 0 8 ==20, àtdak`s ==20 0 8 205;, siàed µ`stb `stb au`strb r`spu`st r`spu`stbb d kdki kdkiàt`. àt`. @a `st` kbsd, tbhoiµ`à, a, jds lµddvà`s và`s sda às`˚ a abeds beds b gbk`r jb eivisiµdda a pdr dreà,, y `j prdoj` dreà prdoj`hb hb brriob kitbe kitbed, d, qu` bkbobh bkbobhds ds e` t`rh t`rhiabr, iabr, bpbr` bpbr`k`r k`rµµĲb, usbaed `j hµ`tded `tded `skdjbr, à jb fdrhb simuiàt`<

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0 ?= ?0 ?2 ?3 20 20 3

Jb eivisiµdda a pdr aµu uh`rds h`rds hbydr`s e` 23 `s hµbbss kdhpjikbeb, buaqu` `j priakipid `s `j hishd. Kdhd jb tbojb e` hujtipjikbr ad vb brriob e` 23, qu` simaiflkb ‚´Kuµbatbs batbs v`k`s4‘). @a `j `l`hpjd 20 y ; `j kdkiàt`.

ý 0 8 ;, 20 `s `j eivieàed, 0 `j eivisdr

;; tà`hds qu` kdal`turbr bjmuabs e` jbs r`spu`stbs y ‚prdobrjbs‘, pbrb v`r kdhd r`sujtba. @sb `s jb eiflkujtbe r`bj kda `s` `spbatbld e` jb ‚mrba eivisiµdda‘. a‘. (Ad dostbat`, à r`bjiebe, jds `stueibat`s eòà kdasie`rbrs` bfdrtuabeds. Bat`s e` jb bedpkiµdda a e` jds aµ u uh`rds h`rds brµbboimds, oimds, jb ‚mrba eivisiµdda‘ a‘ `rb hbt`rib e` hbt`hµbtikbs btikbs huy bvbazbebs qu` sdjbhàt` uads kubatds hbt`hµbbtikds tikds pde pd eµĲba edh edhiab iabr.) r.)

Jb adrhb e` jds simads Gbstb bgdrb, à `st` kbp kbpµµĲtujd, ad s` gba hàkidabed jds aµu uh`rds h`rds a`mbtivds. Kdhd `jjds àkb àkblba lba tba sutijhàt` à jb suhb y jb r`stb, s`r s`rµµĲb bmrbeboj` àkdatrbrj`s tbhoiµ`à a ua jumbr à jb hujtipjikb hujtipjikbkiµ kiµdda a y jb eivisiµdda. a. [updambhds, pdr `l`hpjd, qu` s` e`s`b hujtipjikbr (+;) y ( 5). ´Kµddhd hd fuakida fuak idabrµ brµĲb Ĳb `std `std44

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Kdasie`r`hds b tr`s p`rsdabs, kbeb uab e` jbs kubj`s tià` uab eùeb e` 5 eµddjbr`s. jbr`s. Sbrb sbo`r `j tdtbj e` jb eùeb eò`hds suhbr 5 + 5 + 5, y àkdatrbr qu` kdaluatbhàt`, au`strds tr`s bhimds tiàà uab eùeb e` 2? eµddjbr`s. jbr`s. Bgdrb oià, jb suhb e` ; kubtrds `s `xbktbhàt` jd qu` g`hds e`kieied jjbhbr ; 5. Kdhd `j 5, à `st` kbsd qu` `stbhds kdasie`rbaed, `s uab eùeb e` 5 eµddjbr`s jbr`s r`bjhàt` r`pr`sàtb ua 5, pdr jd qu` au`strd prdoj`hb s` rèuk` b (+;) ( 5), y pu`std qu` g`hds e`kieied qu` jb r`spu`stb `s uab eùeb e` 2? eµddjbr`s, jbr`s, àkdatrbhds qu` (+;) ( 5) 8 ( 2?). @j hishd r`sujtbed s` guoi`rb dotàied si `stuviµ ``rbhds rbhds kdasie`rbaed b 5 p`rsdabs qu` eòi`rba ; eµddjbr`s jbr`s kbeb uab. @a dtrbs pbjborbs (+5) ( ;) 8 ( 2?). Y kdhd ad ihpdrtb `j dreà à qu` s` hujtipjiquà jds aµ u uh`rds, h`rds,

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( 5) (+;) 8 ( 2?) y ( ;) (+5) 8 ( 2?). Sde`hds mà`rbjizbr. Kubaed s` hujtipjikba eds aµu uh`rds, h`rds, uad pdsitivd y dtrd a`mbtivd, a`m btivd, jb r`spu`stb `s si`hp si`hpr` r` a`mbt a`mbtivb. ivb. Auhµ``rikbhàt`, rikbhàt`, jb r`spu r`spu`stb `stb `s jb hishb, kdhd si bhods aµu uh`rds h`rds fu`rba pdsitivds. Bs BsµµĲ, `j prdeuktd e` (+5) y (+;) (+ ;) `s (+ (+2?). 2?). J Jbb pr`sàk pr`sàkib ib e` ua a`mbtivd ad bf`ktb `j vbjdr auhµ``rikd rikd e`j prdeuktd> sdjbhàt` su simad. S`rd, ´Ruµ` suk` sukè` e` si bhods   aµu uh`rds h`rds sda a`mbtivds4 E`smrbkibebhàt`, `s eif eifµµĲkij pàsbr à ua kbsd prµbktikd bktikd qu` puei`rb kdhpràe`r eds aµ u uh`rds h`rds a`mbtivds. [da fµbbkij`s kij`s e` ihbmiabr eùebs iaeivieubj`s e` ; y 5 eµddjbr`s, jbr`s, p`rd, ´Kµddhd hd pd pder erµµĲb uad àkdatrbr 5 d ; gdhor`s qu` jbs tuvi`rba4 @a v`z e` `sd, vbhds b btbkbr `j prdoj`hb p pdr dr dtrd kbhiad. Gbstb bqu bquµµĲ,

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KBS µ IUPJD ;. SB[BAED B[BAED SD_ BJUD BJUD JB [PHB

à jb hujtipjikbkiµdda, a, `j kbhoid e` ua simad e` eds aµu uh`rds h`rds qu` `stµbba a siàed hujtipjikbeds gb s`rvied pbrb kbhoibr `j simad e` jb r`spu`stb. @j kbhoid e` jds simads e` bhods aµu uh`rds h`rds eò` kbhoibr `j simad e` jb r`spu`stb edoj`hàt` < e` pdsitivd p dsitivd b a`mbtivd y, e`sp e`spuµ uµ``s, s, e` a`mbtivd au`vbhàt` b pd pdsitivd. sitivd. (E`spuµ``ss e` tded, si uad eb hèib vu`jtb eds v`k`s, s` àkdatrbrµ b hirbaed à jb eir`kkiµdda a drimiabj). @a kda kdas`k s`kuà uàkib kib,, `s jµ ddmikd mikd (bu (buaqu aqu`` ua pdkd `xt `xtrb˚ rb˚ a ad) d) hba hbaif` if`stb stbrr qu` ( ;) ( 5) 8 (+2?). Sdr jd tbatd, à jb hujtipjikbkiµdda, a, jb adrhb e` jds simads `s kdhd simu`<

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pdsitivd pdsitivd 8 pdsitivd a`mbtivd pdsitivd 8 a`mbtivd pdsitivd a`mbtivd 8 a`mbtivd a`mbtivd a`mbtivd 8 pdsitivd Bor`vibaed, hujtipjikbkiµ dda a eèbsimads imubj`sa`mbtivds. eb r`sujtbeds pdsitivds> jb hujtipjikbkiµ dda a e`jbsimads e`simubj`s r`sujtbeds @s hµbs, bs, `stbs hishbs r`mjbs rimà pbrb jd kdatrbrid e` jb hujtipjikbkiµdda< a< jb eivisiµdda. a. B BssµĲ ((+2 +2?) ?) (+5) 8 (+;)> (+2 (+2?) ?) ( 5) 8 ( ;)> ( 2?) (+5) 8 ( ;) y ( 2?) ( 5) 8 (+;). [` puè` v`riflkbr jd bat`ridr iavirtiàed jbs eivisida`s au`vbhàt` b hujtipjikbkiµ da. da. [i à kbeb uab e` jbs eivisida`s brriob kitbebs s` hujtipjikb `j kdkiàt` pdr `j aµ u uh`rd h`rd hµbbss p`qu`˚ a add e` jb izqui`reb (`j eivisdr), s` dotàerµb `j aµ u uh`rd h`rd hµbs bs mrbae` (`j eivieàed) kdhpj`td, kda `j simad prdpid. [i, pdr `l`hpjd, s` iavi`rt` jb s`muaeb eivisiµdda, a, s` dotàerµb ( ;) ( 5) 8 (+2?).

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Eivieir d ad eivieir Kubaed s` e`fla` jb eivisiµda da kdhd r`stb r`p`tieb gbstb jj`mbr b k`rd r`sujtb b v`k`s qu` `stb dp`rbkiµdda a `s ihpdsioj`. Sdr `l`hpjd, ´Ruµ` `s = ?4 Sdr r`stb r`p`tieb pde`hds e`kir qu` = ? ? ? 8 22,, p`rd bg bgµµĲ tà`hds qu` e`tà`rads. [i r`stbhds ua kubrtd ?, ads vbhds bobld e` k`rd. Bua si r`kdadk`hds jb `xistàkib e` jds aµ u uh`rds h`rds a`mbtivds (jd kubj jds batimuds ad gbkµµĲba) ad pde`hds p`rhiti gbk p`rhitirr qu` jb r`stb s` vbyb byb bobld e` k`rd k`rd.. Jb r`st r`stbb sim simui` uiàt` at` jj`m jj`mbr brµµĲb b -2, jb dtr dtrbb jj`m jj`mbr brµµĲb b - ;;,, e`sp e`spuµ uµ`s `s b -0 - 0 y bbssµĲ suk`s s uk`sivbhàt` ivbhàt`.. Éµdae` dae` ads e`tàerµĲĲbhds4 bhds4 @j sist`hb s` rdhp rdhp`r `rµµĲb tdtbjhàt`. [updambhds qu` jd v`hds à dtrb fdrhb. Sbrb r`sdjv`r = ? pd pde eµĲbhd Ĳb hdss pàsbr à jb tbojb e` hujtipjikbr y pr`muatbrads< ´Ruµ` aµu uh`rd, h`rd, hujtipjikbed

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;0 pdr ? eb =4 _`sujtb qu` ad `xist` tbj aµu uh`rd h`rd à eikgb tbojb, si tdhbhds à kuàtb `j tipd e` aµu uh`rd h`rd qu` g`hds vàied kdasie`rbaed. [bo`hds qu` ; ? 8 6 y 5 ? 8 7, p`rd ad `xi `xist` st` aiamµ u ua a aµ u uh`rd h`rd qu` puèb s`r hujtipjikbed pdr ? y e` =. E` edae` s` kdakjuy` qu`, hiàtrbs e`flabhds jb eivisiµdda a e` `stb hba`rb hba`rb,, s`rµba ba pdsioj`s bjmuabs eivisida`s, p`rd dtrbs s`rµbba a ihpdsioj`s. Jds batimuds, `sp`kibjhàt` jds mri`mds, `stboba pdsitivbhàt` fbskiabeds kda `st` prdoj`hb y ouskboba adrhbs à `j hishd. Sdr `l`h `l`hpjd pjd,, ´Ruµ` a aµ u uh`rds µh`rds puèà s`r eivieieds pdr ?4 _`sujtb qu` uad sµĲ y dtrd ad puèà s`rjd. @j aµu uh`rd h`rd 2 ad puè`, `j ? sµĲ, `j ; ad puè`, `j 5 sµĲ, `j 0 ad pu` puè`, e`, y bs bsµµĲ suk` suk`sivbh` sivbhàt`. at`. Jds aµ u uh`rds, h`rds, p prihitivbhàt`, rihitivbhàt`, s` kjb kjbsiflkbob siflkboba a à eds mru mrupds< pds< jjds ds qu` p pdeµ deµĲĲba ba s`r eivieieds pdr ? y bqu`jjds qu` ad p pde deµµĲba s`rjd. Jds prih`rds hbt`hµbbtikds tikds mri`mds mri`m ds s` e`j` e`jìtboba itboba àkda àkdatrbae trbaedd à `std ua hist` hist`ridsd ridsd simaiflkbe simaiflkbed. d. Bjmuad Bjmuadss kdasie`rboba qu` jds aµu uh`rds h`rds eivisio eivisioj`s j`s p pdr dr ? p p`rtà`k `rtà`kµµĲba bj mà`rd ff`hàiad `hàiad y `rba e`smrbkibeds, hiàtrbs qu` jds dtrds `rba e`j mµ`à`rd a`rd hbskujiad y bfdrtuabeds. (Gby qu` r`kdrebr qu` tdeds jds hbt`hµbbtikds tikds mri`mds `rba gdhor`s y `stboba prèispu`stds à fbvdr e` jds gdhor`s.) @a jb vieb eibrib, `j bsuatd e` jb eivisioijiebe pdr ? ihpr`sidabob bj pµu uojikd ojikd à mà`rbj, `sp `sp`kibjhàt` `kibjhàt` kubaed s` gbkµ gbkµĲb a`k`sbrid r`pbrtir ua e`t`rhiabed auh`rd e` dol`tds àtr` eds p`rsdabs. Jb r`pbrtikiµdda a pdr pbrt`s imubj`s, fr`kuàt`hàt`, `s jb h`ldr fdrhb e` `vitbr pjìtds. Jb fdrhb hµbbss sàkijjb e` r`pbrtikiµdda a ‛à jbs µ``pdkbs pd kbs à qu` `j kdad kdadkihiàkihiàtd e` jb eivisiµdda a àt àtr` r` jb hby hbydr drµµĲb e` jb mà màt` t` eò` e` gbo`r sied obstba obstbat` t` eµ`òij‛ oij‛ gb e` gbo`r kdasistied à jb fdrhbkiµ dda a e` eds pijbs, kdjdkbaed bjt`rabtivbhàt` abtiv bhàt` ua dol`td à kbeb uab. [i guoiµ ``rbhds rbhds `hp `hp`zbed `zbed kda 26 flkgbs (44), gbor gborµµĲb 7 à kbeb pijb. Jbs eds fljbs, kdjdkbebs jbed b jbed, tàer tàerµµĲba jb hishb bjtur bjturb, b, e` bqu bquµµĲ qu` 26 (d kubjq kubjqui`r ui`r a aµ u uh`rd µ h`rd eivisioj` pdr ?) s`b ua aµu uh`rd h`rd ‚pbr‘. [ia `ho `hobrmd, brmd, si guo guoiµ iµ``rbhds rbhds `hp`zbe `hp`zbedd kda 2= flkg flkgbs bs (44), jbs eds pijbs ad tàer tàerµµĲba jb hishb bjturb. Sbrb kubaed guoi`rba quèbed r`pbrtiebs 26 flkgbs, àkdatrbr àkdatrbrµµĲbhds eds pijbs imubj`s, p`rd sdorbr sdorbrµµĲb uab flkgb. Kubjqui`r fljb edae` `skdmiµ``rbhds rbhds kdjd kdjdkbrjb kbrjb s` v`r v`rµµĲb hbs bjtb qu` jb dtrb. FdrhbrµĲb Ĳb ua sbjiàt`, pdr e`kir bs bsµµĲ. Pab batimub pbjborb tùtµddaikb, aikb, simaiflkbaed ‚ua sbjiàt`‘ `rb ‚deei‘, pdr jd tbatd, 2= (d kubjqui`r aµ u uh`rd h`rd ad eivisioj` pdr ?) `s ua aµu uh`rd h`rd ‚ihpbr‘ (‚dee‘ (‚d ee‘ à iam iamjµ jµ``s). s). Jb eivisioijiebe d iaeivisioijiebe e` aµu uh`rds h`rds tbhoiµ`à a fdrhb adrhbs. [ia

Ý

Ý

;6

KBS µ IUPJD ;. SB[BAED B[BAED SD_ BJUD BJUD JB [PHB

(b) Sbr

(o) Ihpbr

Fimurb ;.5

;= `hobrmd, aiamuab adrhb `s tba sàkijjb y s` mrbob kda hbydr fbkijiebe qu` jb bjt`rabkiµdda a ‚pbr d ihpbr‘.

Jds mri`mds s` àtr`tiàà Dtrd e` jds prih`rds e`skuorihiàtds à tdrad b jb eivisiµdda a fu` `j g`kgd e` qu` bjmuads aµu uh`rds h`rds puei`rba s`r eivieieds pdr hµbbss e` ua aµ u uh`rd h`rd p`qu`˚ a ad. d. Sdr `l`hpjd 63 puè` s`r eivieied pdr ?, ;, 5, 0, 6, 23, 2?, 20, ?3 y ;3. Kbeb uad e` `stds aµu uh`rds h`rds `s kbpbz e` eivieir bj 63 pdr pbrt`s imubj`s, y s` jjbhb ua ‚fbktdr‘ e` 63, e` edae` s` eèuk` qu` `j 63 tià` ei`z fbktdr`s eif`ràt`s. [ia eueb, `xistà dtrds fbktdr`s qu` ad s` gba hàkidabed. Pad `s `j hishd 63. E`spu E`spuµµ`s `s e` tded 63 63 8 2. @a r`bjiebe, kubjqui`r aµ u uh`rd h`rd eivieied eivie ied p pdr dr sµĲ hishd `s imubj b 2, pdr jd kubj kubjqui` kubjqui`rr aµu uh`rd h`rd `s uad e` sus prdpids fbktdr`s. @j fbktdr r`stbat` `s `j 2, pu`std qu` 63 2 8 63. Kdhd kubjqui`r aµ u uh`rd h`rd puè` s`r eivie eivieied ied pdr 2, quèb quèbaed aed iat iatbktd bktd à sµĲ hishd, puè` kdasie`rbrs` qu` `j 2 `s uab `sp`ki` e` fbktdr uaiv`rsbj. Eòied b qu` kubjqui`r aµu uh`rd h`rd puè` s`r eivie eivieied ied pdr sµĲ hishd y pdr 2, jds mri`mds (b quià`s àkbatbob lumbr kda jds fbktdr`s) mà`rbjhàt` e`skbrtboba `sds eds aµuh`rds uh`rds kdhd fbktd fbktdr`s. r`s. E`spu E`spuµµ`s `s e` tded, ad `xist `xist`` uab adrhb iat`r`sbat` à bjmd qu` bkdat`k` b tdeds jds aµu uh`rds h`rds sia eistiakiµdda. a. (Iakieàtbjhàt`, pdr kbeb fbktdr qu` ua aµu uh`rd h`rd tià`, `xist` dtrd qu` `s `j a`mbtivd e`j prih`rd. Sdr `l`hpjd, `j 63 puè` s`r eivieied pdr ?, ;, 5 y bs b sµĲ suk`sivbhàt`. [ia `hobrmd, jds mri`md mri`mdss ad r`kdad r`kdadkkµĲba jds aµu uh`rds h`rds a`mbtivds y kdhd r`bjhàt` r`bjhàt`,, ad iatrdeukà abeb au`v au`vd, d, mà`rbjhàt` tbhoiµ`à a jds e`skbrtbhds kdhd fbktdr`s). Bgdrb oià, bua kubaed `j 63 tià` ei`z fbktdr`s, jds aµu uh`rds h`rds k`rkbads bj

ý

ý

∟ ∟ ∟

hishd ad sda tba bfdrtuabeds. @j aµu uh`rd h`rd 07 sdjbhàt` tià` eds fbktdr`s ? y ?9> y `j 6? tbhoi tbhoiµµ`à a tià` u uaikbhàt` µaikbhàt` eds fbktdr`s, ? y ;2. Ad dostbat`, sia ihpdrtbr jd `skbsds qu` s`ba jds fbktdr`s qu` puèb tà`r ua aµ u uh`rd, h`rd, pdr `j sdjd g`kgd e` qu` `xistba s` jjbhb bj hishd ua ‚aµu uh`rd h`rd kdhpu`std‘ yb qu` puè` v`rs` qu` `stµb kdhpu`std e` aµ u uh`rds h`rds hµbbss p`qu`˚ a ads ds qu` puèà hujtipjikbrs` luatds. @j aµ u uh`rd h`rd 07 puè` s`r vistd kdhd ? ?9, y `j aµ u uh`rd h`rd 6? kdhd ? ;2. @j aµ u uh`rd h`rd 63 `s hµbs bs kdhpjikbed pdrqu` tià` vbrids fbktdr`s. Sder SderµµĲb, pdr `l`hpjd, `xpr`sbrs` kdhd ? ;3, p`rd p `rd `j ;3 p pdr dr ssµµĲ hishd `s ua a aµ u uh`rd µh`rd kdhpu`std qu` puè` `xpr`sbrs` kdhd ? 20, siàed b jb v`z `j 20 ua aµu uh`h`rd kdhpu`std qu` puè` `xpr`sbrs` kdhd ; 0. Jb e`skdhpdsikiµdda a flabj,

Ý

Ý

Ý

Ý

Ý

;7

KBS µ IUPJD ;. SB[BAED B[BAED SD_ BJUD BJUD JB [PHB

Ý Ý Ý

àtdak`s `xpr`sb bj 63 kdhd ? ? ; 0. @a jds eds pµbbrrbfds rrbfds bat`ridr`s ad s` gb iatà iatàtbed tbed e`skdhpda`r p pdr dr fbktdr`s jds aµ u uh`rds h`rds ?, (fu`rb ; y 0 d,e`j pbrb `j kbsd `j ?9 y `j ;2. e`j Jb 2). rbzµ dda a dtrbs `stµb à qu` ad tiàà aiamuad prdpid y, abturbjhàt`, @a pbjborbs, ad `xist` aiamµ u ua a aµ uh`rd, uh`rd, fu`rb e` jds prdpids y `j 2, qu` puèba eivieirjds pdr pbrt`s imubj`s. Jds aµ u uh`rds h`rds sia fbktdr`s fbktdr`s,, tbj`s kdhd ?, ;, 0, =, 22, 2;, 2=, 29 y bsµ bsµĲ suk`sivbhàt`, sda jjbhbeds ‚aµ u uh`rds h`rds prihds‘, d sihpj`hàt` ‚prihds‘, e`rivµbbaeds` aeds` `st` adhor` e` uab pbjborb jbtiab qu` simaiflkb ‚prih`rd‘. Sbrb jb màt` qu` s` v` vbjdr`s hist`ridsds à jds aµu uh`rds, h`rds, pbr`k`rµĲb Ĳb kdhd si jds aµuh`rds uh`rds prihds puei`rba gbo`r `xistied prih`rd, hiàtrbs qu` jds aµu uh`rds h`rds kdhpu`stds puei`rba gbo`r sied kdastruieds e`spuµ``s, s, sbkµbbaedjds aedjds e` jds prihds. @a dtrbs pbjborbs, uab v`z qu` `xisti`rda jds aµu uh`rds h`rds ?, ; y 0, pued fdrhbrs` `j aµ u uh`rd h`rd 63 pdr hujtipjikbkiµdda a e` ? ? ; 0.

Ý Ý Ý

Sderµ µĲb pbr`k`r qu` kdafdrh kdafdrh`` iob bbskàeià skàeiàed ed à jbpu`std `sk `skbjb bjb e` aµu uh`h`rds,Sder e`sbpbr`k`r e`sbpbr`k`rµ µĲb q jbu`pdsioijiebe e` uad jds aµ u uh`rds h`rds prihds, qu`jdsgbor gborµ µĲba kbeb v`z hµbbss aµ u uh`rds h`rds p`qu`˚ a ads ds pbrb `skdm`rjds kdhd fbktdr`s pdsioj`s. [ia `hobrmd, `std ad `s ki`rtd. @ukjie`s, hbt`hµbbtikd tikd mri`md, e`hdstrµd gbk` ??33 b˚a ads ds qu` ad gby tbj kdsb kdhd `j aµu uh`rd h`rd prihd hµbbss bjtd. [ia ihpdrtbr jd bjtd qu` fu`rb ua auh`rd prihd, µ`j `j e`hdstrµ d kdhd pd pder erµµĲb uad ie`br ua aµ u uh`rd h`rd hµbbss bjtd bµ u ua, a, qu` tbhoiµ`à a fu`rb prihd. Jds mri`mds s` àtr`tà àtr`tàµµĲba lumbaed kda jd jdss fbktdr`s. Sdr `l`hpjd, suhboba jds fbktdr`s e` jds aµ u uh`rds h`rds (iakjuyàed `j aµu uh`rd h`rd 2 à `stb dkbsiµdda, a, p`rd `xkjuyàed `j prdpid aµu uh`rd), h`rd), pbrb v`r quµ` pbsbob. Bjmuabs v`k`s, jds fbktdr`s e` ua aµu uh`rd h`rd suhboba hàds qu` `j prdpid aµu uh`rd. h`rd. Sdr `l`hpjd, jds fbktdr`s e` 23 (2, ? y 0) suhba sdjbhàt` 7. Bj auh`rd 23, pdr jd tbatd, jd jjbhboba ua ‚aµu uh`rd h`rd e`flkiàt`‘. Jds fbktdr`s e`j 2? (2, ?, ;, 5, y 6), à kbhoid, suhba 26> d s`b hµbbss qu` `j vbjdr e`j aµu uh`rd h`rd hishd y, pdr `stb rbzµdda, a, jjbhboba bj 2? ua ‚aµu uh`rd h`rd bouaebat`‘. [ia `hobrmd, jds fbktdr`s e`j 6 (2, ? y ;) suhba 6 y jds e`j ?7 (2, ?, 5, = y 25) suhba ?7. Jds mri`mds jjbhboba b µ``stds stds ‚a ‚aµ u uh`rds µh`rds p`rf`ktds‘. Hµbbss bµ ua, ua, jds fbktdr`s e` ??3 (2, ?, 5, 0, 23, 22 ?3, ??, 55, 00 y 223) suhba ?75, hiàtrbs qu` jds fbktdr`s e` ?75 (2, ?, 5, =2 y 25?) suhba ??3. y eik` qu` µ``stds std s sda ‚aµ ‚a u uh`rds µh`rds bhimds‘. @stb eivisiµdda a àtr` aµ u uh`rds h`rds prihds, p`rf`ktds, bhimds, `tk., auakb gb sied e` hukgb ihpdrtbakib prµbktikb, bktikb, p`rd gb fbskiabed eurbat` hij`s e` b˚a ads ds b jds hbt`hµbbtikds, tikds, y td tdebv ebvµµĲb jd simu` gbkiàe gbkiàed. d.

;9

Kuàtb y hèikiµ da da Gbstb bgdrb, s` gb vàied gbojbaed e` jds aµu uh`rds h`rds qu` dotà`hds bj kdatbr> 2, ?, ; y bs bsµµĲ suk` suk`sivbhàt`. sivbhàt`. @sd @sdss a aµ u uh`rds µ h`rds y jbs dp`rbkida`s qu` kdhpràeà (gbstb edae` jbs g`hds `xpu`std) sda, b v`k`s, tded jd qu` a`k`sitbhds. Sdr `l`hpjd, jds ai˚ads a ds à ua sbjµdda a e` kjbs`s puèà s`r r`pr`sàtbeds pdr ua aµ u uh`rd. h`rd. Gby 5 ai˚a ads, ds, 0 ai˚ a ads, ds, d bjmµ u ua a dtrd aµu uh`rd. h`rd. [ia `hobrmd, si`hpr` si`hp r` `s ua aµu uh`rd h`rd e e`flaied. `flaied. Pad ad puè` e` e`kir, kir, e`sp e`spuµ uµ``ss e` ua h`tikujd h`tikujdsd sd `skrutiaid< ‚Gby hµbbss e` kubtrd ai˚ a ads, ds, `sd `s hbaifl`std, p`rd, e`flaitiv e`flaitivbhàt`, bhàt`, sda hàds e` kiakd. [updamd qu` `j aµu uh`rd h`rd `stµb àtr` kubtrd y kiakd ai˚a ads‘. ds‘. Ad `xist` ua aµu uh`rd h`rd iat`rhèid àtr` 5 y 0, si `stµb uad kdatbaed dol`tds. [da 5 d 0 ai˚a ads. ds. Hµbbss bµ u ua, a, si uad `hpi`zb kda 5 ai˚a ads ds y jj`mba ? hµbbs, s, `j r`sujtbed `s 6 ai˚ a ads. ds. Ai ua pdkd hµbbss e` 6 ai ua p`jd hàds e` 6. [da `xbktbhàt` 6. Ad dostbat`, supdambhds qu` s` gbk` uab pr`muatb kdhd µ``stb< stb< ´Kuµbbatbs atbs gdrbs gba `stbed `stueibaed `sds ai˚a ads4 ds4 @a tbj kbsd pderµĲb Ĳb uad kdat`stbr fµbbkijhàt`< kijhàt`< ‚@stdy s`murd e` qu` gba `stueibed hµbbss e` uab gdrb, p`rd, e`se` ju`md, hàds e` eds gdrbs‘. BquµµĲ, tbj r`spu`st Bqu r`spu`stbb tà tàer erµµĲb sàtied. @xist` ua jbpsd, ua kdak`ptd e` ‚‚eueurbkiµdda a e`j ti`hpd‘, qu` `s hµbbss e` 2 gdrb y hàds e` ? gdrbs. @j ti`hpd `s bjmd qu` ad s` puè` kdatbr, siad bjmd qu` s` hie`. Jb kuàtb y jb hèikiµdda a eòà s`r hba`lbebs eif`ràt`hàt`. Bj kdatbr, `stµb uad trbtbaed kda dol`tds s`pbrbeds d ‚eiskr`tds‘. Jds aµu uh`rds h`rds qu` g`hds vàied eiskutiàed sda, à sµĲ hishds, s`pbrbeds y eiskr`tds, y kdhd tbj`s, puèà `quipbrbrs` huy oià kda jds dol`tds. [da `jjds tded jd qu` a`k`sitbhds. [ia `hobrmd, `j bktd e` hèir bjmd qu` ad kdastb e` dol`tds s`pbrbeds `s r`bjhàt` dtrd bsuatd. Bqu BquµµĲ `stbhds trbtbaed kda bjmd qu` `s ‚kdatia ‚kdatiaud‘. ud‘. Uà`hds qu` trbtbr, pdr `l`hpjd, kda ua `spbkid e` ti`hpd d kda jb jdamitue e` uab jµĲa`b kdati kdatiaub. aub. Jds aµ uh`rds uh`rds dreiabrids, qu` sda eiskr`tds, ad puèà `quipbrbrs` ‛sia `j p`jimrd e` àkdatrbr eiflkujtbe`s‛ kda bjmd qu` `s kdatiaud, kdhd `j kbsd e`j prdoj`hb brriob hàkidabed hàkidabed,, à r`jbk r`jbkiµ iµdda a kda `j aµu uh`rd h`rd e` gdrbs qu` jds ai˚ads ads gbo gboµµĲba `stbed `stueib `stueibaed. aed. Sbrb `vitbr tbj`s prdoj`hbs, jds aµ u uh`rds h`rds eòà s`r kdatiauds. @j `spbkid àtr` aµ u uh`rds h`rds dreia dreiabrids brids eò` jjàbrs` kda a aµ u uh`rds µh`rds iat`rhèids. Kubaed `std s` gb g`kgd, jds aµuh`rds uh`rds 2, ?, ; y bs bsµµĲ suk`sivbhàt`, suk`sivbhàt`, s` kdavi`rtà, sih-

53

KBS µ IUPJD ;. SB[BAED B[BAED SD_ BJUD BJUD JB [PHB

pj`hàt`,, à uab p`qu`˚ pj`hàt` a abb pbrt` e` ua sist`hb kdatiaud, qu` puè` àtdak`s `quipbrbrs` kda `j ti`hpd, jb jdamitue d kubjqui`r dtrd fàµddhàd hàd kdatiaud qu` eòb hèirs`. B pbrti pbrtirr e`j simui` simuiàt` at` kbp kbpµµĲtujd v` v`r`hds r`hds kdhd surmi` surmi`rda rda `stds aµu uh`rds h`rds iat`rhèids y kµddhd hd sda hba`lbeds.

KbpµĲtujd 5 Aµ uh`rds quòrbeds uh`rds Eivieiàed jb uaiebe @a ua sàtied prµbktikd, bktikd, jb guhbaiebe ad pdeµĲb Ĳb r`kdadk`r jihitbkida`s à jb eivisiµda. da. [updambhds qu` fu`rb a`k`sbrid eivieir eds hbazbabs àtr` kubtrd ai˚ a ads. ds. @s iaµ utij utij e`kir qu` ? ad puè` eivieirs` àtr` 5 pdrqu` ad `xist` aiamµ u ua a aµ u uh`rd h`rd à jb tbojb e` hujtipjikbr qu`, bj hujtipjikbrs` p pdr dr 5, eµ` ?. Jd qu` gbk` uab hber` prµbbktikb, ktikb, à `st` kbsd, `s eivieir kbeb hbazbab à eds pbrt`s imubj`s y ebr b kbeb uad e` jds kubtrd ai˚a ads ds ua pèbzd e` jb hishb (d tbj v`z gbk`r uab kdhpdtb e` hbazbabs). [imuiàed `st` sist`hb, jb guhbaiebe rdhpiµd sus uaiebe`s mà`rbj`s e` hèikiµ dda a à pbrt`s hµbbss p`qu`˚ a abs, bs, eµbbaedj`s ae dj`s adhor adhor`s `s tbhoiµ`à a b µ``stbs. stb s. Sdr `l`hpjd, `l`h pjd, à au` au`strd strd sist` sist`hb hb e` hèie hèiebs, bs, hèid jitrd e` kubjq kubjqui`r ui`r jµĲquie Ĳquiedd puè` s`r eivieied à eds pbrt`s imubj`s jjbhbeds kubrtds y µ``stds stds b su v`z à dktbvds. [i ua kdh`rkibat` tià` eds kubrtds e` jitrd y kubtrd kjiàt`s, kbeb uad e` `jjds s` jj`vb ua dktbvd. Ubhoiµ`à a pu puè` è` e eivieir ivieir jbs uaiebe uaiebe`s `s à hbydr a aµ u uh`rd µh`rd e` suouaiebe`s. Pa nijdmrbhd puè` eivieirs` à kubtrd kubrtds y µ`stds `stds à dkgd ddktbv ktbvds, ds, jds qu`, b su v`z, puèà eivieirs` à 26 ei`kisìsbv ei`kisìsbvds ds y µ``stds stds à ;? trìatbiedsb trìatbiedsbvds. vds. Udebs `stbs kifrbs sda `j r`sujtbed e` jb eivisiµdda a e` bjmd à eds pbrt`s> e`spuµ``s, s, kbeb pbrt` à eds hµbbss y, pdst`ridrhàt`, kbeb p`qu`˚a abb pbrt`, à eds hµbbs, s, y bsµ bsµĲ suk` suk`sivbh` sivbhàt`. at`. Sdr `l`hpjd, s` pd pder erµµĲb eivie eivieir ir ua kubr kubrtd td e` jitrd àtr` eds gdhor`s gdhor`s,, ebaed b kbeb kbe b uad e` `jj `jjds ds ua dktbvd e e`` jitrd e` jµĲquied, Ĳquied , d pd pder erµµĲb eb ebrs` rs` b 5 gd gdhor`s hor`s ua ei`kisìsbvd e` jitrd, d b 7 gdhor`s ua trìatbiedsbvd e` jitrd, d b 26 gdhor`s 52

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µ KBS µ IU IUPJ PJD D 5. 5. A PH@_D[ RP@O_BED[

ua s`sàtbikubtrdbvd e` jitrd, d b ;? gdhor`s ua kiàtd vìatidkgbvd e` jitrd. @std suàb huy oià. S`rd, ´Ruµ` gbk` uad si tià` tr`s kjiàt`s à fljb4 Pa kubrtd e` jitrd ad puè` eivieirs` àtr` ; gdhor`s e` hba`rb qu` kbeb uad dotàmb uab kbatiebe imubj, pu`std qu` ; ad `s fbktdr e` aiamuab e` jbs suouaiebe`s. [ia `hobrmd, `s hµbs bs prdoboj` qu` ; gdhor`s `stµ`à a eispu`stds b kdhpbrtir ua kubrtd e` jitrd, qu` 26 d ;?. @s u utij, µtij, pdr jd tbatd, `skdm`r uab suouaiebe àtr` jds aµu uh`rds h`rds oblds qu` kdhpràeb tbatds fbktdr`s kdhd s`b pdsioj`. Sdr `l`hpjd, tdh`hds `j 2?. Sde`hds fdrhbr kda kubjqui`r kdaluatd e` kdsbs uab edkàb. Y dos`rvbhds kuµbba au utij µ tij `s jb edkàb. [i s` tià` uab edkàb e` hbazbabs, puè` uad eivieirjbs `quitbtivbhàt` à eds mrupds e` 6 d à ; mrupds e` 5, d à 5 mrupds e` ;, d à 6 mrupds e` ?, d à edk` mrupds e` 2 pbrb kbeb mrupd. Jd ihpdrtbat` `s qu` ad sdjbhàt sdjbhàt`` `j ? y `j 5 sda fbktdr`s e` 2?, siad qu` tbho tbhoiµ iµ`à a jd `s `j ;. Sbrb ua fborikbat` qu` vàe` pdr edkàb, tbhoiµ `à a j` `s fµbbkij kij vàe`r à jdt`s hµbbss p`qu`˚ a ads, ds, sia mrba eiflkujtbe à `j blust` e` pr`kids, eòied b qu` eispda` e` huk hukgbs gbs fdrhbs pbrb eivieir jb edkàb pdr pbrt`s imubj`s. Ubhoiµ `à a `xist` jb edkàb e` edkàbs d jb ‚mru`sb‘ qu` r`pr`sàtb 2? 2? d s`b 255 brtµµĲkujds brt Ĳkujds.. Jds fbktd fbktdr`s r`s e` 255 sda< ?, ;, 5, 6, 7, 9, 2?, 26, 27, ?5, ;6, 57 y =?. @a jb brithµ``tikb tikb prµbktikb bktikb `s huy ihpdrtbat` eispda`r e` tbj`s fbktdr`s, y bua `xistà p`rsdabs qu` e`s`br e`s`brµµĲba qu` guoiµ ``rbhds rbhds usbed `j 2? à v`z e`j 23 kdhd obs` e` au`strd sist`hb auhµ ``rikd. rikd. @j aµu uh`rd h`rd 23 sdjbhàt` tià` ? fbktdr`s, `j ? y `j 0, y ad puè` eivieirs` à pbrt`s imubj`s yb s`b àtr` ; d àtr` 5. Jb µu uaikb aikb rbzµdda a pdr jb kubj `j 23 mbaµd bj 2? `s, prdoboj`hàt`, `j bkkieàt` babtµddhikd hikd q qu` u` ads gbk` tà` tà`rr 0 eè eèds ds à kbeb h hbad. bad. [ [ii guoiµ``rbhds rbhds tàied 6 à kbeb hbad... [ia `hobrmd, `j 23 tià` uab vàtblb sdor` `j 2?. @j aµu uh`rd h`rd 0 `s ua fbktdr e`j 23, p`rd ad e`j 2?. Jds batimuds oboijdaids trbtbrda e` kdhoiabr jds kbrbkt`r`s eistiativds e`j 23 y e`j 2?, àkdatrbaed ua aµu uh`rd h`rd pbrb `j kubj ad sdjbhàt` sdjbhàt` fu`r fu`rba ba fbktdr`s `j ?, `j ; y `j 5, siad tbhoi tbhoiµµ`à a `j 0. @j aµu uh`rd h`rd hµbbss p`qu`˚ a add qu` fu` àkdatrbed `s `j 63. Jb bstrda bstrdadh dhµµĲb s` uaiµd b jds oboijdaids à `st` bsp`ktd. @j b˚a add tià` uab eurbkiµdda a e` ;60 eµĲbs (y uabs kuba kubatbs tbs gdrbs), qu` `s `j jbpsd r`qu`ri r`qu`ried ed pdr `j [dj pbrb gbk`r su kirkuitd (bpbràt`) bjrèèdr e`j flrhbhàtd kdatrb jds bstrds fllds. [i `st` kirkuitd tdtbj fu`rb eivieied à ‚ldrabebs eibribs‘ e`j sdj, `xistirµĲĲba ba ;60 pbrt`s e` u ua a ki kirkujd. rkujd.

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Fimurb 5.2< Jb gdrb e` 63 hiautds Bgdrb oià, d jd jdss ob oboijdai oijdaids ds tà tàµµĲba `j auh auh`rd `rd e e`` eµĲĲbs bs jim`rbh jim`rbhàt` àt` `qui `quivdkbed, vdkbed, d, kda hbydr prdoboi prdoboijiebe, jiebe, rèdae` rèdae`brda brda `j aµu uh`rd h`rd b ;63. @a su sist`hb, `rb hbs fµbbkij kij hba`lbr ;63, qu` `s `j r`sujtbed e` 63 6. Bs BsµµĲ pu`s, eiviei eiviei`rda `rda jb `sf`rb k`j`st` à ;63 pbrt`s imubj`s, b jds kubj`s à jb bktubjiebe jjbhbhds ‚mrbeds‘. Hµbbss bµ u ua, a, eiviei`rda kbeb mrbed à 63 pbrt`s imubj`s b jbs kubj`s jjbhbhds ‚hiautds‘ y kbeb hiautd à dtrbs 63 pbrt`s imubj`s b jbs kubj`s jjbhbhds ‚s`muaeds‘(44).2

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Udebv debvµ µĲb `j `hpj`bhds `stè`sist`hb oboijdaikd. bbs, s, kdhd `j au`strb ti`hpd s` hie` pdr hdvihiàtd jds ku`rpds k`j`st`sBe`hµ e`j flrhbhàtd, gdrb `stµb eivieieb à 63 hiautds, jds kubj`s, b su v`z, `stµbba a eivieieds à 63 2

Jb pbjborb jbtiab ‚e`mrbeus‘ simaiflkb‚`skbjdabr‘ y pdr tbatds r`fl`r` b bjmd bsµĲ kdhd uab `skbj`rb d `skbjiabtb pdr jb kubj puè` ustè oblbr. Jbs eivisida`s r`mujbr`s ‛qu` s` bs`h`lba b jds `skbjda`s e` uab `skbj`rb d `skbjiabtb‛ puèà s`r jjbhbebs ‚mrbebs‘siàed `stb pbjborb uab kdrrupkiµda da e` ‚e`mrbeus‘. ‚e` mrbeus‘. Bs BsµµĲ tà`hds ad sdjbh sdjbhàt` àt` mrbeds e` ua kirkujd, ki rkujd, siad tbhoiµà à mrbeds e` t`hp`rbturb. Jbs s`sàtb eivisida`s e` ua mrbed fu`rda prihitivbhàt` jjbhbebs ‚pbrs hiaut` prihb‘ y jbs 63 eivisida`s e` kbeb uad e` `stds ‚pbrs hiaus s`kuaeb‘ simaiflkbaed ‚p`qu`˚ a abb pbrt` prih`rb‘ y ‚p`qu`˚ a abb pbrt` s`muaeb‘ r`sp`ktivbhàt`. Jbs frbs`s bat`ridr`s kda `j ti`hpd s` bkdrtbrda b ‚hiautd‘ y ‚s`muaed‘, r`sp`ktivbhàt`.

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µ KBS µ IU IUPJ PJD D 5. 5. A PH@_D[ RP@O_BED[

s`muaeds. Pa iaeik iaeikid id e`j sist`hb e` 2? tbh tbhoiµ oiµ à à s` e`h e`hu`str u`strbb à jb fdrhb à qu` e`kih e`kihds jb gdrb. Au`str Au`strds ds`s ti`hpds eiurads ds y ad adktur kturads ads `stµ bba a eivieieds à 2? gdrbs.ds Drimiabjh Drimiabjhàt àt`, `, bat bat`s e` quèiura jds r`jdl`s fu`rba iav iavàt àtbeds, beds, `stbs gdrbs vbriboba e` bku`red kda jds kbhoids e` eurbkiµdda a e`j eµĲb y jb adkg`, s`mµua ua jbs `stbkida`s e`j b˚ a ad. d. @a iav iavi`rad i`rad,, jbs gdrbs e`j eµĲb y jb adkg` adkg`,, s`mµ u ua a jbs `stbkida`s e`j b˚a ad. d. @a iavi`rad, jbs gdrbs e`j eµĲb `rba kdrtbs y jbrmbs jbs e` jb adkg`, hiàtrbs qu` à v`rbad `rb bj kdatrbrid. @a jb bktubjiebe au`strbs gdrbs tiàà jb hishb eurbkiµdda a à tdeb µ``pdkb> pd kb> à v`rbad `j eµĲb eurb hµbs bs e` 2? gdrbs y hàds jb adkg` y, à `j iavi`rad, bkdat`k` jd kdatrbrid. Ad dostbat`, dostb at`, au`strds r`jd r`jdl`s l`s simu` simuà a tàiàed ttdebv debvµµĲb sdjbh sdjbhàt` àt` 2? aµu uh`h`rds y `stbhds dojimbeds b eistiamuir àtr` jb 2 B.H. y jb 2 S.H.(Jbs fu`rzbs brhbebs gba kdatia kdatiaubed ubed e` jbs 2? e`j hèideµĲĲbb à be`jbat`, y gbojba e` jbs 2; p`rd, bpbràt`hà bpbràt`hàt`, t`, ad gb brrbimbed jb kdstuhor kdstu hor`, `, bj hàds àtr` jds kivij kivij`s.) `s.)

Sbrt`s e` uad Jd qu` puè` s`r g`kgd pdr `j gdhor` prdhèid kda jbs uaiebe`s kdhua`s e` hèikiµdda, a, puè` s`r g`kgd pdr jds hbt`hµbbtikds tikds kda jds aµu uh`rds h`rds purds. ´Sdr quµ` ad ad e eivi ivieir eir aµu uh`rd h`rd 2, d kubjqui`r dtrd aµu uh`rd, h`rd, à eds, tr`s, d kubtr kubtrdd pbrt`s imubj`s4 B fla e` gbk`r `sd eòiebhàt` `s a`k`sbrid, prih`rd, ebr ua adhor` e` `sbs ‚‚pbrt`s pbrt`s e` uad‘< s`muaed, àkdatrbr ua sµĲhodjd bprdpibed pbrb jbs pbrt`s y, t`rk`rd, e`sbrrdjjbr ua sist`hb pbrb hba`lbr `sbs pbrt`s eàtrd e` jbs dp`rb dp`rbkida`s kida`s brithµ``tikbs. tikbs. E`spuµµ`s E`spu `s e` tded, si jbs pbrt` pbrt`ss e` jds aµu uh`rds h`rds puèà hba`lbrs` kdhd jds aµ u uh`rds h`rds kdhua`s, puèà tbhoiµ `à a kdasie`rbrs` kdhd aµu uh`rds h`rds kdhua`s pbrb tdeds jds dol`tds, tbatd t`µddrikds rikds kdhd prµbbktikds. ktikds. Jds adhor`s sda fµbbkij`s kij`s y prdviàà e`j jàmubl` kdhµu ua. a. Eds pbrt` pbrt`ss imubj`s e` kubjqui`r kdsb sda ‚hitbe`s‘. Jds r`sujtbeds e` jb eivisiµdda a e` ua auh`rd à vbribs pbrt`s sda jjbhbeds e` bku`red kda jb fdrhb ‚dreiabj‘ e` `sds aµ u uh`rds, h`rds, @s e`kir, jbs tr`s pbrt`s e` uad sda ‚t`rkids‘, kubtrd pbrt`s sda ‚kubrtds‘ ‚ku brtds‘,, ‚quiatds ‚quiatds‘, ‘, ‚s`x ‚s`xtds‘ tds‘ y bbssµĲ suk`si suk`sivbhàt`, vbhàt`, gbstb ed edae` ae` s` qu qui`rb i`rb jj`mbr. Pab hitbe `s `j r`sujtbed e` jb eivisiµdda a e` uab uaiebe à eds pbrt`s. @a dtrbs pbjborbs, `s 2 ?. @std ad eb drimà b aiamµu ua a aµ u uh`rd h`rd kdhµ u ua a y ad gby a`k`siebe e` ouskbrjd. Jb dp`rbkiµdda a `s suflkiàt`hàt` sàkijjb pbrb

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50 iaeikbrjb à jb fdrhb hbs eir`ktb pdsioj` y s` rèuk` b `skrioir ?2 d 2/?. @std puè` j``rs` kdhd ‚uad eivieied pdr eds‘, b hàued ‚uad sdor` eds‘ d, hµbs bs fr`kuàt`hàt` bµ u ua, a, sàkijjbhàt`, ‚uab hitbe‘. @a fdrhb sihijbr, 2; `s ‚ua t`rkid‘, 20 `s ua quiatd y bsµ bsµĲ suk`sivbhàt`. Ad `xist` iatàkiµdda a e` r`sdjv`r jb eivisiµda> da> sàkijjbhàt` hbaifl`stb ustè jb 2 eivisiµdda. a. @j e`kir qu` 2 ; `s imubj b ; , ad `s e`kir hbs qu` ‚uad eivieied pdr tr`s `s imubj b uad eivieied pdr tr`s‘. @std puè` sdabr b frustrbkiµdda. a. @j j`ktdr puè` pr`muatbr< S`rd, ´´Ruµ Ruµ` `s 2 uad eivieied pdr tr`s4 Jb r`spu`stb `s< ´Kuµbbjj `s jb eif`ràkib4 [i ; puè` s`r hba`lbed kdhd si fu`rb jb r`spu`stb, ´B quµ` ouskbr hµbs4 bs4 B `stbs pbrt`s e` jb uaiebe s` j`s jjbhb ‚frbkkida`s‘ d ‚quòrbeds‘ (e`j jbtµµĲa ‚quòrb jbt ‚quòrbr‘) r‘) pu`std qu` s` doti dotiàà àà bj ‚q ‚quòrbr‘ uòrbr‘ 2 à pbrt`s. Jds a aµ u uh`rds µh`rds kdhua`s sda jjbhbeds pdr kdatrbst` ‚àt`rds‘ (e` uab pbjborb jbtiab qu` simaiflkb ‚iat`mrd‘). Jbs frbkkida`s d quòrbeds puèà, tbhoi tbhoiµµ`à, a, s`r vistbs kdhd pbrt`s e` aµ u uh`rds h`rds àt`rds d e` ua tdtbj e` kdsbs à mà`rbj. Y, bgdrb, vbhds b kdasie`rbr `j hba`ld e` jds quòrbeds. @hp`zbr`hds pdr pr`muatbr si puèà suhbrs` jds quòrbeds y, si `s bsµĲ, kµddhd hd puè` gbk`rs` jb dp`rbkiµdda. a. [updambhds qu` qu`r`hds suhbr ;2 y 2; . [i kbh kbhoibhds oibhds `j prdoj prdoj`hb `hb b pbjbor pbjborbs bs pbr`k pbr`k`` obstba obstbat` t` sàk sàkijjd. ijjd. Bqu BquµµĲ gby ua t`rkid y bjjµb dtrd t`rkid. [uhbaed bhods, dovibhàt` tàerµb ustè eds t`rkids. (@xbktbhàt` kdhd uab hbazbab hµbbss dtrb hbazbab sda eds hbazbabs.) Jb simuiàt` pr`muatb `s< ´Kµddhd hd sihodjizbr eds t`rkids4 [i ua t`rkid s` 2 `skrio` ; , `s jµddmikd mikd supda`r qu` eds t`rkids eò`rµbba a `skrioirs` ;? . @sd jd pde`hds kdhprdo kdhprdobr. br. @j sµĲhodjd 2; simaiflkb ‚uad eivieied pdr tr`s‘> pdr jd tbatd, `j sµĲho Ĳhodj djdd ?; eò` simaiflkbr ‚eds eivieied pdr tr`s‘. S`rd, ´Kµddhd hd gbk`hds pbrb eivieir eds àtr` tr`s4 [i tuviµ``rbhds rbhds eds p`qu`˚a ads ds pbst`j`s y tr`s ai˚ a ads, ds, pder pderµµĲbhds gbk`rjd eivieiàed kbeb pbst`j à tr`s pbrt`s imubj`s qu` ads ebr ebrµµĲba 6 pbrt`s imubj`s, kbeb uab imubj b ua t`rkid e` pbst`j. @stbs 6 pbrt`s p pder derµµĲba s`r eivieiebs àtr` jds ; ai ai˚ a ads ˚ ds ebaed b kbeb uad ? pèbzds, kda jd kubj bkbobr bkbobrµµĲba r`kioiàed eds t`rkids e` pbst`j. (44) @a kdas`kuàkib, eds t`rkids puèà  kdasie`rbrs` kdasie`rbrs` kdhd `j kdkiàt` e` ? ; ? y puèà  r`pr`sàtbrs` kdhd ; . Psbaed `j hishd tipd e` rbzdabhiàtd, s` puè` e`hdstrbr qu` kubjqui`r eivisiµda da puè` r`pr`sàtbrs` à fdrhb e` quòrbed. Kubràtb y tr`s pbst`j`s eivieieds pdr pbrt`s imubj`s àtr` s`tàtb y tr`s p`rsdabs ebrµbba a kdhd 5; r`sujtbed `j qu` kbeb p`rsdab r`kiob =; e` pbst`j. S`rd vdjviàed b au`strb suhb y r`stb. @a jb hishb fdrhb à qu` rbzd-

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µ KBS µ IU IUPJ PJD D 5. 5. A PH@_D[ RP@O_BED[

Fimurb 5.? 2

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8 0. abhds kda + , pde`hds rbzdabr qu` + + 8 y qu` Sde`hds fllbr uab r`mjb mà`rbj. Bj suhbr d r`stbr quòrbeds kda ‚eàdhiabedr`s‘ (adhor` ebed bj aµ u uh`rd h`rd e`j quòrbed qu` s` àkuàtrb bobld e` jb jµĲa`b d b jb e`r` e`r`kgb kgb e` jb eibmda eibmdabj bj qu` s` usb bjmuabs v` v`k`s) k`s) imubj`s sdjbhàt` `s a`k`sbrid suhbr d r`stbr jds ‚auh`rbedr`s‘ (adhor` ebed bj aµuh`rd uh`rd sup`ridr e`j quòrbed, d bj e` jb izqui`reb e` jb eibmdabj). Jd hishd sukè` kda jb hujtipjikbkiµdda a y eivisiµdda a e` quòrbeds pdr àt`rds> sdjbhàt` s` bf`ktb `j auh`rbedr e`j quòrbed. @j prdeuktd e` =2 kda 6 ? `s imubj b 6= . @j kdkiàt` e` 27 eivieieds àtr` 9 `s ?; . @xbktbhàt` kdhd, si ?; hujtipjik hujt ipjikbrb brb 2 hbazba hbazbabb pdr 6, `j r`su r`sujtbed jtbed s`r s`rµµĲb 6 hbazb hbazbabs, abs, d si eiviei eiviei`rb `rb 27 hbazb hbazbabs abs àt àtr` r` 9, `j kdkià kdkiàt` t` s`r s`rµµĲb e` ? hbazba hbazbabs. bs. Suè` sukè`r qu` ;

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5= eurbat` jbs suhbs d hujtipjikbkida`s e` jds quòrbeds `j auh`rbedr r`sujt` tba mrbae` d kubaed hàds hµbs bs mrbae` qu` `j eàdhiabedr. Sdr `l`hpjd, à 2

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; ; suhb e` ; + ; + ; jb r`spu`stb `s ; , bsµ jb kdhd . ´Ruµ` ysim simaifl aiflkba kba 4 @vieàt`hàt`, si s` eivie` uab uaiebeĲ à tr`sàpbrt`s; imubj`s e`spuµ `s `s ; s` suhba jbs pbrt`s, vdjv`rµb b tà`r su uaiebe drimiabj. @a dtrbs pbjborbs, ;; `quivbjà b 2 àt`rd> y `stµb e` bku`red kda jb e`flaikiµdda a qu` g`hds g`kgd e` ua quòrbed, pu`std qu` ; ; 8 2. 2395=6 @a jb hishb fdrhb ?? , 55 , ?= d 2395=6 sda tdeds, `quivbjàt`s b 2. ?= S`rd, ´Ruµ` suk` sukè` e` si 2; `s hujtipjikbed pdr 5 pbrb qu` jb r`spu`stb s`b ;5 4 ´Kuµbbjj `s jb simaifl simaiflkbkiµ kbkiµdda a e` 5; 4 Ouàd, 5; puèà kdasie`rbrs` kdhd ;; + ;2 . Yb sbo`hds qu` ;; sda imubj`s b 2, pdr jd tbatd, 5; eòà s`r imubj imubj`s `s b 2 + ;2 y, à aeds` ‚uad ua t`rkid‘. `sb fdrhb, s` `skrio` 2 2; d 2 2/;, j`yµ`àeds` Mà`rbjhàt`, à jb `sku`jb ads às`˚ a aba ba b kbhoibr à àt`rds tbatd kdhd s`b pdsioj` e` ua quòrbed. Sdr `s` hdtivd, `stbhds gboitubeds b kbhoibr 5 b 2 , ?= b ? y bs bsµµĲ suk`siv suk`sivbhàt`. bhàt`. Ad `xist` uab rbzµdda a r`bj pbrb gbk`r ; ; 0 0 ? 2 5 ?= `std.2@a r`bjiebe, kij hba`lbr ; y 0 qu` 2 ; y 0 0 . @s sdjbhàt` `j 0 `s hµbbss fµbbkij kdas`rvbtishd abturbj e` ua huaed qu` gb tàied qu` bk`ptbr jbs frbkkida`s, gboiàed `stbed bkdstuhorbed b hba`lbr eurbat` b˚a ads ds u uaikbhàt` µaikbhàt` àt`rds. _`bjhàt`, jds quòrbeds kda vbjdr`s hàdr`s e` 2 (`std `s, kda `j auh`rbedr auhµ ``rikbhàt` rikbhàt` hàdr qu` `j eàdhiabedr) sd sda a jjbhbeds ‚quòrbeds prdpids‘. Bqu`jjds imubj`s d hbydr`s e` 2 (`s e`kir, bqu`jjds kda `j auh`rbedr imubj d hbydr qu` `j eàdhiabedr) sda jjbhbeds, jjb hbeds, ad dostbat`, kda eµ`òij oij e`sbprdobkiµ dda a ‚quòrbeds ihprdpids‘. [ia `hobrmd, `s ihpdrtbat` r`kdrebr qu` tdeds jds quòrbeds puèà hba`lbrs` e` imubj hba`rb. E`se` `j puatd e` vistb hbt`hµbbtikd, tikd, tdeds sda imubjhàt` ‚prdpids‘.

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@j eàdhiabedr s` uab bj lu`md [updambhds qu` kdasi [updambhds kdasie`rbh e`rbhds ds jb frbkk frbkkiµ iµda da 6; . @a vbjdr `s imubj b ?, kdhd pderµb v`rs` bj kdav`rtir 6; à 6 ;. Bgdrb oià, ´Ruµ ` suk` sukè` e` si huj hujtipjik tipjikbhds bhds tbat tbatdd `j auh auh`rbed `rbedrr kdhd `j 6 2? eàdhiabedr pdr ?, kbhoibaed ; b 6 4 @vieàt`hàt`, abeb j` sukè` bj vbjdr e`j quòrbed pu`std qu` 2? 6 simuà sim uà ssiàed iàed ?. D pd pder erµµĲbhds hujtipjikb hujtipjikbrr 6 27 `j auh`rbedr y `j eàdhiabedr e` ; pdr ; pbrb dotà`r 9 > d pdr ?= pbrb dotà`r 26? > d pdr 232 pbrb dotà`r 636 . @a tdeds jds kbsds, `j vbjdr e`j 72 ;3; quòrbed, kdhd si s` dotuvi`rb pdr eivisiµdda, a, simu` siàed ? y, pdr jd tbatd,

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57

µ KBS µ IU IUPJ PJD D 5. 5. A PH@_D[ RP@O_BED[

p`rhba`k` iatbktd. Jd hish hishdd bbkdat` kdat`k`rµ k`rµĲĲbb si guoiµ``rbhds rbh ds `hp `hp`zbe `zbedd kkda da 2?3 (imubj b ?) y eivi63 63 ;3 53 ;3 ) `rbhds `rbhds tbatd `j auh`rbedr eàdhiabedr (pbrb dotà`ràt`. deiµ àtr` ; (pbrb dotà`r ) d kdhd àtr` `j 5 (pbrb dotà`ràtr` ) y?bsµ Ĳ suk suk`sivbh `sivbhàt`. ?3 20 @a tdeds jds kbsds, `j vbjdr p`rhba`k`r p`rhba`k`rµµĲb imubj. Ai s`r s`rµµĲba eif`ràt eif`ràt`s `s jbs kdsbs si `stuviµ ``rbhds rbhds trbtbaed kda ua quòrbed qu` ad fu`rb imubj à vbjdr b ua àt`rd. [i ;2 `s hujtipjikbed sup`ridr ` iaf`ridrhàt` pdr ?, pbrb fdrhbr ?6 , ad gby kbhoid. [i s` eivieiµd ua pbst` pbst`jj à ; pbrt`s imubj` imubj`ss y s` tdhµ d uab pbrt`, d s` eivieiµd à 6 pbrt`s imubj`s y s` tdhbrda ?, s` t`rhiabrµb kda jb hishb kbatiebe e` pbst`j à kbeb dkbsiµdda. a. Kdas`kuàt Kdas`kuàt`hàt`, `hàt`, ;2 y 6? sda s da ieµ`àtikd at ikds. s. @atdak`s, jb r`mjb mà`rbj `s qu` puèà hujtipjikbrs` d eivieirs` `j auh`rbedr y `j eàdhiabedr e` kubjqui`r quòrbed pdr ua hishd aµu uh`rd h`rd sia qu` s` bjt`r` su vbjdr. @stb puè` s`r uab r`mjb µu utij tij à eiv`rsbs fdrhbs. Sdr `l`hpjd, p`rhit` pr`skiaeir bjmuabs v`k`s (ad si`hpr`) e` jds aµu uh`rds h`rds bjtds. Sdr `l`hpjd, `j quòrbed 2?6 puè` s`r eivieied sup`ridr ` iaf`ridrhàt`, 279 àtr` 6; y s` kdavi`rt` à `j fbhijibr ?; . Ubho Ub hoiiµà 200 eivieieds sup`ridr ` iaf`ridrhà iaf`ridrhàt` t` àtr` ;2, s` kdavi`rtà à ;2 0 qu` `quivbjà b 0, pu`std qu` 0 2 8 0. 2 @stb `s jb prih`rb v`z qu` hàkidad quòrbeds kda ua eàdhiabedr e`j 2 y µ``stds stds sda `xk`pki `xk`pkidabjhàt` dabjhàt` ihpd ihpdrtbat`s. rtbat`s. _`kuµ``re`s` re`s` qu` kubjqui kubjqui`r `r aµu uh`rd h`rd eivieied pdr 2 p`rhba`k` sia kbhoid bjmuad à su vbjdr. @std simaiflkb qu` kubjqui`r quòrbed kda eàdhiabedr e` 2 `s imubj à vbjdr b su auh`rbedr. @a `stb fdrhb, ?=; 8 ?=;> 0399; 8 0399; 0399;,, y bs bsµµĲ suk`si suk`sivbhàt`. vbhàt`. Sdr `stb rbzµdda, a, 2 2 ad tà`hds qu` eivieir jds aµu uh`rds h`rds kdhd quòrbeds, kdavirtiàed tdeds jds àt`rds à quòrbeds y pdaiàed bj hishd àt`rd kdhd auh`rbedr y b 2 kdhd eàdhiabedr. Jds quòrbeds kda eàdhiabedr e` 2 puèà hba`lbrs` à imubj fdrhb qu` jds e`hµbbs< s< 20 20 ;3 5 ; 2? + 8 > 8 2 2 2 2 2 2 @j j`ktdr pr`muatbrµb qu` kuµbbjj `s jb vàtb vàtblb lb e` bmr`mbr uab jµĲĲa`b a`b y ua eà eàddhiabedr bpbràt`hàt` iaa`k`sbrids b àt`rds p`rf`ktbhàt` sbtisfbktdrids. Jb r`spu`stb `s qu`, à `sb fdrhb s` p`rhit` b jds àt`rds y jds quòrbeds tdhbr pbrt` pb rt` kdaluatbh kdaluatbhàt` àt` y kda hbydr fbkiji fbkijiebe ebe à jjbs bs dp dp`rbkida `rbkida`s `s brithµ``titikbs. Sdr `l`hpjd, ´Kµddhd hd suhbr ;2 y 20 4 Gbstb bgdrb, sµddjd jd g`hds vàied suhbaed quòrbeds kda jds hishds eàdhiabedr`s, p`rd pbr`k` a`k`sbrid àkdatrbr

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59 jb fdrhb e` kdav`rtir `stds quòrbeds à dtrds qu` tàmba jds hishds eàdhiabedr`s. hiabed r`s. Bqu BquµµĲ `s edae` jb huj hujtipjikb tipjikbkiµ kiµda da sup`ridr ` iaf`ridr `s au`vbhàt` utij. u µtij.Srih`rd hujtipjikbhds 2 0, sup`ridr ` iaf`ridrhàt`. @std ads p`rhit` ; 0 dotà`r 20 si sia a kbhoibr `j vbjdr e`j quòrbed quòrbed.. E`sp E`spuµ uµ``s, s, hujtip hujtipjikbhds jikbhds 02 ;, à ; jb hishb fdrhb bat`ridr, pbrb dotà`r 20 , au`vbhàt` sia kbhoibr su vbjdr. 2 2 0 ; Sdr jd tbatd, ; + 0 sda imubj`s qu` 20 + 20 , y `stds u ujtihds µjtihds sda hbs fµbbkij`s kij`s 7 e` hba`lbr. Jb r`spu`stb, dovib, `s< 20 . Bgdrb oià, pde`hds s`muir `j hishd sist`hb à jbs suhbs qu` kdhpràeba tbatd àt`rds kdhd quòrbeds, si kbhoibhds jds àt`rds b quòrbeds e` ‚eàdhiabedr uad‘. Sbrb suhbr ; + 2; + 2 25 , prih`rd kdav`rtihds tded b quòrbeds< 2; + 2; + 05 . E`spuµ``s, s, hujtipjikbhds kbeb quòrbed sup`ridr ` iaf`ridrhàt`, à fdrhb tbj qu` t`rhia`hds kda quòrbeds e` imubj eàdhiabedr> `j prih`r quòrbed pdr 2?, `j s`muaed pdr 5 y `j t`rk`rd pdr ;. @std 5 20 ads eb< ;6 + 2? , qu` suhba 00 . 2? 2? 2? 00 puèà s`r kdav`rtieds [i `j j`ktdr e`sbpruòb jds quòrbeds ihprdpids, 2? 57 = à 2? + 2? , qu` suhba 5.

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Ruòrbeds obld k`rd Uded jd qu` gbstb bgdrb g` eikgd sdor` jds quòrbeds tià` p`rf`ktb vbjie`z à jds aµ u uh`rds h`r ds a`mb a`mbtivds tivds.. Bs BsµµĲ 2 ; puè` `skrioirs` ∟;2 y, 2 ; puè` 2 `skrioirs` ∟; . @a kubjqui`r kbsd (pu`std qu` a`mbtivd eivieied àtr` pdsitivd y vik`v`rsb eba kbeb uad kdkiàt` a`mbtivd) jb tdtbjiebe e`j quòrbed puè` kdasie`rbrs` a`mbtivb y sàkijjbhàt` `skrioir 2; . Bqu B quµµĲ `j ` j simad s imad e` h hàds àds s` bpjikb b tded `j quòrbed y ad à pbrtikujbr bj auh`rbedr d bj eàdhiabedr.

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∟; Sdr dtrd jbed, 2 ; puè` `skrioirs` y pu`std qu` a`mbtivd eivieied ∟2 àtr` a`mbtivd eb ua kdkiàt` pdsitivd, ∟ puè` kdasie`rbrs` kdhd + 2; . ; Jds quòrbeds a`mbtivds s` hba`lba à jb suhb y jb r`stb sdor` jbs hishbs obs`s qu` jds quòrbeds pdsitivds. Sdr `l`hpjd, ´Ruµ` simaiflk simaiflkbb 2 2 2; 4 Urbasfdrhbaed bhods b quòrbeds tà`hds 2; 5; . Sbrb dotà`r eàdhiabedr`s s`h`lbat`s, hujtipjikbhds 22 pdr ;, sup`r sup`ridr idr ` ia iaf`ridrhàt`. f`ridrhàt`. Bs BsµµĲ ddotà`hds otà`hds ;; 5; siàed jb r`spu`stb 2; .

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03

µ KBS µ IU IUPJ PJD D 5. 5. A PH@_D[ RP@O_BED[

KbpµĲtujd 0 Frbkkidabaed pdr ei`k`s @vitbaed jb eivisiµ da da Sdr aiamµ ua ua hdtivd g`hds t`rhiabed kda jds quòrbeds. Gbstb bgdrb sµddjd jd g`hds hujtipjikbed y eivied b jds quòrbeds pdr jds àt`rds. ´@s pdsioj` hujtipjikbr y eivieir quòrbeds pdr quòrbeds4 Y si bsµĲ `s, ´Kuµbbjj `s jb fdrhb fdrhb44 2 [updambhdss qu` pbrti [updambhd pbrtihds hds ; à eds pbrt`s imubj`s. Ur`s t`rkids, tdhbeds à kdaluatd, `quivbjà b 2 àt`rd. [i pbrtihds kbeb t`rkid à eds, s` a`k`sitbrµbba a 6 p`qu`˚a abs bs pbrt`s pbrb gbk`r ua àt`rd. Sdr jd tbatd, kbeb uab e` `stbs p`qu`˚ a abs bs pbrt`s `quivbj` b ua s`xtd y, pdr àe`, pde`hds e`kir qu` jb hitbe e` ua t`rkid `s ua s`xtd. [imuiàed `j hishd rbzdabhiàtd, jb hitbe e` ua kubrtd `s ua dktbvd y ua t`rkid e` ua kubrtd `s ua edk`bvd. Pab fdrhb kdavàiàt` e` sihodjizbr `std `s `skrioir<

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[` usb `j simad e` hujtipjikbkiµdda, a, pdrqu` pbr`k` qu` dotà`hds jb r`spu`stb kdrr`ktb à kbeb kbsd si hujtipjikbhds jds eds eàdhiabedr`s. [ia `hobrmd, ´Ruµ` eir`hds e` jds auh`rbedr`s4 @j 2 pbr`k` ad gbo`rs` tdkbed, p`rd, pdr dtrd jbed, 2 2 8 2. Jb r`spu`stb b jb pr`muatb `s< vbhds b àsbybr quòrbeds kda auh`rbedr`s eistiatds b 2. [updambhds qu` pbrtihds `j 23 à 0 pbrt`s imubj`s. Kbeb pbrt` `quivbj` b ua quiatd e` 23. Kdhd 23 0 8 ?, `s` quiatd e` ei`z `s imubj b ?. Jb frbs` ‚ua quiatd e` ei`z‘ puè` `skrioirs` 20 23 , p`rd vbhds b frbkkidabrjd. Srih`rd, 2 ? puè` kdav`rtirs` à 23 bj hujtipjikbrjd pdr ?, sup`ridr ` iaf`ridrhàt`, 0

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KBS µ IUPJD 0. F_BKKIDA F_BKKIDABAED BAED SD_ EI@K@[ EI@K@[

b fla e` kbh kbhoibr oibr `j auh auh`rbed `rbedrr e` 2. E`spu E`spuµµ``s, s, `j 23 puè` kdav kdav`rtir `rtirs` s` à 23 y hujtipjikbaed tbatd `j auh`rbedr kdhd `j eàdhiabedr pdr ;, puè` 2 ;3

trbasfdrhbrs` à ; . e`kir qu` 2 23 `s jd hishd qu` ? ;3 . [i hujtipjikb@atdak`s, pde`hds 0 23 ; hds `stds eds µu ujtihds jtihds quòrbeds, auh`rbedr pdr auh`rbedr y eàdhiabedr 63 pdr eàdhiabedr, dotà`hds ?23ÝÝ;3; d s`b ;3 , d ? àt`rds. @st` `s `j r`sujtbed qu` `sp`rµbbobhds. obhds. D supdambhds qu` à jb hujtipjikbkiµda da ;2 ?2 gbo gboµµĲbhds kdav`rtied jds 5 6 quòrbeds à 2? y 2? ‛kdhd pdeµĲĲbhds bhds gbo`rjd g`kgd‛ hujtipjikbaed `j pri5 6 h`rd sup`ridr ` iaf`ridr pdr 5 y `j s`muaed pdr 6. Bgdrb oià, 2? `s 2? ?5 imubj b 255 , si s`muihds `j sist`hb e` hujtipjikbkiµdda a e` auh`rbedr pdr au?5 h`rbedr y eàdhiabedr pdr eàdhiabedr. Eivieiàed jb r`spu`stb 255 ?5 2 (?5 ?5 8 2) y (255 ?5 8 6) dotà`hds 2 6 , qu` g`hds e`kieied `s `j r`sujtbed kdrr`ktd e` 2; 2? . Jb eivisiµdda a dp`rb à fdrhb s`h`lbat`. Jds auh`rbedr`s eòà s`r eivieieds pdr jds auh`rbedr`s y jds eàdhiabedr`s pdr jds eàdhiabedr`s. 0 23ý0 BsµĲ, 23 `s imubj b 22 d ;? . [ia `hobrmd, bqu bquµµĲ s` pr`sàtb uab kd kdhpjikbhpjikbý= ?2 = kiµdda. a. ´Ruµ` sukè` kuba kubaed ed bj eivi eivieir eir ua auh`rbe auh`rbedr dr pd pdrr dtrd d ua eà eàdhiabe dhiabedr dr pdr dtrd (d bhods) `j kdkiàt` ad eb uab kifrb k`rrbeb4 [updambhds qu` s` trbtb e` eivieir ?= pdr ?; . Ubatd `j auh`rbedr kdhd `j eàdhiabedr brrdlbrµbba a frbkkida`s> `xistir `xistirµµĲba frbkkida`s eàtrd e` jbs frbkkida`s. Bfdrtuabebhàt`, puè` `vitbrs` tbj eivisiµdda. a. Vdjvbhds b au`strd prdoj`hb e` frbkkidabr `j 23 à 0 pbrt`s imubj`s. Dotàer`hds jb hishb r`spu`stb, ?, oià s`b qu` `skriobhds 23 0, d 23 20 . Bgdrb oià, 0 puè` `skrioirs` 02 , y 02 , pdr e`kirjd bsµ bsµĲ, `s 02 bj bj r`vµ r` vµ`s. `s . Eds frbkkida`s s`h`lbat`s uab b jb dtrb, `xk`ptd qu` `j auh`rbedr e` uab `s `j eàdhiabedr e` dtrb y vik`v`rsb, s` eik` qu` sda ‚frbkkida`s d quòrbeds iav`rsds‘. iav` rsds‘. Jb pbjborb ‚iav ‚iav`rsd‘ `rsd‘ vià` e` ua tµ``rhiad rhiad jbtiad qu` simaiflkb ‚bj 0 2 kdatrbrid‘. Ki`rtbhàt` 2 sda 0 ‚bj kdatrbrid‘. Bs BsµµĲ, 0 `s `j iav` iav`rsd rsd e` 02 y 02 00 `s `j iav`rsd e` 0. Hµbbss bµ u ua, a, ?; sda jds iav`rsds e` ?; > 2?6 sda jds iav`rsds e` 2?6 , y bs bsµµĲ suk suk`sivbh `sivbhàt`. àt`. 00 @atdak`s, kubaed e`kihds qu` 23 0 eb `j hishd r`sujtbed qu` 23 02 , pbr`k` qu` fu`rb imubj eivieir pdr ua aµu uh`rd h`rd qu` hujtipjikbr pdr su iav`rsd. (Uµ`àmbs` ambs` pr`sàt` qu`, à `st` prd prdkèihiàt kèihiàtd, d, `j iav` iav`rsd rsd `s si`hpr` `j eivisdr y auakb `j eivieàed.) Srdo`hds dtrd kbsd. Brriob bkbod e` hbaif`stbr qu` 23 0 23 = `rb imubj b ?; . [updambhds qu`, à v`z e` `std, tà`hds ?2 . Jb ?2 = 0 =3 r`spu`stbb s`r r`spu`st s`rµµĲb e` 230 . Eivieiàed `sb frbkkiµdda a ‛sup`ridr ` iaf`ridrhàt`‛ ? àtr` ;0, dotà`hds ; d s`b jb hishb r`spu`stb.

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0; Bgdrb pde`hds eivieir 0= àtr` ?; , sia `j p`jimrd e` jbs frbkkida`s eàtrd 20 e` jbs frbkkida`s, hujtipjikbaed 0= ;? . @j r`sujtbed `s 25 .

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Hµbbss bµ u ua, bj hujtipjikbr frbkkida`s, eò`hds r`kdrebr qu` ad tià` ihpdrtbakib `ja,dreà à qu` s` gbmba jbs dp`rbkida`s. Sdr `l`hpjd, hujtipjikbr 23 = = `s jd hishd qu` hujtipjikbr 23 . @a `j prih`r kbsd, jb dp`rbkiµda da ?2 0 0 ?2 23Ý= 23Ý= =3 `s ?2Ý0 y à `j s`muaed 0Ý?2 . @a kubjqui`rb e` jds kbsds, dotà`hds 230 , d (eivieiàed auh`rbedr`s y eàdhiabedr`s àtr` ;0) ;? . 23 S`rd `xist` uab vàtblb à jb s`muaeb dp`rbkiµdda. a. Jds quòrbeds ?2 y 0= ad puèà s`r rèukieds à fdrhb hµbbss sàkijjb, p`rd 23 puèà kdav`rtirs`, 0 ? = b sihpj` vistb, à 2 y ?2 qu`, kdhd fµbbkijhàt` kijhàt` s` v`, `quivbjà b ;2 . = @j prdoj`hb e` 23 `s kbhoibed b ?2 2; , dot` dotàià aiàed ed bsµ bsµĲ ;? . 0 ?2 Jb utijiebe e` trboblbr, si`hpr` qu` s`b pdsioj`, kda auh`rds hbs p`qu`˚a ads, ds, kdaeuk` b jb rutiab e` jb eivisiµdda a sup`ridr ` iaf`ridr e` jds quòrbeds, sia siqui`rb hdj`stbrs` à kbhoibr `j dreà e` jds hishds. Bs BsµµĲ, à `j prdoj`-

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2=

Ý

hb 23 59 , `j auh`rbedr e` uad e` jds quòrbeds y `j eàdhiabedr e`j dtrd 2 2= sda eivieieds àtr` =, kda jd kubj quèb kbhoibed `j prdoj`hb b 23 . @a = 2= kubjqui`r kbsd, jb r`spu`stb `s jb hishb< =3 , p`rd `s hµbbss fµbbkij kij dotà`rjb pdr hèid e` jb s`muaeb v`rsiµdda. a. Jb kdav kdavàiàkib àiàkib e` rèukir ‚b jds tµ`rhiads `rhiads hµbbss oblds‘ d ouskbr jds ‚fbktdr`s‘ kubaed s` hujtipjikba jds quòrbeds, kdaeuk` bj kdaflbed `stueibat` b iatàtbr `j hishd breie bj `f`ktubr suhbs e` quòrbeds. Abeb hµbs qu` à = 2 ad `s jb hishb qu` jb e` 23 jb suhb ad dp`rb. Jb suhb e` 23 + 2= + 2= . Jb 59 = 02; 2?;9 prih`rb suhb jj`mb b 593 y jb s`muaeb b 593 . Jb eiflkujtbe `striob bqu bquµµĲ à qu`, bat`s e` qu` puèb uad gbk`r kubjqui`r kdsb, à jb suhb e` quòrbeds, eò` imubjbr jds eàdhiabedr`s. @a `j kb= sd e` 23 + 2= , puè` gbk`rs` hujtipjikbaed `j prih`r quòrbed ‛sup`ridr ` 59

Ý

;5;

2=3

iaf`ridrhàt`‛ pdr 59 y `j s`muaed pdr 23, b fla e` dotà`r 593 + 593 . Pab v`z qu` s` gbyb g`kgd `std, s` v`rµb jb iautijiebe e` ouskbr fbktdr`s pdrqu` `j r`sujtbed s`rµb qu` jds eàdhia eàdhiabedr`s bedr`s quèbrµbba a au`vbhàt` fu` fu`rb rb e` jµĲa`b. Sdr jd tbatd, à jb suhb e` quòrbeds, gby qu` djviebrs` e` jds fbktdr`s.

Fdrzbaed jds quòrbeds b bjia`brs` [ia `hobrmd, eò` behitirs` qu` gby bjmd qu` ad `s huy ouàd à jds a rdhp` quòrbeds. Yb s`b qu` ‚uad y hèid‘ s` `skriob 2 2/?, d 2 2? , jb frbkkiµdda jb kdrriàt` uaifdrh` y jb g`rhdsb jµddmikb mikb e` jb badtbkiµda da e` pdsikida`s.

05

KBS µ IUPJD 0. F_BKKIDA F_BKKIDABAED BAED SD_ EI@K@[ EI@K@[

@j aµ u uh`rd h`rd ;275 ;5 simaiflkb, e` bku`red kda au`strds vbjdr`s e` pdsikida`s, ; ‚hij`s‘ hµbs bs 2 ‚kiàtd‘ hµbbss 7 ‚edkàbs‘ hµbbss 5 ‚uaiebe`s‘ hµbbss ; ‚kubrtds e` uaiebe‘. adsuaàkdatrbhds kda `sb da da g`hds e`lbed qu` kbebGbstb jumbrqu` tàmb vbjdr e`kupjikbed kdahis`rboj` r`sp`ktd frbkkiµ bj jumbr b su izqui`reb. ´Sdr quµ` ad pbsbhds `j jumbr e` jbs ‚uaiebe`s‘4 2 2 2 @a dtrbs pbjborbs, 2333 23 8 233> 23 2333 23 8 23 y 23 23 8 2. Gbstb 2 2 bq bqu uµĲ `sd `s `stµ tµb huy oià, p`rd, ´Sdr quµ` ad kdatiaubr kd kdhd hd si simu`< mu`< 2 23 8 23 > 2 2 2 2 2 2 8 233 > 233 23 8 2333 , y bs bsµµĲ suk` suk`sivbhàt`4 sivbhàt`4 @a `stb fdrhb, usbaed ua 23 23 sist`hb kdastbat`hàt`, pde`hds pbsbr jb pdsikiµdda a e` jbs ‚uaiebe`s‘, kdavi virtiµ rtiµ`àedj ae djbs bs à ‚eµ``kihd ki hds‘ s‘,, ‚k` ‚kàtµ atµ``sihd si hds‘ s‘,, ‚h ‚hijµ ijµ`sihd `si hds‘ s‘,, y bsµĲ suk`s su k`sivbh ivbhàt àt`. `. 2 Vbhds b kdasie`rbr jb frbkkiµda da ? . Hujtipjikbaed tbatd `j auh`rbedr kdhd 0 `j eàdhiabedr pdr p dr 0, r`sujtb qu` pde`hds `xpr`sbrjb tbhoiµ`à a eikiàed 23 0 d b 00.0, j`yµ`àeds` aeds` `st` Pa aµ u uh`rd h`rd kdhd 00 2? puè` kbhoibrs` b 00 23 ujtihd u µjtihd kdhd ‚kiakuàtb y kiakd puatd kiakd‘, kdjdkµbaeds` baeds` `j puatd iahèibtbhàt eibtbh àt`` e`spu e`spuµµ``ss e`j jumbr e` jbs uaiebe uaiebe`s, `s, pbrb s`pbr s`pbrbr br jds àt àt`rds `rds e` jbs frbkkida`s. @atdak`s, jb situbkiµdda a e` jds aµ u uh`rds h`rds puè` j``rs` kdhd 0 ‚e`kàbs‘ hµbbss ‚uaiebe`s‘ hµbs bs 0 ‚eµ``kihds‘ kih ds‘.. ; Jb frbkkiµda da 5 puè` s`r kdav`rtieb ‛hujtipjikbaed tbatd `j auh`rbedr =0 =3 0 = 0 kdhd `j eàdhiabedr eàdhiabedr pdr ?0‛ à 233 . @stds sda imubj`s b 233 + 233 d b 23 + 233 . ; Sdr jd tbatd, `j aµ uh`rd uh`rd 00 5 puè` `skrioirs` kdhd 00.=0 (0 ‚e`kàbs‘ hµbs bs 0 ‚uaiebe`s‘ hµbs bs = ‚eµ``kihds‘ kih ds‘ hµbs bs 0 ‚kàtµ``sihds si hds‘). ‘). Jbs frbkkida` frbkkida`ss à fdrhb e` ‚eµ``kihds‘, kihds‘, ‚kàtµ``sihds‘ sihds ‘ d ‚hijµ``sihds‘, sihds‘ , sda jjbhbebs frbkkida`s e`kihbj`s (e` uab pbjborb jbtiab pbrb ‚ei`z‘). Kubaed jbs frbkkida`s e`kihbj`s `stµba ba uaiebs b uab kifrb, kdhd r`sujtbed e` uab dp`rbkiµdda a brithµ``tikb, tikb, jds prdeuktds kdhd 00.0 y 00.=0 sda jjbhbeds e`kihbj`s y `j puatd usbed pbrb s`pbrbr jds àt`rds e` jbs frbkkida`s `s `j ‚puatd e`kihbj‘. = , eò` `skrioirs` Pab frbkkiµda da e`kihbj qu` s`b hàdr qu` uad, tbj kdhd 23 si`hpr` kdhd .= pdaiàed b jb e`r`kgb e` jb hishb `j puatd e`kihbj. @j p`jimrd e` p p`re`r `re`r `j jumbr e`j puatd e`kihbj `s kdasie`rboj` y jb `quivdkbkiµda da e` pda`r .= à v`z e` = puè` prdeukir ua mrba `rrdr. [` bkdstuhorb, pdr `stb rbzµdda, a, `skrioir .= kdhd 3.= (k`rd ‚uaiebe`s‘ hµbbss = ‚‚eµ eµ``kihds‘ kih ds‘,, qu` ``ss jd h hishd ishd qu` e`kir e `kir ssihpj`h` ihpj`hàt` at` = ‚eµ``kihds‘) kihds‘ ) kda `j u uaikd µ aikd dol`td e` prdt`m`r `j puatd = e`kihbj e`ki hbj.. SderµĲb Ĳb uad `skr `skrioir ioir tbhoiµ`à a 23 kdhd 3.=3, 3.=33, d 3.=33333333. @j buhàtd e` 3 ‚kàtµ``sihds‘ sihds‘ hµbbss 3 ‚hijµ``sihds‘, sihds‘, ad kbhoib `j vbjdr drimiabj auhµ`rikd `ri kd e` 3. 3.=. =. Jb mrba vàtblb e`j sist`hb e`kihbj kdasist` à qu`, bj suhbr y r`stbr, uad puè` djviebrs` e` jbs frbkkida`s y dp`rbr kdhd si sdjbhàt` s` trbtbrb e`

Ý

Ý

Ý

Ý

Ý

Ý

00 àt`rds. @a `j µbbobkd, obkd, pdr `l`hpjd, ad s` a`k`sitb gbk`r qu` `j obrrdt` iaf`ridr s`b `j e` jbs ‚uaiebe`s‘. Suè` gbk`rs` qu` `j obrrdt` e` à hèid s`b `j kdrr`spdaeiàt` b jbs uaiebe`s, e`lbaed qu` jds qu` simuà gbkib brriob s`ba ‚e`kàbs‘, ‚kàtàbs‘, ‚hijjbr`s‘ y bsµ bsobrrdt`s µĲ suk`siv suk`sivbhàt`, bhàt`, hiàtrbs qu` jds qu` vba gbkib bob bobld ld s`ba ‚eµ``kihds‘, kihds‘, ‚kàtµ``sihds‘, sihds‘ , ‚hijµ``sihds‘, sihds ‘, y bs bsµµĲ ssuk`sivbhàt`. uk`sivbhàt`. Jbs dp`r dp`rbkida`s bkida`s à ``jj µbbobkd obkd kdhµ u ua a y kdrriàt` trboblbrµba ba oià s`b à hijjbr`s d à dp`rbkida`s kda pbp`j y jµbpiz. bpiz.

06

KBS µ IUPJD 0. F_BKKIDA F_BKKIDABAED BAED SD_ EI@K@[ EI@K@[

[updambhds qu` e`s`bhds suhbr 2 2? + 2 ;5 , kdas`rvbaed jbs frbkkida`s. [updambhds Srih`rd, kbhoibr kbhoibrµµĲbhds jds aµu uh`rds h`rds b ;? + 5= , e`s e`spuµ puµ`s `s b 56 + 5= , pbrb dotà`r 2; 5

2

5 , S`rd jds kubj`s kbhoibr kbhoibrµ µĲbhds pdst pdst`ridrhàt` b; . supµdambs` dambs` qu`, à v`z e``ridrhàt` `sd, usbhds e`kihbj`s. @j

à 2.0 y `j

2

; 5

2

2 ?

s` kdavi`rt`

b 2.=0. Jds suhbhds kdhd simu`< 2.03 + 2.=0 ;.?0

(Beviµ``rtbs` (Beviµ rtbs` qu` g` `skr `skritd itd 2.0 kdhd 2.03 kda do dol`td l`td e` tà` tà`rr bjmd à jb kdjuhab e` jds kàt kàtµµ``sihds, sihds, pu`std qu` gby ua aµu uh`rd h`rd à `sb kdjuhab kdrr`spdaeiàt` bj dtrd e`kihbj.) @j e`lbr fu`rb `s` 3 buhàtbr buhàtbrµµĲb jb pdsioijiebe e` qu` jds priakipibat`s `skrioi`rba< 2.03 +2.=0 jd kubj dkbs dkbsidabrµ idabrµĲĲbb eiflkujt eiflkujtbe`s. be`s. ? 0 Jb r`spu`stb b jb v`rsiµdda a e`kihbj `s ;.?0, jb kubj `s ; hµbs bs 23 hµbs bs 233 . 2 @std, pdr si e`s`b `j j`ktdr v`riflkbrjd, `s imubj b ; 5 , r`spu`stb qu` sbo`hds `s kdrr`ktb. @a r`bjiebe ad gby a`k`siebe e` `stbr kbhoibaed e` quòrbeds b e`kihbj`s. Pab v`z qu` jb badtbkiµdda a e`kihbj s` fllb à jb hàt`, `s pdsioj` trboblbr tdtbjhàt` kda e`kihbj`s y sàtirs` b mustd kda `jjds.

Btiads y e`sbtiads adrt`bh`rikbads Pa `l`hpjd e`j kµddhded hd ed usd e`j hµ``tded tde d e`kihb e`kihbjj s` àk àkuàtrb uàtrb à `j sist`hb hda`tbrid adrt`bh`rikbad. Jbs hdaèbs sda e`kihbj`s pdr e`kihbj`s pdr abturbj`zb, abturbj `zb, yb qu` 23 hijµ``sihds sihds `quivbjà b 2 kàtbvd> 23 kàtbvds `quivbjà b ua eµ``kihd kih d e` eµddjbr> jbr> (2 eih`), 23 eµ``kihds kihds (ei`z eih`s) `quivbjà b 2 eµddjbr. jbr. (Kda tdeb td eb s`muriebe pde`hds bflrhbr qu` auakb s` gba usbed jds hijµ``sihds sihds 2 y jbs µbmuijbs, bmuijbs, p`rd s` gb kdas`rvbed `j priakipid e` j`y). 2

@j tµ`rhiad `rhi ad ‚µbbmuijb‘ muijb‘ s` bpjikbob à jds @stbeds Paieds `sp`kµ `sp`kµĲflkbhàt` b jb j b hdaèb e` drd e` 23 edjbr`s. Dotuvd su adhor` e`j g`kgd e` qu` `j µbmuijb, bmuijb, `hoj`hb fbhijibr e` jds @stbeds Paieds, bpbr`zkb b su r`v`rsd. (@a fdrhb s`h`lbat`, jb jiorb drd bku˚abeb abeb

0= Sdr jb bat`ridr `xpjikbkiµdda, a, si`hpr` pde`hds r`f`rirads bj eia`rd à e`kihbj`s. @l`hpjiflkbaed, si uad tià` 2;.?6 eµddjbr`s jbr`s à su odjsijjd, à tdtbj, µ`stds èds stdseµ``kihds puèà `stbr eivieieds e` ei`z ddjbr`s, ; oijj`t`s jbr, kihds e` edjbr (eih`s)ày 26 oijj`t` kàtµ``sihds sihds eèµ eµ djbr djbr`s, jbr (kàtb (kàtbvds). vds). e` 2 eµddjbr, Kjbrd qu` puè` ad tà`rjds à `sb fdrhb, siad tà`r, à jumbr e` `sd j oijj`t` e` 2 eµddjbr, jbr, 0 hdaèbs e` hèid eµddjbr, jbr, 9 kubrtds e` eµdjbr dj br,, 5 eµ``kihd ki hdss (eih`s), ? aµĲqu`j`s (3.30 kàtb kàtbvds) vds) y 2 kàtb kàtbvd(pàay). vd(pàay). [ia `hobrmd, `stbs hdaèbs frbkkidabribs si`hpr` s` `skrioà kdafdrh` bj sist`hb e`kihbj. Auakb s` `skrio` 2? eµddjbr, jbr, siad 3.03. @a imubj fdrhb, ua kubrtd e` eµddjbr jbr s` `skrio` 3.?0> ua eµ``kihd kihd (eih`), 3.23> ua aµĲqu`j, 3.30 y ua kàtbvd (p` (pàay) aay) 3. 3.32. 32. Kdhd bjt`rabtivb, jbs hdaèbs puèà `xpr`sbrs` kdhd 2, 0, 23, ?0 y 03, p`rd s` kdas`rvb `j sist`hb e`kihbj. @stbhds tba bkdstuhorbeds b `st` sist`hb, qu` auakb ads pda`hds b pàsbr à jbs hukgbs vàtblbs qu` tià`. [ia `hobrmd, r`﬏`xida`hds eurbat` ua hdhàtd sdor` `j sist`hb hda`tbrid oritµbbaikd, aikd, à `j kubj 5 kubrtds e` pàiqu` gbkà 2 pàiqu` (pàay), 2? pàiqu`s (pàk`), ua kg`j kg`jµµĲa (skgijjiam)> y 23 kg`jia`s, uab jiorb. Ki`rtbhàt`, ua iamjµ``ss s` àfràt àfràtbb b uab breub tbr`b pbrb suhbr 5 jiorbs, 7 kg`jia`s, ? pàiqu`s> hµbbss 20 jiorbs, 29 kg`jia`s y 22 pàiqu`s. pàiqu` s. (@j r`suj r`sujtbed tbed `s ?3 jiorbs jiorbs,, 7 kg` kg`jia`s jia`s y 2 pàiqu` pàiqu`>> p`rd e`lbr e`lbrµµ` qu` `j j`ktdr trbt` e` àkdatrbr kdhd s` jj`mµd bj hishd). @a r`bjiebe, ua ldvà iamjµ``ss eèikb ua ti`hp ti`hpdd kdasie`rboj` b bpràe`r b suhbr jbs kbatiebe`s e` eia`rd, kdatrbribhàt` bj ldvà adrt`bh`rikbad, d bj e` dtrbs abkidabjiebe`s edae` pr`vbj`k` `j sist`hb hda`tbrid e`kihbj, qu` ad r`qui`r` `hpj`br tbatd ti`hpd pbrb `s` dol`td. Uba prdatd kdhd bpràeà brithµ``tikb, tikb, puèà hba`lbr `j sis sist`hb t`hb h hda`tbrid da`tbrid.. [ia `hobrmd, jds @stbeds Paieds y jb Mrba Or`tb˚a abb kdas`rvba bµu ua a `j e`sdreàbed `xtr`hd e` jb obtutb à jd qu` b hèiebs kdhua`s s` r`fl`r`. @j sist`hb adrhbj e` hèiebs qu` s` usb à tded `j huaed, `xk`ptd à jds pb pbµµĲs`s e` g gbojb bojb iamj`s iamj`sb, b, `s `j ‚‚sist`hb sist`hb hµ`trikd‘, `trikd‘, qu` fu` iavàtbed à Frbakib pdr Iamjbt`rrb `s jjbhbeb ‚ua sdo`rbad‘ eòied b qu` bpbr`k` à jb hishb jb kbo`zb e`j hdabrkb iamjµ`s.) `s.) Gboiµ`àeds` aeds` r`tirbed jb kirkujb kirkujbkiµ kiµdda a jbs hdaèbs e` drd à jds @stbeds Paieds, s` gb djviebed `j tµ``rhiad rhiad ‚µbbmuijb‘, muijb‘, p`rd à `j pbsbed, jb hdaèb e` drd e` ?3 eµddjbr`s jbr`s `rb ua ‚µbmuijb bmuijb edoj`‘, jb e` 0 eµddjbr`s, jbr`s, uab ‚hèib µbbmuijb‘ muijb‘ y jb e` ?.03 eµdjbr`s, djbr`s, ua ‚kubrtd e` µbmuijb‘. bmuijb‘. @j sist`hb hda`tbrid adrt`bh`rikbad b obs` e` e`kihbj`s, puè` kdatiaubr gbkib brriob si usbhds bjmuads e` jds tµ``rhiads rhiads vujmbr`s kda jds kubj`s s` e`simaba jds oijj`t`s e` hbydr vbjdr. Sdr `l`hpjd, ei`z µbmuijbs bmuijbs `quivbjà b ua oijj`t` e` 233 eµddjbr`s jbr`s y ei`z oijj`t`s e` 233 edjbr`s b ua ‚mrbae`‘, d s`b ua oijj`t` e` 2333 eµdjbr`s.

07

KBS µ IUPJD 0. F_BKKIDA F_BKKIDABAED BAED SD_ EI@K@[ EI@K@[

à 2=92. @j sist`hb hµ``trikd trikd `s e`kihbj. Sbrb pda`r ua `l`hpjd, pd pde`hds e`hds kdasie`rbr uaiebe uaiebe`s `s e` jdamit jdamitue. ue. uaiebey hµ tri kb e`tdhb jdamit jdamitue ueadhor` `s `j ‚h`tr ‚h`trd‘ d‘ (qu`jbs tià` ;9.;= pujmbebs e` Jb jdamitue e```trikb edae` su `j sist`hb.) Ei`z h`trds `quivbjà b ua ‚e`kµbbh`trd‘, h`trd‘, ei`z e`kµbbh`trds h`trds `quivbjà b ua ‚g`ktµddh`trd‘ h`trd‘ y ei`z g`ktµdh`trds dh`trds sda ua ‚nijµddh`trd‘. h`trd‘. @a fdrhb e`skàeàt`, ua eµ``kihd kihd e` h`trd `s ua ‚e`kµ ‚e`kµĲh`trd,‘ ua eµ``kihd kihd e` µ``st` st` `s ua ? ‚k ‚kàtµ àtµĲĲh`tr h`trd‘ d‘,, y ua eµ``kihd ki hd e` µ``st` st ` `s ua ‚h ‚hijµ ijµĲh`tr Ĳh`trd‘ d‘.. @std simaiflkb qu` bjmd qu` tià` ? nijµddh`trds, h`trds, 0 g`ktdh`trds, 2 e`kµbbh`h`trd, = h`trds, 7 e`k e`kµµĲh`trds, ? kàt kàtµµĲh`trds y 9 hij hijµµĲh`trds e` jdamit jdamitue, ue, hie` ?02=.7?99 h`tr ?02=.7? h`trds. ds. Bs BsµµĲ e` sàki sàkijjd jjd `s rùai rùairjds. rjds. [i tià` uad eds dol`tds, uad e` jds jd s kub kubj`s j`s h hie` ie` ? h`trds 7 e`kµĲĲh`trds h`trds y 9 kàt kàtµµĲh`trds y `j dtrd 0 h`t h`trds rds 0 e`kµµĲh`trds y 0 kàt e`k kàtµµĲh`trds Ĳh`trds,, jb j b jdamit jdamitue ue kdhoiabeb e` bhods `s e` ?.79+0.00, d s`b 7.55 h`trds, pueiµ`àeds` aeds` j``r tbhoiµ`à a kdhd 7 h`trds 5 e`k e`kµµĲh`trds y 5 kàtµµĲh`trds (d, sihpj`hàt`, kàt sihpj`h àt`, 7 h`trds 55 kàt kàtµµĲh`trds, kdhd s` pr`fl`rb.) Kdhpµbbr`s` r`s` `std kda `j sist`hb oritµbbaikd aikd y adrt`bh`rikbad e` hèikiµda da e` jdamitue`s, `hp`zbaed pdr jb pujmbeb, kdhd vbhds b e`hdstrbrjd. 2? pujmbebs 8 2 pi` ; pi`s 8 2 ybreb 2 breb ebss 8 2 pµ`rti ` rtikb kb (26 2? pi`s) 0 ? ybr 53 pµ``rti rtikb kbss 8 2 `stb `stbei eidd (2 27 e` hijjb) 7 `stbeids 8 2 hijjb @vieàt`hàt`, `std `s e`hbsibed kdhpjikbed. @a jb prµbbktikb ktikb auakb s` usba jb pµ``rtikb rtikb y jds `stbeids. @a v`z e` `jjds s` supda` qu` 2=63 ybrebs ( 2 0 ? 7) gbkà uab hijjb.

Ý

S`rd Jb bu bua ar`spu`stb bbssµĲ, ´Ku ´Kuµµbbatd atd5suhba hijjb y 26;? ybrebs, hµbbssjb?dotuv` hijjbs kda 705 ybrebs4 `s< hijjbs 2=?6 ybrebs, p`rd, ´Kµddhd hd y kµddhd hd pd pder erµµĲb ``jj j` j`ktdr ktdr r`sdjv`r `j p prdoj`hb rdoj`hb44 Ptijizbaed uaiebe`s hµbbss p`qu`˚ a abs, bs, ´Kuµbbatd atd suhba ; ybrebs, ? pi`s 7 pujmbebs, y 0 ybrebs, ? pi`s, = pujmbebs4 _`spu`stb< 9 ybrebs ? pi`s ; pujmbebs. ´Kµddhd hd s` dotuvd4 ?

Jb pbjborb ‚h`trd‘ (hµ`tr` `tr` à frbakµ`s) `s) vià` e`j jbt jbtµµĲa ‚h`truh‘ qu` simaifl simaiflkb kb ‚hèir‘. Jds pr`fllds pbrb jds hµ u ujtipjds jtipjds e`j h`trd viàà e` pbjborbs mri`mbs, hiàtrbs qu` jds kdrr`spdaeiàt`s b jbs suoeivisida`s e`j h`trd viàà e` pbjborbs jbtiabs. Sdr tbatd, jds pr`fllds ‚e`kb-‘, ‚g`ktd-‘ y ‚nijd-‘ viàà e` jbs pbjborbs mri`mbs ‚e`nb‘ (ei`z). ‚g`nbtda‘ (kià), y ‚nijd‘ (hij). Jds pr`fllds ‚e`ki-‘ ‚kàti-‘, y ‚hiji-‘, viàà e` jbs jbtiabs ‚e`k`h‘ (ei`z). ‚kàtuh‘ (ki`h) y ‚hijj`‘ (hij).

09 Jds ai˚ a ads ds adrt`bh`rikbads tiàà qu` eèikbr mrba kbatiebe e` ti`hpd bj bpràe`r kdhd hba`lbr tbj`s uaiebe`s. Uiàà qu` bpràe`r, bsihishd, y àt àtàe àeµµ``rs`jbs rs`jbs kdauab jbseùaiebe uaiebe`s `s e`tià` vdj vdjuhà, uhà, p`sdvbrièbe y sup`rflk sup`rflki`, i`, y bsµ bsµĲ suk`sivbhàt`, kbeb jbs kubj`s su prdpib trbeikidabj e` e`sbtiad. Mà`rbjhàt`, auakb bpràeà kdhpj`tbhàt` tbj hba`ld. @a kbhoid, jds ai˚a ads ds sdviµ``tikds tik ds (jb Pai Paiµµdda a [dviµ``tikb tik b gb bed bedptb ptbed ed `j sist`hb sis t`hb hµ``trikd) tri kd) ad tiàà aiamuab eiflkujtbe. Hba`lba tded tipd e` uaiebe`s pdr hèid e` suhbs kdhua`s y kdrriàt`s. ´Sdrr quµ ´Sd ` s` bp`mba jds adrt` adrt`bh`ri bh`rikbads kbads b `st` sist`hb e` hèie hèiebs bs tba kdhpjikbed, à v`z e` bedptbr `j sist`hb hµ``trikd4 trikd4 @a pbrt`, p pdrqu` drqu` simaiflkbrµĲb Ĳb uab mrb mrba a iav`rsi i av`rsiµµdda a iaikibj, yb qu` td tdeds eds jds iastruhàtds y g`rrrbhià g`rrrbhiàtbs tbs tàerµµĲba qu` e`s`kgbrs` y rèis` tàer rèis`˚ a abrs` ˚ brs` pbrb bebptbrjds b jbs au`vbs hèiebs, p`rd, priakipbjhàt`, pdrqu` s` r`sp`tb jb trbeikiµda. da. Jbs màt`s `stµbba a bkdstuhorbebs b jds hµ``tdeds tdeds batimuds y jds kbhoiba sdjbhàt` b r`mb˚a abeiàt`s. beiàt`s. @a uaa kdhd µ``st`, st`, Paieds tàer tàerµµĲbatbhoiµ qu` `à s`r fdrzbebs `j Mdoi`rad, y jb Mrba Or`tb˚ abbkbsd y jds @stbeds a tiàà uabpdr trbeikiµ dda< a< ad fdrzbr b jbs p`rsdabs. @atr` pbrµ`àt`sis, at`sis, jds kiàt kiàtµµĲflkds adrt`bh`r adrt`bh`rikbads ikbads y oritµbbaikds, aikds, qu` vbjµ u uba ba jb sihpjikiebe e` dp`rbkiµdda a bua sdor` jb kdhdeieb kdhdeiebe e e` uab rutiab, gba bedptbed uaifdrh` uaifdrh`hàt` hàt` ``jj sist`hb hµ`trikd. `trikd. @a r`bjieb r`bjiebe, e, jds kiàt kiàtµµĲflkds adrt`bh`rikbads jd usba kda fr`kuàkib à fdrhb kbsi irr`v`ràt`. Sdr `l`hpjd, jds qu` trboblba pbrb `j mdoi`rad y tiàà bjmuabs v`k`s qu` trbtbr kda mrbae`s kifrbs e` eia`rd, s` r`fl`rà ldkdsbhàt` b hij eµddjbr`s jbr`s kdhd ‚Nijdeµddjbr`s‘ jbr`s‘ (Nijdoukn). (Oukn, kdhd s` sbo`, `s jb fdrhb vujmbr e` e`simabr bj eµddjbr, jbr, tbj v`z pdrqu` ‚oukn‘ `rb jb `xpr`siµdda a vujmbr pbrb e`simabr b uab flkgb e` pµddqu`r). qu`r). Imubjhàt`, s` r`fl`rà b ua hijjµdda a e` eµddjbr`s jbr`s kdhd ‚h`mbeµddjbr`s‘, jbr`s‘, pu`st pu`stdd qu` `j pr`flld ‚h`mb-‘ (e`j mri`md ‚h`mbs‘, qu` simaiflkb ‚mrbae`‘) `s usbed à `j sist`hb hµ``trikd trikd pbrb eàdtbr ua hijjµdda a e` bjmd.

Jdkbjizbkiµ da da e`j puatd e`kihbj Gbstb bquµ bquµĲ `j sist` sist`hb hb e`kih e`kihbj bj puè` pbr`k`r kdhd `j ki`jd à jb ti`rr ti`rrb, b, kdhpbrbed kda jbs frbkkida`s dreiabribs> p`rd, `f`ktivbhàt`, kdhd tdeds jds ki`jds à jb ti`rrb, tià` sus e`svàtblbs. Sdr `l`hpjd, si`hpr` `xist` `j prdoj`hb e` pda`r `j puatd e`kihbj à jb pdsikiµdda a kdrr`ktb. Kdhd `l`hpjd, kdasie`r`hds `j prdoj`hb 3.? 3.? Pad pderµĲb Ĳb trbtbr e` r`sdjv`r `stb hujtipjikbkiµ dda a rbzdabaed kdhd simu`<

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KBS µ IUPJD 0. F_BKKIDA F_BKKIDABAED BAED SD_ EI@K@[ EI@K@[

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? + ? 8 5 y ? ? 8 5> pdr jd tbatd, kdhd 3 .? + 3.? 8 3.5, ád eòà s`r 3.? 3.? 8 3.54

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Ouàd, (qu` pu`s yb ad g`hds eòà s`rjd, y vbhds b v`r kuµbbj jb rbzµdda. a. [i quòrbeds bpràeied b hba`lbr), `jj `s e`kihbj 3.? s`kbhoibhds kdavi`rt` àb ? Bgdrb oià si hujtipjikbhds auh`rbedr y eàdhiabedr e` bhods aµu uh`rds, h`rds, 23 ? ? 5 5 tà`hds 23 23 8 233 > y 233 à e`kihbj`s `s imubj b 3.35. @a kdas`kuàkib, 3.? 3.? ad `s imubj b 3.5, siad b 3.35. Sde`hds àsbybr dtrbs hujtipjikbkida`s e` e`kihbj`s, v`riflkbaed jds r`sujtbeds pdr hèid e` jds quòrbeds `quivbjàt`s y kdaeukirµbba, a, pdr `l`hpjd, b qu` 3.7? 3.?2 8 3.2=??, hiàtrbs qu` 3.7? ?.2 8 2.=?? =??(E`s (E`spuµ puµ`s `s e` tded, td ed, 7? ?2 2=?? 7? ?2 2=?? 8 23333 hiàtrbs qu`, 233 23 8 2333 ) 233 233 Bj flabj `s pdsioj` e`kieir sdor` uab r`mjb mà`rbj< bj hujtipjikbr e`kihbj`s, à jb r`spu`stb, `j aµu uh`rd h`rd e` kifrbs b jb e`r`kgb e`j puatd e`kihbj `s imubj bj aµ uh`rd uh`rd tdtbj e` kifrbs b jb e`r`kgb e` jds puatds e`kihbj`s e` jds aµ u uh`rds h`rds qu` s` `stµbba a hujtipjikbaed. BsµĲ 3.? 3.?, àtr` jds eds, tiàà ua tdtbj e` eds kifrbs b jb e`r`kgb e`j puatd e`kihbj, jd hishd qu`, 3.35 (s` kuàtb `j k`rd b jb e`r`kgb e`j e`kihbj kdhd uab kifrb). Abturbjhàt`, si uad e` jds aµu uh`rds h`rds qu` `stb siàed hujtipjikbed `s ua àt`rd, àt `rd, `std ad bf`k bf`ktb tb jb pdsikiµ dda a e`j pua puatd td e`ki e`kihbj. hbj. @a `s` kbsd `j puat puatdd e`kihbj dkupb `j hishd jumbr à jb r`spu`stb qu` `j qu` tià` à `j aµuh`rd e`kihbj qu` `stµb siàed hujtipjikbed. E` edae`, 3.? ? 8 3.5> 2.0 0 8 =.0 y 2.2 205 8 269.5. @std, kdhd puè` v`rs`, vb e` bku`red kda jb r`mjb. @a kbeb kbsd, `j aµ u uh`rd h`rd e` kifrbs b jb e`r`kgb e`j puatd e`kihbj à jb r`spu`stb `s imubj bj aµuh`rd uh`rd tdtbj e` kifrbs b jb e`r`kgb e`j puatd e`kihbj à jds aµu uh`rds h`rds qu` `stµbba a siàed hujtipjikbeds. Jb kdjdkbkiµdda a e`j puatd e`kihbj à jb eivisiµdda a e` e`kihbj`s puè` gbk`rs` à fdrhb s`h`lbat`, bua kubaed bj kdatrbrid. @a r`bjiebe s` bkdstuhorb sihpjiflkbr `j bsuatd quitbaed `j puatd e`kihbj e`j eivisdr (d e`j eàdhiabedr, kubaed jb eivisiµda da s` `skrio` à fdrhb e` quòrbed). Sdr `l`hpjd, supdambhds qu` s` e`s`b eivieir 2.7? àtr` 3.?. @std puè` `xpr`sbrs` kdhd 23..7? y, ? kdhd quòrbed, r`tàerµb su vbjdr si tbatd `j auh`rbedr kdhd `j eàdhiabedr sda hujtipjikbeds pdr 23. Bgdrb 2 .7? 23 (simuiàed au`strb r`mjb e`kihbj) s` kdavi`rt` à 27.?3 d 27.? pu`std qu` `j µu ujtihd jtihd k`rd ad bmr`mb abeb bj vbjdr auhµ``rikd rikd y puè` `jihiab `jihiabrs`. rs`. E` imu imubj bj h hba`rb ba`rb 3.? 23 8 ?.3 y `s lustbhàt` ? (pu`std qu` ? hµbs bs 3 eµ``kihds, kihds, ad `s eif`ràt eif`ràt`` b e`kir sihpj`hàt` ?). @a kdas`kuàkib, `j quòrbed puè` `skrioirs` kdhd 27?.? kd kdavi avirtiµ rtiµ`àeds ae ds`` `j eàdhiabedr à ua àt`rd. Y pu`std qu` `s ua àt`rd, puè` jj`vbrs` b

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62 kbod jb eivisiµdda a sia kbhoibr à `j auh`rbedr jb pdsikiµdda a e`j puatd e`kihbj (kdhd à jds kbsds s`h`lbat`s hàkidabeds à r`jbkiµda da kda jb hujtipjikbkiµdda). a). 27.? @xistiàed uab kifrb b jb e`r`kgb puatdye`kihbj `xistir uab kifrb b jb e`r`kgb à `j e`j kdkiàt` 8 9à .2. `j auh`rbedr, eò` ? @std ads eb ua hµ``tded tded pbrb jb kdav kdav`rsiµ `rsiµda da e` frbkkida`s dreiabribs b e`kihbj`s usbaed jb eivisiµdda. a. [updambhds qu` e`s`bhds dotà`r `j `quivbjàt` 2 333 e`kihbj e` 53 . Sder SderµµĲbhds `skrioi `skrioirjd rjd kdhd 2.53 y kdhd `stbh `stbhds ds eiviei eivieiàed àed àtr` ua àt`rd, jb pdsikiµdda a e`j puatd e`kihbj ad kbhoibrµbb.. Jb eivisiµdda a prds`muirµb kdhd s` iaeikb b kdatiaubkiµdda< a<

3.3?0  53 2.333

73 ?33 ?33 3

2 [` e`hu`strb qu` `j `quivbjàt` e`kihbj e` 53 `s 3.3?0. [` puè` v`riflkbr `std kdavirtiàed 3.3?0 à quòrbeds dreiabrids. @s ? 0 ?3 0 ?0 + 2333 , d 2333 + 2333 , d 2333 y si `st` u ujtihd µjtihd `s eivieied `j auh`rbedr kdhd 233 2 `j eàdhiabedr àtr` ?0, ki`rtbhàt` e`hu`strb s`r 53 .

Hdviàed `j puatd e`kihbj Vbhds b `f`ktubr ua `xbhà hµbs bs kdakiàzued bj bsuatd e` jb hujtipjikbkiµda da pdr 23. @a ua pµbrrbfd brrbfd bat`ridr hujtipjikbhds 2.7? pdr 23 y dotuvihds 27.?. Dosµ``rv`s` rv`s` qu` jb hujtipjikb hujtipjikbkiµ kiµda da ad tuvd hµbbss `f`ktd qu` hdv`r `j puatd e`kihbj e`ki hbj ua jumbr b jb e`r` e`r`kgb kgb.. @a imubj fdrhb jd hdv hdv`r `rµµĲb eds jumbr`s b jb e`r`kgb si jb hujtipjikbkiµdda a fu`rb pdr 233 y tr`s jumbr`s b jb e`r`kgb si fu`rb pdr 2333. (Urbt` `j j`ktdr e` gbk`rjd y s` kdavàk`rµbb.) .) B jb iav`rsb, jb eivisiµda da pdr 23, sihpj`hàt` kdhpràe`r kdhpràe`rµµĲb hdv` hdv`rr b jb izqui`reb `j puatd e`kihbj. Bs BsµµĲ 2.7? 23 s`r s`rµµĲb `quivbjàt`, pdr r`mjb iav`rsb, 2 b 2.7? 23 . @std s`r s`rµµĲb, à e`kihbj` e`kihbj`s, s, 2.7? 3.2, y si `stds fu`rba jj`vbeds kda btàkiµ dda a b jb r`mjb pbrb jdkbjizbr `j puatd e`kihbj à jbs hujtipjikbkida`s, jb r`spu`stb s`rµĲb Ĳb 3.27?. Kdhd puè` v`rs` `j puatd e`kihbj, v`rebe`r v`rebe`rbhàt` bhàt` `s hdvied ua jumbr b jb izqui`reb.

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KBS µ IUPJD 0. F_BKKIDA F_BKKIDABAED BAED SD_ EI@K@[ EI@K@[

Eivieiàed p Eivieiàed pdr dr 233, `j pua puatd td e`ki e`kihbj hbj s` hdv hdv`r `rµµĲb, à jb hishb fdrh fdrhb, b, eds jumbr` jumbr`ss b jb izqui izqui`reb> `reb> eivieiàe eivieiàedd pdr 2333, s` hdv hdv`r `rµµĲb tr`s jumbr`s b jb izqui`re izqui`reb, b, y bs bsµ µĲ suk`sivbhàt`. Y pu`std qu` `j kbhoid e` uab kifrb pdr hujtipjikbkiµdda a d eivis eivisiµ iµdda a e` 23 eb kdhd r`sujtbed `j kdas`rvbr jds aµ u uh`rds h`rds y abeb hµbbss hdv`r `j puatd e`kihbj, `s pbrtikujbrhàt` kdavàiàt` qu` ads fbhijibrik`hds kda `j ‚pdrkàtbl`‘. Mà`rbjhàt`, s` bkdstuhorb, kda jbs p`rsdabs d iastitukida`s qu` pdr a`mdkid pr`stba eia`rd pbmbrj`s e`t`rhiabeb kbatiebe kdhd ‚iat`r`s`s‘, beikidabeds bj vbjdr e`j prµ`stbhd `stbhd kubaed s` `f`ktµ u ubb jb e`vdjukiµdda a e`j hishd. @std s` gbk` pbrb kdhpàsbrjds e` jb iakdavàiàkib e` ad eispda`r e` su eia`rdd eurba eia`r eurbat` t` ua p`rided e`t` e`t`rhiab rhiabed ed e` ti`hpd, y tbh tbhoiµ oiµ `à a pdr `j ri`sm ri`smdd qu` bsuhi`rda e` jb pdsioj` ad r`kup`rbkiµdda a e`j hishd. Kdhd ua `l`hpjd, jb p`rsdab d iastitukiµda da qu` pr`stb, puè` pèir 6 eµddjbr`s jbr`s bj b˚ a add pdr kbeb 233 eµddjbr`s jbr`s pr`stbeds. Kdhd `j iat iat`rµ `s, `s, usubjh usubjhàt àt`, `, s`pdr kbjku kbjkujb pdrkbeb uaiebe uaiebe`s e` kià (`sy e`ki e`kir, r, tbatds eµddjbr`s, jbr`s, d`rµ p`sd p`sds, s, d odjµ ĲĲvbr`s vbr`s b˚a addjbpdr 233`spr`stbeds), kdhd jb pbjbor pbjborbb jbtiab `s ‚p`r kà kàtuh‘ tuh‘ e` µ``stb stb e`riv e`rivbb au` au`strd strd ‚pdr kià kiàtd‘ td‘ `j kubj s` sihodjizb kdhd tbatd pdr kiàtd (qu` à r`bjiebe `s uab fdrhb e`j simad e` eivisiµdda) a) y, bj gbojbr e` ua iamr`sd e` 6 eµddjbr`s jbr`s pdr b˚ a add pdr kbeb 233 pr`stbeds, r`bjhàt` `stbhds `xpr`sbaed ua ‚iat`r ‚iat`rµµ`s `s e` sìs pdr kiàtd baubj.‘ Sdr jd mà`rbj, à jds a`mdkids, jbs utijiebe`s, hµbrmà`s brmà`s e` mbabakib, kdhisida`s, e`r`kgds e` butdr y dtrds bsuatds s`h`lbat`s, s` kbjkujba b obs` e` pdrkàtbl`. Bgdrb oià, 2 pdr kiàtd simaiflkb 2 eµddjbr jbr d 2 p`sd d 2 e` kubjqui`r dtrb 2 hdaèb pdr kbeb 233 uaiebe`s e` jb hishb, y s` r`pr`sàtb kdhd 233 .Sbrb sbkbr `j 2 pdr kiàtd e` kubjqui`r kifrb, µu uaikbhàt` aikbhàt` s` r`qui`r` hdv`r `j puatd e`kihbj eds jumbr`s b jb izqui`reb. Bs BsµµĲ, `j 2 p pdr dr kiàtd e` 2;03 eµddjbr`s jbr`s `s 2;.03. Jb kbatiebe r`pr`sàtbeb pdr `j 6 pdr kiàtd e` 2;03 eµdjbr dj br`s `s s`r s` rµĲb 2 6 233 , d 6 2;.03 8 72.33 eµddjbr`s. jbr`s. 23 2 Pab kdhisiµdda a e` 23 pdr kiàtd s`r s`rµµĲb 233 e` jb kifrb drimiabj d 23 . @a `st` kbsd, `j puatd e`kihbj s` hu`v` ua jumbr b jb izqui`reb. Pab kdhisiµdda a e` 23 pdr kiàtd sdor` 2;03 eµddjbr`s jbr`s `quivbjer `quivbjerµµĲb b 2;0. B v`k`s s` pr`sàtba eiflkujtbe`s. Pab kdhisiµdda a e` 2 pdr kiàtd sdor` 6=0.;= eµddjbr`s jbr`s s`r s`rµµĲb e` 6.=0;=. @a jds a`md a`mdkids kids s` gb bedptbed, kdhd sdjukiµda da prµbbktikb, ktikb, jb adrhb e` qu` jbs kifrbs kda hµbbss e` eds aµu uh`rds h`rds b jb e`r`kgb e`j puatd e`kihbj, à jd r`f`ràt` b eia`rd, s` rèdae`à, y à `st` kbsd jb kdhisiµdda a s` kdasie`r kdasie`rbr brµµĲb s`r e` 6.=0.

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6; Uded `std dp`rb huy oià à jds sist`hbs hda`tbrids e`kihbj`s, p`rd ad tbatd à `j sist`hb oritµbbaikd. aikd. Pa 23 pdr kiàtd sdor` 2;0 jiorbs y 23 kg`jia`s r`sujtb s`r 2; jiorbs y 22 kg`jia`s. (´Suè` bj j`ktdr r`sdjv`r `sd4)

E`kihbj`s siafla @a `j sist`hb e`kihbj, uab iakdhdeiebe hµbbss s`rib qu` `j hishd prdoj`hb e` àkdatrbr jb kdjdkbkiµdda a e`j puatd, `s `j g`kgd e` qu` bjmuabs frbkkida`s ad puèà `xpr`sbrs` kdhd e`kihbj`s à fdrhb dreiabrib. Sdr `l`hpjd, ´kdhd pde`hds `xpr`sbr 2; kdhd e`kihbj4 Sbrb àkdatrbr jb fdrhb `skrioir`hds 2; kdhd 2.33333333 y prdkè prdkè`r`hd `r`hdss b ; eivieir eivie ir kdhd simu` simu` e`spuµ``ss 6, qu` `s 2 + ? + ;> e`spuµ`s `s 23, qu` `s 2+ 2 + ? + ; + 5> e`spuµ``ss 20, qu` `s 22+ + ? + ; + 5 + 0> y bs bsµµĲ suk`siv suk`sivbhàt`. bhàt`. Uàiàed pr`sàt` `stb r`jbkiµdda, a, s` puè` gbk`r jb jistb e` aµu uh`rds h`rds tribamujbr`s iae`flaiebhàt`, sia siqui`rb tà`r qu` gbk`r `j triµbbamujd amujd y kdatbr jds puatds. Ubhoiµ`à a s` p puè` uè` e`kir si ua aµuh`rd uh`rd `s tr tribamuj ibamujbr br d ad, eivieiµ`àedaedjd à uab s`ri` e` suhbs kdhd jbs brriob iaeik iaeikbebs. bebs. [i s` tià` µ``xitd, xitd, `s ua aµu uh`rd h`rd tribamujbr. Kubjqui`r mrupd e` aµuh`rds uh`rds qu` puèb kdastruirs` suk`sivbhàt` pdr hèid e` ua sist`hb kdhd µ``st` st` `s jjbhbed ‚s`ri`‘. Jds aµ u uh`rds h`rds qu` r`pr`sàtba jds puatds y fdrhba ua kuberbed tbhoiµ `à a fdrhba uab s`ri s`ri`. `. Au`vbhàt`, ua ssdjd djd puatd puè` kda kdasie`rbr sie`rbrs` s` pd pdrr sµ sµĲ hishd kdhd ua kuberbed ujtrb-hikrdskµddpikd. pikd. B p`sbr e` tded, s` a`k`sitba kubtrd puatds pbrb gbk`r ua kuberbed kda eds puatds à kbeb jbed> au`v` puatds, pbrb gbk`r uad kda tr`s puatds à kbeb jbed, y bsµĲ suk` suk`sivbhàt`. sivbhàt`. Jb s`ri` e e`` a aµ u uh`rds µ h`rds kuberbeds `s< 2, 5, 9, 26, ?0, ;6, 59, 65, y bs bsµµĲ suk`sivbhàt`, gbstb edae` s` qui`rb jj`mbr. [i s` dos`rvb `stb s`ri` kuiebedsbhàt`, s` pderµb adtbr qu` kbeb aµu uh`rd h`rd `stµb fdrhbed e` jb suhb suk`sivb e` aµu uh`rds h`rds ihpbr`s. Sbrb `hp`zbr, 2 `s 2> p`rd 5 `s 2 + ;> 9 `s 2 + ; + 0> 26 `s 2 + ; + 0 + =, y bsµ bsµĲ suk`siv suk`sivbhàt`. bhàt`. Jb r`jbkiµdda a àtr` aµ u uh`rds h`rds à jbs s`ri`s tribamujbr`s y à jbs s`ri`s e` kuberbeds kuberbe ds p puè` uè` tbhoiµ`à a e` e`hdstrbr hdstrbrs` s` ei eibmrbhµ bmrbhµbbtikbhàt`, tikbhàt`, kdhd s` dos`rvb à (44) . Jds mri` mri`mds mds tbhoiµ`à a tàµĲĲba ba aµu uh`rds h`rds pàtbmdabj`s, kdhd puè` v`rs` à (44). @stds sda uab `sp`ki` e` fusiµdda a e` aµu uh`rds h`rds tribamujbr`s y kuberbeds.

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Fimurbb 6.;< _`jbk Fimur _`jbkiµ iµdda a àtr` aµ u uh`rds h`rds tribamujbr`s y aµu uh`rds h`rds kuberbeds

Fimurb 6.5< Aµ u uh`rds h`rds pàtbmdabj`s

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Fimurb 6.0< Kuods

[i s` kdastruy` ua pàtµbbmdad mdad b obs` e` puatds, s` àkdatrbrµb qu` fdrhba uab s`ri` e` aµu uh`rds h`rds kdhd simu`< 2, 0, 2?, ??, ;0, 02, =3, y bsµĲ suk`sivbhàt`. @stds aµ u uh`rds h`rds s` fdrhba suhbaed b iat`rvbjds e` ;. Sdr `l`hpjd, 2 `s 2> 0 `s 2 + 5> 2? `s 2 + 5 + => ?? `s 2 + 5 + = + 23, y bsµ bsµĲ suk`siv suk`sivbhàt`. bhàt`. Jds mri`mds tdebvµĲĲbb tà tàµµĲba dtrbs flmurbs m`dhµ``trikbs, trikbs, jbs kubj`s gbk gbkµµĲba b obs` e` puatds y, à mà`rbj, jds aµu uh`rds h`rds r`sujtbat`s e` tbj mbrrbpbt`d hbt`hµbbtikd tikd sda jjbhbeds ‚aµu uh`rds h`rds flmurbtivds‘. Bjmuabs e` sus flmurbs `rba sµdjiebs. djiebs. Sdr `l`hpjd, puèà fdrhbrs` kuods b obs` e` puatds. Ubj`s kuods sda eif eifµµĲkij`s Ĳkij`s e` hds hdstrb trbrr à jb pr` pr`sà sàt` t` gdlb, p`rd pu` puè` e` kbp kbptbr tbrs` s` jb ie` ie`bb dos`rvbaed (44). Jb s`ri` e` aµ u uh`rds h`rds kµ u uoikds oikds `s< 2,7, ?=, 65, 2?0, y bsµ bsµĲ suk`siv suk`sivbhàt`. bhàt`. Ubhoiµ`à a bquµĲ `xis `xist` t` uab r`jb r`jbkiµ kiµdda a qu` kdhpràe` jb suhb e` aµu uh`rds h`rds ihpbr`s, kdhd à `j kbsd e` jds aµu uh`rds h`rds kuberbeds, p`rd jb suhb ad `hpi`zb à kbeb kbsd kda `j 2. 2 `s 2, p`rd 7 `s ; + 0> ?= `s = + 9 + 22> 65 `s 2; + 20 + 2= + 29, y bs bsµµĲ suk`siv suk`sivbhàt`. bhàt`. Kbeb mrupd e` aµu uh`rds h`r ds qu` s` `stµ` suhbaed kdaluatbhàt` `hpi`zb edae` t`rhiaµd `j bat`ridr, y kbeb mrupd kdatià` ua aµ u uh`rd h`rd hµbbss qu` `j mrupd bat`ridr.

69

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Gbstb bgdrb, jbs s`ri`s e` aµu uh`rds h`rds qu` g` pr`sàtbed puèà fdrhbrs` pdr hèid e` r`p`tikiµ dda a e` suhbs. [ia `hob `hobrmd, rmd, tbhoiµ`à a pu` puèà eà ia iakjuir kjuir hujtipjikbkida`s. Pab e` jbs s`ri`s qu` kdhpràeà hujtipjikbkida`s surm` kdhd simu`<

[updambhds qu` s` eispda` e` 5 kuàtbs e` kdjdr`s eif`ràt`s, qu` trbtbhds e` àsbrtbr, ´Kubatbs kdhoiabkida`s e` kdjdr`s s` puèà kr`br kda `sbs kubtrd kuàtbs4

[i jbs kuàtbs sda rdlbs, a`mrbs, bzuj`s y v`re`s (kubjqui`r dtrd kdjdr s`rvirµ b pbrb `j dol`td qu` trbt trbtbhds bhds e` `xpji `xpjikbr) kbr) p pder derµµĲb `hp`zb `hp`zbrs` rs` àsbr àsbr-tbaed kubjqui`rb e` jbs kubtrd. Yb tà`hds bqu bquµµĲ 5 pdsioijiebe`s. Gboiàed àsbrtbed uab, s` puè` àsbrtbr kubjqui`rb e` jbs tr`s r`stbat`s, kda jd qu` bjkbazbhds bjkba zbhds 5 ;, d s`b 2? pdsioijiebe`s gbstb bgdrb. Yb àsbrtbeds eds kdjdr`s pde`hds pda`r kubjqui`rb e` jbs eds r`stbat`s, y `sd ads eb 5 ; ?, d s`b ?5 pdsioijiebe`s. Fiabjhàt`, kda tr`s kuàtbs àsbrtbebs, quèb sdjbhàt` uab, y s` tià` 5 ; ? 2, d s`b ?5 pdsioijiebe`s. Jb simuiàt` tbojb e`hu`strb jd bat`ridr y kbtbjdmb jbs ?5 kdhoiabkida`s e` kdjdr`s.

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Kdhd puè` v`rs`, `j aµ u uh`rd h`rd ?5 fu` fdrhbed kdhd prdeuktd e` jb hujtipjikbkiµ dda a e` 5 ; ? 2. Psbaed `j hishd brmuhàtd, puè` e`hdstrbrs` qu` jbs kdhoiabkida`s pdsioj`s e` = kuàtbs e` eif`ràt`s kdjdr`s `quivbjà b = 6 0 5 ; ? 2, d s`b 0353, y à fdrhb sihijbr pbrb kubjqui`r aµu uh`rd h`rd e` kuàtbs. Jds aµ u uh`rds h`rds fdrhbeds pdr jb hujtipjikbkiµdda a e` vbrids eµĲmitds Ĳmi tds ssuk`s uk`sivbhà ivbhà-t`, `hp`zbaed pdr `j 2, sda jjbhbeds ‚fbktdribj`s‘. @j prdeuktd e` 5 ; ? 2 `s jjbhbed ‚fbktdribj e` 5‘ pdr s`r `j aµu uh`rd h`rd hbs bjtd kdhpràeied à jb hujtipjikbkiµ da. da. Imubjhàt`, `s ‚fbktdribj e` =‘ `j prdeuktd e` = 6 0 5 ; ? 2. Jds fbktdribj`s mà`rbjhàt` s` sihodjizba usbaed `j puatd e` `xkjbhbkiµdda. a. BsµµĲ `j ‚fbktdribj e` 5‘ puè` r`pr`sàt Bs r`pr`sàtbrs` brs` kdhd 5! @j puatd e` `xkjbhbkiµdda a tbj v`z `st `stµµ` lusti lustiflkbed, flkbed, pu`st pu`stdd qu` jbs s`ri` s`ri`ss buhàtba huy rµbbpiebhàt`. piebhàt`. Jbs s`ri`s e` fbktdribj`s r`pr`sàtbeds pdr 2!,

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=2 ?!, ;! , 5! bs bsµµĲ suk`sivbhàt` sda, r`sp r`sp`ktiv `ktivbhàt`, bhàt`, 2, ?, 6, ?5, 2?3, =?3, 0353, 53;?3,, ;;6?773, 53;?3 6?773, y bsµĲ kbeb v`z hµbbs. s. @j vìatbvd aµ u uh`rd h`rd à jb s`ri` (?3!) s`r s`rµµĲb ?5;?9;?3372=6653333.

Kube Ku berb rbed eds, s, kuo ku o ds y hµ bs bs bjjµ bj jµ b @a r`bjiebe, si r`trdkè`hds b jds aµu uh`rds h`rds kuberbeds y b jds aµ u uh`rds h`rds kµuoikds, uoikds, pde`hds fµbbkijhàt` kijhàt` v`r qu`, bua kubaed `xist` uab r`jbkiµdda a r`mujbr qu` iakjuy` jb suhb, tbhoiµ `à a `xist` uab r`jbkiµ dda a r`mujbr qu` kdhpràe` jb hujtipjikbkiµ dda. a. Bj priakipid e`j Kbp KbpµµĲtujd ; s` gbojµd e` jb sup`rflki` e` jds kuberbeds. [` r`kdrebrµ b qu` ua kuberbed kda ua jbed imub imubjj e` 2 uaiebe (yb s`b kàt kàtµµĲh`trd y hijµĲĲh`trd h`trd d kubjqui`r dtrb hèieb) tià` uab sup`rflki` e` 2 2, d s`b 2 uaiebe kuberbeb. Pa kuberbed kda ua jbed imubj b ?, tià` uab sup`rflki` imubj e` ? ?, d s`b 5. _`bjhàt`, si kdasie`rbhds uab s`ri` e` kuberbeds kda jbeds imubj`s b 2, ?, ;, 5, 0, 6, = uaiebe`s (y bs bsµµĲ suk`sivbhàt suk`sivbhàt`) `) jbs sup`rflki`s à jds kuberbeds r`sujtba r`sp`ktivbhàt` 2, 5, 9, 26, ?0, ;6, 59, `tk. @stb `s, kdhd s` v`, jb s`ri` e` aµu uh`rds h`rds kuberbeds qu` s` hàkidaµd bj priakipid e` `st` kbpµ kbpµĲtujd. @a v`z e` `skrioir jb hishb kdhd 2, 2 + ;, 22+ + ; + 0, 2 + ; + 0 + =, y bs bsµµĲ suk suk`si `siv vbh` bhàt at`, `, pde`hds `skr `skrioi ioirr 2 2, ? ?, ; ;, 5 5, 0 0, `tk. Bgdrb oià, ua kuod `s ua sµdjied djied r`mujbr jihitbed pdr sìs kuberbeds imubj`s y qu`, p pdr dr jd tbatd, tiàà tbhoiµà à imubj`s sus tr`s eihàsida`s. Pa pbr e` ebeds sda, à r`bjiebe, ua pbr e` kuods. @j vdjuhà e` ua kuod s` dotià` hujtipjikbaed `j bakgd e` uad e` jds jbeds, jb bjturb y jb jdamitue. @std puè` e`hds e`hdstrbrs trbrs`` kda `j hishd rbzda rbzdabhià bhiàtd td usbed à `j Kbp KbpµµĲtujd ;, r`sp`ktd b qu` jb sup`rflki` e` ua kuberbed (d e` kubjqui`r kuberijdamd) puè` dotà`rs` hujtipjikbaed su bakgd pdr su jbrmd. Pa kuod qu` hieb 2 uaiebe à kbeb jbed tià` ua vdjuhà e` 2 2 2, d s`b 2 uaiebe kµu uoikb. oikb. Pa kuod qu` hieb ? à kbeb jbed, tià` ua vdjuhà e` ? ? ? d s`b 7. Kdatiaubaed `std, s` v` iahèibtbhàt` qu` jds vdjµu uhà`s hà`s e` jds kuods qu` hieba à jds jbeds 2, ?, ;, 5, 0, 6, `tk., ebrµba jbs [`ri`s e` aµ u uh`rds h`rds kµ u uoikds oikds 2, 7, ?=, 65, 2?0, ?26 y bsµĲ suk`siv suk`sivbhàt`. bhàt`. Sdr jd tbatd, puèà r`pr`sàtbrs` kdhd 2 2 2> ? ? ?> ; ; ; > 5 5 5> 0 0 0> `tk. Jds adhor`s e` jds kuberbeds y jds kuods `stµbba a oià pu`stds, p`rd e`lbaed b ua jbed jbs bktubj`s flmurbs m`dhµ``trikbs, trikbs, pd pde`hds e`hds ihbmiabrads jb fdrhbkiµ da da

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µ KBS µ IUPJD IUP JD 6. JB FD_H FD_HB B E@ JD[ A PH@_D[

e` dtrbs s`ri`s e` aµu uh`rds, h`rds, à jbs kubj`s kbeb aµu uh`rd h`rd `s `j prdeuktd e` kubtrd d hµbbss aµ u uh`rds h`rds hujtipjikbeds e` kdatiaud. Jbs hujtipjikbkida`s suk`sivbs e`j hishd aµu uh`rd h`rd r`sujtba s`r iaflaitbhàt` u utij`s µ tij`s à jbs hbt`hµbbtikbs tikbs y s` a`k`sitba au`vds sµĲhodjd Ĳhodjds. s. _`kuµ``re`s` re`s` qu` `j usd fr`kuàt` e` dp`rbkida`s tbj`s kdhd 6 + 6 + 6 + 6 drimiaµd jb iavàkiµda da e` jb kdrtb y hµbbss fµbbkijhàt` kijhàt` mrboboj` hujtipjikbkiµdda a e` 6 5. @a jb hishb fdrhb, `j usd fr`kuàt` e` dp`rbkida`s tbj`s kdhd 6 6 6 6, eid kdhd r`sujtbed `j usd e` ua au`v au`vdd tipd e` sµĲĲhodjd< hodjd< 6 5. Bj priakipid, 65 sihpj`hàt` simaiflkbob ‚kubtrd sìs`s hujtipjikbeds e` kdatiaud‘> imubjhàt` 230 simaiflkbr simaiflkbrµµĲb ‚kiakd ei`k`s hujtipjikbeds e` kdati? aud‘ hiàtrbs qu` ; sim simaifl aiflkbrµ kbrµĲĲbb ‚ed ‚edss ttr`s` r`s`ss hu hujtip jtipjikb jikbeds eds àtr` sµĲ‘. Jbs s`ri`s e` aµ uh`rds uh`rds kuberbeds puèà `skrioirs` kdhd simu`< 2? , ?? , ;? , 5? , 0? y bs bsµµĲ suk`siv suk`sivbhàt`. bhàt`. Jbs s`ri`s e` aµu uh`rds h`rds kµ u uoikds oikds puèà `skrioirs` 2; , ?; , ;; , 5;, 0; , `tk.

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@j aµ u uh`rd h`rd a add àqu` jb pbrt` sup`ridr e`r`kgb kdhd e`j aµ u uh`rd `s ‚aµ jjbhbed 5 ‚`xpda` ‚`xpdaàt` at`‘ ‘ y p`qu`˚ ua aµu uh`rd h`rd kdatià` ua `xpdaàt` 6h`rd `s ua u uh`rd h`rd 5 `xpdaàkibj‘. @j aµ u uh`rd h`rd hujtipjikbed, à `st` kbsd `j 6 e`j 6 , `s jb ‚obs`‘ e`j aµ u uh`rd h`rd `xpdaàkibj. Jb hujtipjikbkiµdda a e` kdatiaud e` ua aµu uh`rd h`rd s` eàdhiab ‚`j`vbkiµdda a e` 5 `s` aµ u uh`rd h`rd b uab e e`t`rhiabe `t`rhiabebb p pdtàkib‘ dtàkib‘ . BsµĲ 6 `s ‚sìs `j`vbed b jb kubrtb pdtàkib‘ d, hµbs bs sàkijjbhàt`, ‚sìs b jb kubrtb‘. Imubjhàt`, 230 `s ‚ei`z b jb quiatb.‘ [` puè`, si s` e`s`b, gbojbr e` ;? d ;; kdhd ‚tr`s b jb s`muaeb‘ d ‚tr`s b jb t`rk`rb‘ sia iakurrir, à r`bjiebe, à uab iakdrr`kkiµdda. a. [ia `hobrmd, `s hµbbss iakjiabrs` b jb batimub kdstuhor` mri`mb y r`f`rirs` b `stbs eds `xpr`sida`s kdhd ‚tr`s bj kuberbed‘ y ‚tr`s bj kuod.‘

@vitbaed jb hujtipjikbkiµ da da Jb utijiebe e` jds aµ u uh`rds h`rds `xpdaàkibj`s rbeikb à `std< ads dfr`kà ua hµ`tded `tded pbrb kdav`rtir jb hujtipjikb hujtipjikbkiµ kiµdda a à uab suhb (y `s hbs fµbbkij kij suhbr qu` hujtipjikbr). Sdr `l`hpjd, supµddambs` ambs` qu` s` hujtipjikµd 26 65. @j prdeuktd `s 23?5. S`rd 26 `s 5 5 y 65 `s 5 5 5. [i gbk`hds `stds kbhoids, àtdak`s 26 65 s` kdavi`rt` à 5 5 5 5 5, qu` ki`rtbhàt`, ads eb 23?5, kdhd puè` v`riflkbrs`. Ubhoiµ`à a 2266 pu puè` è` kd kdasi asie`rb e`rbrs` rs` kkdhd dhd ? ? ? ?, y 65 kdhd ? ? ? ? ? ?. Ad dostbat`, 26 65 puè` `xpr`sbrs` kdhd ? ? ? ? ? ? ? ? ? ?

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=; qu`, kdhd puè` v`riflkbrs` v`riflkbrs`,, ads eb tbhoi tbhoiµµ`à a 23?5. Psbaed aµ u uh`rds h`rds `xpdaàkibj`s pde`hds sbo`r qu` 26 `s 5? d ?5 > 65 `s ;

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0

23

5 µd ? y 23?5 `s 5 µd ? . Sde`hds `skrioir `j prdoj`hb 26 x 65 823?5, `xpdaàkibjh `xpda` akibjhàt àt`, `, à kubjq kubjqui`r ui`rbb e` jbs eds fdrhbs< 5? 5; 8 50, µd ?5 ?6 8 ?23. Dosµ ``rv`s` rv`s` qu` à kbeb kbsd `j `xpdaà `xpdaàt` t` e` jb r`spu r`spu`stb `stb `s jb suhb e` jds `xpdaàt`s e` jds eds aµ u uh`rds h`rds qu` `stµbba a siàed hujtipjikbeds. [i s` àsbyb kda dtrds aµ uh`rds uh`rds qu` puèba `xpr`sbrs` à fdrhb `xpdaàkibj, s` àkdatrbrµb qu` mà`rbjhàt`, `stb r`jbkiµdda a s` hbatàerµbb,, si`hpr` qu` tdeds jds `xpdaàt`s kdhpràeieds `j`và `j hishd aµuh`rd. uh`rd. @a `stb fdrhb, sia tà`r qu` r`kurrir r`bjhàt` b jbs hujtipjikbkida`s, pde`hds `stbr s`murds e` qu` ?5 ?? ?25 8 ??3 y qu` 75 7= 8 722. @j sist`hb s ist`hb tbhoiµ`à a dp dp`rb `rb p pbrb brb jb e eivisiµ ivisiµdda, a, kda jb sbjvèbe e` qu qu`` à µ``stb stb 0 ; ? jds `xpdaàt`s eòà r`stbrs`. Bs BsµµĲ ? ? 8 ? , qu` à aµu uh`rds h`rds dreiabrids `s ;? 7 8 5, s`mµ u ua a puè` v`riflkbrs`.

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puè` pbr`k`r ua breie obstbat` r`striamied, pbrbe` s`rs`r e``xpr`sbeds mrba utijiebe.@std S`rd, à prih`r jumbr, jds aµ u uh`rds h`rds eòà s`r kbpbk`s kdhd `xpdaàkibj`s. `xp daàkibj`s. @j hujtipjikbr 7 y 26 `s hbmaµĲĲflkd, flkd, pu`std qu` `quivbj` ; 5 b? ? , p`rd, ´Ruµ` eir`hds e` jb hujtipjikbkiµ dda a e` = y 2=4 Iakjusd si jds aµuh`rds uh`rds sda kbpbk`s e` s`r `xpr`sbeds kdhd `xpdaàkibj`s, puèà ad tà`r jb hishb obs`. Sdr `l`hpjd, 7 9 `s ?; ;? y bqu bq uµĲ ad s` p puè` uè` usbr `j bbreie reie 0 0 e` suhbr jds `xpdaàt`s. Jb r`spu`stb ad `s ai ? ai ; , ai s` àkuàtrb à hèid e` `stbs eds kifrbs. @atdak`s, ´Sbrb qu` pr`dkupbrs` kda `std4 @a r`bjiebe, kubjqui`r breie qu` kdavi`rt` à suhb jb hujtipjikbkiµdda a s`rµĲb Ĳb tba vbjids vbjidsdd qu` vbjerµĲb Ĳb jb p` pàb ab `stueibrjd. Sbrb gbk`rjds trboblbr à fdrhb mà`rbj, eò`hds mà`rbjizbr `j kdadkihiàtd e` jds `xpdaàt`s.

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Gbstbe`bgdrb, g`hdshujtipjikbed kdasie`rbede`qu` ua aµu uh`rd h`rd `xpdaàt` r`pr`sàtb ‚tbatd‘ ua aµu uh`rd h`rd kdatiaud. @a `sbs kdaeikida`s, `j `xpdaàt` hµbbss p`qu`˚ a add pdsioj` `s `j ?, pu`std qu` ad s` puè` tà`r hàds e` ? kifrbs pbrb s`r hujtipjikbebs àtr` ssµµĲ. [ia `hobrmd, pd pde`hds e`hds kdasie`rbr qu` `sb e`flaikiµdda a gb s`rvied yb su prdpµddsitd sitd y pde`hds, pdr hèid e` jd qu` gbstb bgdrb g`hds bpràeied e` jds `xpdaàt`s, ir hbs bjjµb e`j ?.

B k`rd k`rd y hµ bs bs bjjµ bjjµ b ... au`vbhàt au`vbhàt` ` Sdr `l`hpjd 26 7 8 ?. Kdhd 26 8 ?5 y 7 8 ?; , puè` `xpr`sbrs` jb eivisiµdda a kdhd ?5 ?; 8 ?2 . [djbhàt` pde`hds e`kieir qu` ? 2 y ? sda jb

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hishb kdsb> à dtrbs pbjborbs ?2 8 ?. Jb hishb jµddmikb mikb dp`rbr dp`rbrµµĲb pbrb jds `xpdaàkibj`s qu` `j`v `j`vbrba brba b uab pdtàkib kubjqui`r aµu uh`rd h`rd obs`. Sdr jd tbatd, pde`hds fllbr jb r`mjb mà`rbj e` qu` kubjqui`r aµ u uh`rd h`rd `j`vbed b jb prih`rb pdtàkib p`rhba`k` iabjt`rbed. BsµĲ 02 8 0> ?=2 8 ?=, `tk. S`rd jbs kdsbs s` gbrµba ba hbs `xtrbdreiabribs. Sdr `l`hpjd `l`h pjd,, ´Ruµ` `s 7 74 @vieàt`hàt` `s 2. S`rd 7 `s ? ; y bsµĲ ?; ?; 8 2. [i, pdr supu`std, p`rd pdr r`stb e` `xpdaàt`s r`sujtb qu` ? ; ?; 8 ?3 ´[imaiflkb `std qu` ?3 8 24 Sbr`k` qu` bs bsµµĲ eò` s`r. @std puè` sdrpràe`r bj j`ktdr. Ubj v`z bk`pt` `j qu` ? 2 s`b imubj b ? pu`std qu`, bua kubaed jb `xpr`siµdda a e` ‚ua eds hujtipjikbed e` kdatiaud‘ suà` rbrb, puè` ihbmiabrs` bj ? bjjµĲ sdjd y r`bjhàt` `s imubj b ?. [ia `hobrmd, ?3 pbr`k pbr`k`` simaifl simaiflkbr kbr ‚aiamµ u ua a ? hujtipjikbed e` kdatiaud‘ y, s`murbhàt`, 3 sµddjd jd puè` simaiflkbr qu` ? `s imubj b k`rd. @std ad puè` pbr`k`r ki`rtbhàt` jµddmikd, mikd, p`rd jds hbt`hµbbtikds tikds ad `stµbba a à r`bjiebe iat`r`sbeds à jb jµddmikb mikb

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eibrib (Ád s` gdrrdrizb `j j`ktdr4) siad à jb kdasistàkib iat`rab. @a dtrbs pbjborbs, jds hbt`hµbbtikds tikds `stµbba a bauàt`s b fllbr kubj`squi`r r`mjbs ‛ sia ihpdrtbr kuµbba a jdkbs pbr`zkba b jb fdrhb dreiabrib e` pàsbr ‛ si`hpr` qu` `sbs r`mjbs ad s` kdatrbeimba àtr` si à kubjqui`r kdsb qu` r`sujt`. @j breie e` suhbr y r`stbr `xpdaàt`s dp`rb tba oià, qu` si t`rhiab e`hdstrµbbaedads aedads qu` ?3 `s imubj b 2, jd e`flaihds  kdhd kdhd imubj b 2 y t`rhiabhds jb kdatrdv`rsib. [i à v`z e` ?; ?; àsbybhds kda 6; 6; , àkdatrbr`hds qu` 63 `s tbhoiµ tbh oiµ à à imubj b 2. @a r`bji r`bjiebe, ebe, si àsb àsbybh ybhds ds aµu uh`rd h`rd trbs aµu uh`rd, h`rd, prdatd `hp`zbr`hds b kdhpràe`r qu` kubjqui`r aµu uh`rd h`rd `j`vbed b jb pdtàkib k`rd `s imubj b 2. Bgdrb vbhds b ir hµbbss be`jbat`. [i eivieihds 65 àtr` 2?7, `j r`sujtbed `s

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65 d s`b 2 . @xpdaàkibjhàt` `j prdoj`hb `s ?6 ?= y jb r`spu`stb `s ?∟2 . 2?7 ? 2 @std simaiflkb qu` ?∟2 eò` s`r imubj b d, d , pd pdaiµ aiµ`àedjd aed jd tbhoiµ`à a à fdrh fdrhbb ? 2 `xpdaàkibj, `xpdaà kibj, ( )2 . ? 2 Imubjhàt`, ;? 2?7 8 ( ). @std, pdaiµ`àedjd aedjd à fdrhb `xpdaàkibj, `s 5 2 2 ?0 ?= 8 ?∟? , pdr jd qu` ?∟? . eò` s`r imubj b , qu`, b su v`z, `s imubj b ( )? . 5 ? [i `j j`ktdr àsbyb dtrds `l`hpjds, àkdatrbrµb qu`, à mà`rbj, ua `xpdaàt` a`mbtivd puè` s`r kdav`rtied à `xpdaàt` pdsitivd tdhbaed `j iav`rsd e`j aµu uh`rd. h`rd.

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=0 2 2 @a dtrbs pbjborbs, 5∟= 8 ( )= y 23∟; 8 ( ); . Sbrb `s` bsuatd dp`rb à 5 23 2 ∟5 5 fdrhb kdatrbrib. @j `xpdaàkibj 6 `s imubj b ( 6 ) . Sde`hds ebr `l`hpjds bgdrb hishd, e`hdstrbaed qu` `s kdasistàt` `st` 2 puatd e` vistb sdor` jds `xpdaàt`s. Sdr `l`hpjd, ´@s ( )5 r`bjhàt` imubj 6 2 b 6∟5 4 Vbhds b v`r. Jb `xpr`siµda da ( )5 puè` pda`rs` à `stb fdrhb< 2 65 . 6 S`rd kdhd 2 `s imubj b 63 jb `xpr`siµdda a s` kdavi`rt` à 63 65 , pdr jd qu`, r`stbaed jds `xpdaàt`s, `j r`sujtbed pruòb s`r ki`rtbhàt` 6∟5 . Au`vbhàt` surm` jb pr`muatb, ´@s r`bjhàt` 63 imubj b 24 Ouàd, ;6 2 8 2, kdhd sia eueb `j j`ktdr `stµb eispu`std b kdakè`r. Kdhd ;6 8 6 ? , ;6 2 2 2 àtdak`s 8 ( )? d s`b 6∟? . Jb `xpr`siµdda a ;6 s` kdavi`rt`, pdr tbatd, ;6 6 ;6

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6 qu`, abeb à 6@a r`bjiebe, pdr suhb, s` pruòb kdavi`rt` à pu`std 6 , qu`qu` `s imubj b 2. jd qu` jds e` `std bjmd, kdhpràe` hbt`hµbbtikds tikds jjbhba ‚rbzdabhiàtd kirkujbr‘. (@s jd qu` pbsb kubaed uad eik`< ‚Pa mbtd `s kubjqui`r baihbj qu` gbk` hibu‘. Ju`md, ‚`s` baihbj qu` gbk` hibu `s, pdr tbatd, ua mbtd‘). Ad dostbat`, jd bat`ridrhàt` `xpu`std e`hu`strb qu` `j sist`hb e` `xpdaàt`s si dp`rb. Ubhoiµ `à a pde`hds e`hdstrbrjd à dtrb fdrhb, jistbaed bjmuabs kifrbs `xpdaàkibj`s. Sde`hds `hp `hp`zbr `zbr kda jb batimub e`flaikiµdda a e` jds aµu uh`rds h`rds hujtipjikbeds e` kdatiaud, pdr jd qu`< ?6 8 ?

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?∟? 8 2 5 2 ?∟; 8 7 [i kdatiaubhds, s` v`rµb qu` jds `xpdaàt`s hàdr`s e` ? simaiflkba lustbhàt` jd qu` g`hds e`kieied qu` eòba simaiflkbr. @std puè` dp`rbr kda kubjqui`r aµ u uh`rd h`rd obs`. [` àkdatrbrµb qu` jds `xpdaàt`s e`j aµ u uh`rd h`rd obs` 2 ; iakjuirµbba a vbjdr`s qu` s` rèukà kbeb v`z qu` `xpdaàt` eishiauy` uab ; 2 uaiebe. Jds `xpdaàt`s e`j aµ u uh`rd h`rd obs` 6 s` rèukirµba ba kbeb kbe b v`z, y bsµĲ 6 suk`sivbhàt`. S`rd, à kbeb kbsd, `j sist`hb dp`rbrµb. b. Uded `std simaiflkb qu` g`hds buhàtbed jbs dpdrtuaiebe`s e` kbhoibr uab hujtipjikbkiµdda a b uab suhb. Bgdrb pde`hds hujtipjikbr 2 y 23?5 pdr 7 `xpdaàt`s. [ia `hobrmd, bua ad g`hds e`kieied kµddhd hd hujtipjikbr = y 2=. ´Sde`hds bhpjibr bµu ua a hµbbss jds `xpdaàt`s4 Bgdrb, qu` tà`hds jd hishd `xpdaàt`s a`mbtivds qu` pdsitivds, ´Ruµ` eir`hds sdor` jds `xpdaàt`s frbkkidabrids4 Bat`s e` qu` pdebhds kdasie`rbr jd qu` puèba simaiflkbr jds `xpdaàt`s frbkkidabrids, eò`hds kdasie`rbr jd kdatrbrid e` jb ‚`j`vbkiµdda ab uab pdtàki pdtàkib‘. b‘.

KbpµĲtujd = Ouskbae ba ed rbµĲk`s E`skàeiàed e` uab pdtàkib Gbstb bgdrb, kbeb fdrhb e` dp`rbkiµdda a hbt`hµbbtikb tikb qu` s` gb `xpu`std tià` su kdatrbrib. Sbrb jb suhb `xist` jb r`stb y pbrb jb hujtipjikbkiµda jb eivisiµdda. a. Yb qu` g`hds kdasie`rbed jb `j`vbkiµdda a b uab pdtàkib, `s abturbj pr`muatbr kuµbbjj `s su kdatrbrib. Kdhd jb `j`vbkiµdda a b uab pdtàkib kdasist` à hujtipjikbr ua aµ uh`rd uh`rd e` kdatiaud, su kdatrbrib eò` s`r jb eivisiµda da e` kdatiaud. Sdr `l`hpjd, ;? puè` s`r eivieied pdr ? pbrb ebr 26, qu` puè` s`r eivieied pdr ? pbrb ebr 7, qu` puè` eivieirs` pdr ? pbrb ebr 5, qu` puè` eivieirs` pdr ? pbrb ebr ? y qu`, b su v`z, puè` eivieirs` pdr ? pbrb ebr 2. (Kdhd à jb eivisiµdda, a, qu` `xpus `xpus`` à `j Kbp KbpµµĲtujd ;, `j breie `s t`rhi t`rhiabr abr `xbktbhàt` à 2.) Kdhd `stb eivisiµdda a pdr ? fu` r`p`tieb 0 v`k`s, puè` gbojbrs` e` `jjb kdhd ‚jb quiatb rb rbµµĲz e` ;?‘. Sdr dtrb pbrt`, siàed 72 ; ; ; ; 8 2, ``ss àtdak àtdak`s `s ; ‚jb kubrtb rbµ rbµĲz e` 72‘, 7 2‘, (´Sdr q quµ uµ` rbµĲĲz4 z4 Sdrqu` ;;?? ss`` e`s e`sbrrdjjb brrdjjb à o obs` bs` b jds j ds a aµ u uh`rds µh`rds ?, y 72 à obs` b jds aµu uh`rds h`rds ;, tbj y kdhd uab pjbatb s` e`sbrrdjjb e` sus rb rbµµĲk`s.) @j sµĲho Ĳhodj djdd e` uab ua b rbµĲz Ĳz `s . Jb vbr vbrièbe ièbe e` jb rbµ rbµĲz `s iaeikbeb pdr hèid e` ua p`qu`˚a add aµ u uh`rd h`rd à jb pbrt` sup`ridr izqui`reb e`j sµĲhodjd. Bs BsµµĲ, jb quiatb rb rbµµĲz e` ;? s` `skrio` ;? y jb kubrt kubrtbb rb rbµµĲz e` 72 s` `skri `skrio` o` 72. Bj simad s` j` jjbhb ‚simad rbeikbj‘ y jds aµu uh`rds h`rds qu` kdatiàà rbµ rbµĲk`s sda jjbhbeds ‚rbeikbj`s‘, viaiàed `st` tµ``rhiad rhiad e` jb pbjborb jbtiab ‚rbeix‘ qu` sima si maifl iflkb kb ‚rbµĲĲz‘. z ‘. Jb Jbss rb r bµĲk`s Ĳk` s hµbbss bjtbs sdjbhàt` s` àkuàtrba dkbsidabjhàt`. @j tipd hbs kdhµ u ua a e` rb rbµµĲz `s bqu`j qu` `s `j iav iav`rsd `rsd e`j kuber kuberbed. bed. @a

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dtrbs pbjborbs, ?0 `s `j kuberbed e` 0 y pdr jd tbatd 0 `s jb ?0 . @striktbhàt` gbojbaed jb ?0 eò`rµĲĲbb s`r jjbhbe jjbhbebb jb ‚s`muae ‚s`muaebb rbµ rbµĲz e` ?0‘ pu`std

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qu` ?0 0 0 8 2. [ia `ho `hobrmd, brmd, jb fbhiji fbhijibrieb briebe e e`j tµ``rhiad rhiad ‚kuber ‚kuberbed‘ bed‘ s` sdor`pusd b jb kdasistàkib. Jb ‚s`muaeb rb rbµµĲz e` ua a aµ u uh`rd µ h`rd kuber kuberbed‘ bed‘ s` bkdrtµd kdavirtiµ`àeds` aeds` à ‚rb ‚rbµµĲz kuberbe kuberbeb‘, b‘, y bsµ bsµĲ gb sied bedptb bedptbeb eb uaiv`rsbjhàt`. Jb J b ‚s ‚s`muaeb `muaeb rb rbµµĲz e` ?0‘ `s si si`hpr` `hpr` jjbhbe jjbhbebb ‚j ‚jbb rbµĲĲzz kub kuberbeb erbeb e` ?0 ?0‘. ‘. (@a imubj fdrhb, jbs ‚t`rk`rbs rb rbµµĲk`s‘ sda jjbhbeb jjbhbebss ‚‚rb rbµµĲk`s kµu uoikbs‘). oikbs‘). @s tba fbhijibr jb rb rbµµĲz kuberbeb qu` iakjusd s` bkdstuhorb uaiv`rsbjhàt` djviebrs` e`j p`qu`˚a add ? y `skr `skrioir, ioir, pdr `l`h `l`hpjd, pjd, jb rbµ rbµĲz kuber kuberbeb beb e` ?0 sihpj`hàt` kdhd ?0. Jb simuiàt` pr`muatb qu` surm` `s kµddhd hd e`t`rhiabr kuµbj bj `s jb rbµ rbµĲz e` ua aµuh`rd. uh`rd. Gbstb bgdrb s` gb vàied trboblbaed gbkib btrµbbs. s. [` kdadk` qu` ?0 `s ;?, pdr jd qu`, si s` eivie` ;? àtr` ? kiakd v`k`s, s` jj`mbrµb b 2. (E`s (E`spuµ puµ`s `s e` tded, gboiàed `j`vbed ua aµu uh`rd h`rd b uab pdtàkib `s sàkijjd iav`rtir jds

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pbsds e`skàe`r e```tded jb ed pdtàkib uabrhiabr v`z bbs). s).hµ``tikbhàt @ayr`bj r `bjieb iebe, e, ``jj hµ td pbrb e`t` e`t`rhi abrhµ brit brithµ tik bhàt`` jb jbss rbµĲĲk`s k`s utij utijizb izb `st` sist`hb e` trboblbr ‚gbkib btrµbbs‘, s‘, hµbbss qu` à jbs mrbae`s eivisida`s. Sdr `l`hpjd, pbrb `xtrb`r jb rb rbµµĲz kuberbeb e` 6?0 `j prdoj`hb s`r s`rµµĲb kdhd simu`<

[` pda` `j prih`r ? e` jb r`spu`st r`spu`stbb trboblbaed gbkib btrµbbss y sboiàed qu` ? ? sda 5, qu` `s jd hµbbss k`rkb qu` s` puè` jj`mbr b 6. [i s` pruòb kda `j ; s` jj`mb b ; ; 8 9, qu` s` pbsb e` jb kifrb hbrkbeb. Ju`md, gby qu` r`stbr y oblbr eds kifrbs, à v`z e` uab, kdhd s` gbk` à jb eivisiµdda, a, dp`rbki dp`rbkiµµdda a b jb kubj s` pbr`k` ua pdkd. ([i s` `stuvi`rb `xtrbyàed rb rbµµĲk`s kkµ u uoikbs, µoi kbs, gbo gborrµĲb qu` oblbr tr`s kifrbs> `xtrby `xtrbyàed àed kubtrd rb rbµµĲk`s, gbor gborµµĲb qu` oblbr kubtrd kifrbs, y bs bsµµĲ suk`siv suk`sivbhàt`). bhàt`). Sbrb dotà`r jb simuiàt` kifrb à jb r`spu`stb, gby qu` eivieir ??0 àtr` 50. [i jj`mb b 50 edojbaed jb prih`rb kifrb e` su r`spu`stb, jd kubj eb 5. Jb s`muaeb kifrb eò` s`r imubj b jb s`muaeb kifrb à jb r`spu`stb

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=9 y gby qu` prdobr vbribs kifrbs pbrb v`r kubj ebrµb `j aµ u uh`rd h`rd hµbbss k`rkbad b ??0. @j aµ u uh`rd h`rd 0 jd `s, pu`std qu` 0 50 gbk` `xbktbhàt` ??0.

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@std puè` pbr`k`r suhbhàt` kdhpjikbed, y jd `s, ad gby qu` a`mbrjd, p`rd `j prdk`sd eb jumbr b bjmuads r`sujtbeds vitbj`s. Sdr `l`hpjd, ´kuµbbjj `s jb rb rbµµĲz kuberbe kuberbebb e` ?, d, à sµĲhodjd Ĳhodjds, s, ? 4 ´R ´Ruµ uµ` aµu uh`rd h`rd hujtipjikbed pdr sµĲ hishd eb kdhd r`spu`stb ?4 Sbrb `hp`zbr, pde`hds v`r, e`se` ju`md, qu` aiamµu ua a àt`rd s`rvirµb pbrb jb r`spu`stb pu`s 2 2 8 2 y ? ? 8 5. @j prih`rd `s huy p`qu`˚a add y `j s`muaed huy mrbae`. Jb r`spu`stb eò` `stbr àtr` 2 y ? y, pdr tbatd, s`r uab frbkkiµdda. a. ´Sde`hdss gbojbr e` uab rb ´Sde`hd rbµµĲz kuberbe kuberbebb frbkkidab f rbkkidabrib4 rib4 ´Sdr q quµ uµ` a ad4 d4 [`mµu ua a ? ? ? si maifl iflkb kbrrµĲb 2 0 2 ?0 au`strb e`flaikiµdda a drimiabj e` kifrbs `xpdaàkibj`s, ( 2 0 ) sima , si uad qui`r` tdhbrs` jb hdj`stib e` dotà`rjd. y `j r`sujtbe r`sujtbedd s`r s`rµµĲb 2 ?5 ?0

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@std simaiflkb qu` 2 ?5 8 2 ?0 . Ad sdjbhàt` tà`hds uab rbµ rbµĲz kuberbeb ?0 frbkkidabrib, frbkkida brib, siad qu` tb tbhoiµ hoiµ`à a tà` tà`hds hds u uab ab frb frbkkiµ kkiµdda a à `j pbp`j e` ua aµ u uh`rd h`rd kuberbed, y bhods àkblba eàtrd e` jbs r`mjbs qu` rimà b jds àt`rds. uh`rd h`rd qu` s` bprdIaikibjhàt`, 2 ?0 hujtipjikbeb pdr sµĲ hishd, eb ua aµu ? xihb obstbat` b ?, pdr jd kubj 2 0 eò` bprdxihbrs` obstbat` b jb ?. [d2 jbhàt` ads fbjtb ?0 pbrb jj`mbr b jb h`tb, pu`std qu` `j kuberbed e` 2 ?0 `s ?0 ?5 2 ?0 à v`z e` 2 ?0 qu` s`rib ?. 52 , si s` hujtipjikb pdr si hishb, Sde`hds bk`rkbrads hµbbs. s. Jb frbkkiµda da 2 233 9772 Sder erµµĲb ss`r `r qu qu`` si blustµ bl ustµbbrbhds rbhds eb kdhd r`spu`stb 2 23333 qu` s` bk`rkb hµbbss b ?. Sd jd suflkiàt` jbs kdsbs, dotàer dotàerµµĲbhds à r`bjiebe uab frbkkiµ dda a (tbj v`z huy kdhpjikbeb) qu` fu`rb `xbktbhà `xbktbhàt` t` jb rb rbµµĲz kuberbeb e` ?. S`rd, ´jd gbr`hds4

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Kdhp Kd hpbr brba baed ed jµĲa`b Ĳa`bss @st` prdoj`hb e` àkdatrbr jb rb rbµµĲz kuberbeb e` ? s` j`s pr`sàtµ d prih`rbhàt` b jds mri`mds. Kdhd `rba m`µddh`trbs h`trbs bat` tded, kdhd yb s` `xpusd bat`ridrhàt`, s` pr`dkupboba hukgd hµbbss à kdhpbrbr à kifrbs jbs jdamitue`s mitue `s e` jbs jµĲa`bs Ĳa`bs.. Sdr `l`h `l`hpjd, pjd, supdambh supdambhds ds qu` `stbh `stbhds ds kdasi kdasie`rba e`rbaed ed ua kuberijdamd qu` tià` uab eibmdabj à jb fdrhb iaeikbeb à `j simuiàt` eiould.(44) ´Kµddhd hd s` kdhpbrbr kdhpbrbrµµĲb jb jdamitue e` jb eibmdabj kda jb jdamitue e` jds jbeds e`j kuberijdamd4 @vieàt`hàt`, jb eibmdabj `s hµbbss jbrmb qu` kubjqui`rb e` jds jbeds. S`rd, ´kuµbbatd atd hµbbs4 s4 Jds mri`mds e`s`boba sbo`rjd.

73

KBS µ IUPJD IUP JD =. OP[KBA OP[KBAED ED _Bµ IK@[

Fimurb =.2< Kuberijdamd kda eibmdabj

Bgdrb oià, kubaed kdhpbrbhds jbs jdamitue`s e` eds jia`bs uab e` jbs kubj`s tià` ? kàt kàtµµĲh`trds e` jdamitue y jb dtrb 2, e`kihds qu` ‚jbs jdamitue`s sda kdhd ? `s b 2‘. [i uab tià` 5 kàt kàtµµĲh`trds e` jdamitue y jb dtrb ?, p pde`hds de`hds e`kir qu` ‚jbs jdamitue`s sda kdhd 5 `s b ?‘. [ia `hobrmd, à bhods kbsds, jb jµĲa`b hµbbss jbrmb `s `xbktbhàt` `j edoj` e` jb jdamitue e` jb hµbbss kdrtb. Sbrb `j hbt`hµbbtikd, tikd, `j g`kgd e` qu` uab jµĲĲa`b a`b s`b jd edoj` e` jdamitue qu` jb dtrb `s e` hbydr iiat`rµ at`rµ`s `s y simaifl simaiflkbkiµ kbkiµdda a qu` `j e` qu` uab jµĲa`b hie hiebb ? kàtµĲĲh`trds h`t rds y jb dtrb 5, d qu` uab hieb ?5 kà kàttµĲh`tr Ĳh`trds ds y jb dtrb 57. @j pr`fl`r` e`kir à tdeds jds kbsds qu` ‚jbs jdamitue`s sda kdhd ? `s b 2‘. Jb h`ldr hba`rb e` btàe`r `std butdhµbbtikbhàt` tikbhàt` `s r`pr`sàtbr jb eif`ràkib e` jdamitue`s kdhd uab frbkkiµdda. a. @a dtrbs pbjborbs, si uab jµĲa`b hie` ? ? kàtµ kàtµĲh`trds y jb dtrb 2 p pde`hds de`hds e`kir qu` jbs jdamitue`s sda . Y si au`s2 trbs trb Ĳa`bss hie hieà à 57 kàt kàtµµĲh`t Ĳh`trds rds pd pdrr ?5 kàt kàtµµĲh`t Ĳh`trds rds jb frbk frbkkiµ kiµdda a s` kdavi`rt` s57jµĲa`b à qu`, kdhd frbkkiµdda, a, puè` kbhoibrs` bj eivieir àtr` ?5 `j auh`rbedr ?5 ? y kdav`rt kdav`rtirs irs`` bs bsµµĲ à 2 Jb frbkkiµdda a qu` r`pr`sàtb uab kdhpbrbkiµda da e` eds hèiebs sihijbr`s (jbs jdamitue`s e` eds jµĲa`bs, jds vdjµ u uhà`s hà`s e` eds r`kipiàt`s, jds p`sds e` eds gdhor`s) `s jjbhbeb ‚rbzµdda‘. a‘. Sdr supu`std, eds jdamitue`s puèà ad `stbr tba r`jbkidabebs kdhd pbrb ebr uab rbzµda da tba sihpj` kdhd ? b 2. [updambhds qu` uab jia`b tià` 2 ? 9 . Jb rbzµdda a ad `s pr`kisbhàt` . @s kàtµµĲh`trd e` jdamitue y jb dtrb 2 23 kàt 2

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`hobrmd, eàtrdàe``st` jbs kbsd, frbkkida`s sda uab iaa`k`sbrib [ia kdhpjikbkiµ dda, a, jbs qu`frbkkida`s puè` `vitbrs`, hujtipjikbaed pdr 23 `j 29 auh`rbedr y `j eàdhiabedr y bs bsµµĲ t`rhiabr`hds kda uab rbzµdda a imubj b . 23 @a `stb fdrhb, eds aµu uh`rds h`rds kubj`squi`rb `xpr`sbeds kdhd frbkkiµdda a puèà kdav`rtirs` b uab rbzµdda a e` aµ u uh`rds h`rds àt`rds. Sdr `l`hpjd, si uab jµĲa`b 5 , jb rbzµdda a e` jbs eds jdamitue`s h`t rds e e`` jda jdamitu mitue e y jb ddtrb trb 2 2; hie` ? 2= kàtµĲĲh`trds 20

s`rµĲb ? 2

5 2= 2;

. [i hujtipjikbhds ‛sup`ridr ` iaf`ridrhàt`‛ `stb bpbràt`hàt`

20

?70 kdhpjikbeb frbkkiµdda a pdr 2?= 2? , jb rbzµdda a s` kdavi`rt` à kdhpràeiàed ?;7 sdjbhàt` àt`rds. (@std (@ std s`rµĲĲbb hµbs bs sàkijjd usbaed uab badtbkiµdda a e`kihbj, p`rd jds mri`mds ad

jb tà tàµµĲba y jb gistdrib s` gbk` hµbbss bpbsid bpbsidabat abat`` si s`muih s`muihds ds `j prdkèi prdkèihià hiàtd td usbed pdr `jjds pbrb `st` dol`td.) Yb `stbhds jistds pbrb vdjv`r b au`strd kuberijdamd, edae` e`lbhds b au`strds mri`mds trbtbaed e` e`skuorir jb rbzµdda a e` jbs jdamitue`s e` jb eibmdabj y e` jds jbeds. Sbrb sihpjiflkbr jbs kdsbs, eir`hds qu` kdhd jbs eds pbrt`s e`j kuber kuberijdamd ijdamd,, b kbeb jbed e` jb eibmda eibmdabj, bj, sda p`rf`k p`rf`ktbhà tbhàt` t` sihµ ``trikbs, trikbs, pde`hds e`lbr fu`rb uab e` `sbs pbrt`s. Jb qu` quèb, kdhd s` puè` v`r à jb ihbmà (44) `s jjbhbeb ‚triµbbamujd amujd r`ktd‘ y jd qu` à `j kuberijdamd `rb uab eibmdabj s` jjbhb bgdrb jb ‚gipdtàusb‘.

Fimurb =.?< Uriµbbamujd amujd r`ktd kda gipdtàusb Jds `mip `mipkids kids sbo sboµµĲba, e` hukgds simjds btrµbbss y sihpj`hàt` pdr `xp`riàkib r`bj, qu` si ua jbed e` ua triµbbamujd amujd r`ktd hè hèµµĲb ; uaiebe`s e` jdamitue y `j

7?

KBS µ IUPJD IUP JD =. OP[KBA OP[KBAED ED _Bµ IK@[

dtrd jbed 5, jb gipdtàusb hè hèµµĲb `xbktbhàt` 0 uaiebe`s . @a ua kbsd kdhd ; `s` jb rbzµdda a e` jb gipdtàusb bj jbed hµbbss jbrmd `rb e` y bj jbed hµbbss kdrtd 0 5 e` . ; Jds mri`mds mà`rbjizboba hµbs. bs. E`s`boba kdadk`r jb rbzµdda a e` kubjqui`r triµbbamujd amujd r`k r`ktd. td. Pad à `j kub kubjj jds jbe jbeds ds tuv tuvi`r i`rba ba 7 y ?; k` kàt atµµĲh` Ĳh`trd trdss e` jdamitue, d 2 y =, d ?7;== y ;396?2 ... kubjqui`rb. E` bku`red kda jb gistdrib, Sitµbbmdrbs, mdrbs, `j hbt`hµbtikd btikd mri`md e`skuoriµd qu`, à kubjqui`r   triµbbamujd amujd r`ktd, `j kuberbed e` uad e` sus jbeds hµbbss `j kuberbed e`j dtrd jbed imubjboba bj kuberbed e` jb gipdtàusb. @a su gdadr, b `std s` j` gb jjbhbed `j t`dr`hb e` Sitµbmdrbs, bmdrbs, bua kubaed à r`bjiebe jds kgiads pbr`kà gbo`rjd kdadkied 633 b˚ a ads ds bat`s e` Sitµbbmdrbs. mdrbs.

Fimurb =.;< U`dr`hb e` Sitµbbmdrbs mdrbs @stb `s jb fdrhb à qu` dp`rb `j t`dr`hb e` Sitµbbmdrbs. mdrbs. [updambhds qu` ua triµbbamujd amujd r`ktd tià` uab jdamitue e` ; kàt kàtµµĲh`trds à uad e` sus jbeds y 5 kàt kàtµµĲh`trds à `j dtrd (44). @j kuberbed e`j prih`r jbed `s ; ; d s`b 9> `j kuberbed e`j s`muaed jbed `s 5 5 d s`b 26. @j kuberbed e` jb gipdtàusb

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7; `s jb suhb e` jds kuberbeds e` jds jbeds, d s`b 9 + 26 8 ?0. [i `j kuberbed e` jb gipdtàusb `s ?0, àtdak`s jb gipdtàusb tià` uab jdamitue imubj b jb rb rbµµĲz kuberbeb e` ?0 qu`, abturbjhàt`, `s 0. G` bqu bquµµĲ dtrd kbsd. Pa jbed e` ua triµbbamujd amujd r`ktd hie` 0 kàt kàtµµĲh`trds y `j dtrd 2?. @j kuberbed e`j prih`rd `s 0 0, d s`b ?0 y `j e`j s`muaed 2? 2?, d s`b 255. [uhbaed ?0 y 255 tà`hds 269 qu` `s `j kuberbed e` jb gipdtàusb. Jb jdamitue e` jb gipdtàusb `s, pdr jd tbatd, 269, d s`b 2; kàt kàtµµĲh`trds, pu`std qu` 2; 2; 8 269. @a `st` triµbbamujd, amujd, jb rbzµdda a e` jb gipdtàusb bj 2; 2; jbed hµbbss jbrmd `s e` y bj jbed hbs kdrtd e` . 2? 20 Psbaed `j t`dr`hb e` Sitµbmdrbs bmdrbs ki`rtbhàt ki`rtbhàt`` pbr`k`r pbr`k`rµµĲb qu` jb rbzµdda a e` jb gipdtàusb y jds jbeds pder pderµµĲb dp`rbr pbrb kubjqui`r   triµbbamujd amujd r`ktd, kda jd kubj `stbr `stbrµµĲba f`jik`s jds hbt`hµbbtikds tikds mri`mds. @sp`kibjhàt` eò` dp`rbr pbrb `j hµbbss sàkijjd e` tdeds jds triµbamujds bamujds r`ktds, bqu`j à `j kubj eds e` sus

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jbeds tiàà jb hishb jdamitue (`j triµbbamujd amujd isµddsk`j`s). sk`j`s). Eikgd triµbbamujd amujd bpbr`k` eioulbed b kdatiaubkiµdda a (44). Sbrb sihpjiflkbr jd hµbbss p pdsioj`, dsioj`, vbhds b supda`r qu` kbeb jbed hie` 2 kàt kàtµµĲh`trd e` jdamitue. @j kuberbed e` kbeb jbed `s 2 2, d s`b 2. [uhbaed jds eds kuberbeds tà`hds ? kdhd kuberbed e` jb gipdtàusb. Jb jdamitue e` jb gipdtàusb s`rib àtdak`s ?, kubjqui`rb qu` s`b.

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Fimurb =.5< Uriµbbamujd amujd isµddsk`j`s sk`j`s

Uded jd qu` jds mri`m mri`mds ds tà tàµµĲba qu` gbk`r `rb dotà dotà`r `r uab frbk frbkkiµ kiµ dda a qu` puei`r r`pr`sàtbr jb ? y bbssµĲ `stbr kbpbkitbeds pbrb e`kir qu` jb rbzµdda a e` jb gipdtàusb bj jbed e` ua triµbbamujd amujd isµddsk`j`s sk`j`s `rb `sb frbkkiµdda. a. Y pdr hèid e` ua blust` be`kubed, p pder derµµĲba gbo gbo`r `r kdav`rtied `sb rbzµdda a b uab qu` sµddjd jd kdhpràei`rb àt`rds... y pderµĲĲba ba gbo`rs` ied b k`jòrbr su e`skuorihiàtd.

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75

KBS µ IUPJD IUP JD =. OP[KBA OP[KBAED ED _Bµ IK@[

Jb ia`xistàkib e` frbkkida`s ?

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Bj priakipid e`j kbp kbpµµĲtujd, eil`? qu` 2 0 `rb kbsi `j vbjdr e` ?. ?. [i fu`rb pderµ derµĲĲbb kdav`rtirs kdav`rtirs`` b a aµ u uh`rds µh`rds àt`rds `j vbjdr `xbktd, jb rbzµda s` s`rµ rµĲĲbb 2 0 qu` p 2 hujtipjikbaed pdr 0 `j auh`rbedr y `j eàdhiabedr pbrb qu` s` kdavirti`rb = à . 0 52 s` bk`rkb hµbs bs da 2 233 S`rd, ¡by!, 2 ?0 ad `s `xbktbhàt` `j vbjdr. Jb frbkkiµda 52 252 y ebrµĲb Ĳb uab rbzµdda a e` 2 233 µd `s à aµ u uh`rds h`rds àt`rds. Jb frbkkiµda da 2 ?3= 033 2 233 ?3= =3= 2 à aµ u uh`rds h`rds àt`rds. hbs k`rkbab bua, y ads ebr ebrµµĲb uab rbzµdda a e` 033 d 2 033 tbhpdkd tbhp dkd `s `xbktb. Eurbat` uab µ``pdkb, pdkb, jbs hbt`hµbbtikds tikds mri`S`rd 2 ?3= 033 mds eèikbrda su ti`hpd b ouskbr y trbtbr e` àkdatrbr jb frbkkiµdda a `xbktb qu` r`pr`sàtbrb b jb ?, ?, b fla e` `stbr à pdsioijiebe e` pda`r jb rbzµda da ? 2 à aµ uh`rds uh`rds àt`rds. [` `stuvi`rda bk`rkbaed obstbat`, p`rd auakb jj`mbrda b dotà`r jb frbkkiµdda a `xbk `xbktb. tb. Fiabjhàt`, `j hb hbt`hµ t`hµbbtikd tikd mri`md @ukjie`s e`hdstr e`hdstrµµd qu` pdr hukgd qu` e`s`brb uad `sfdrzbrs `sfdrzbrs`, `, s`r s`rµµĲb iaµutij utij trbtbr e` dotà`r jb frbkkiµdda. a. A Add ``xi xistµ stµĲb Ĳb tbj tbj frbkkiµ dda. a. ([` ku kuàtb, àtb, bua ku kubaed baed pr prdoboj`h doboj`hàt` àt` jb gistd gistdrib rib ad `s v`r v`rµµĲeikb, qu` Sitµbbmdrbs mdrb s yb gbo gboµµĲb e`skuoi e`skuoi`rtd `rtd `std, p`rd qu` gbo gboµµĲb quèbed tba gdrrdri gdrrdrizbed zbed bat` g`kgd g `kgd tb tba a `x `xtrbdreia trbdreiabrid, brid, qu` gbo gboµµĲb g`kgd lu lurbr rbr b ssus us eisk eiskµµĲpujds qu` ad r`v`jbrµµĲba `j s`kr`td, y µ``jj hishd auakb jd kdhàtµdd). r`v`jbr ). ? [i ad puè` `xp `xpr`s r`sbrs brs`` kdhd uab rbz rbzµµdda a qu` sdjbhàt` kdhpràeb 2

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àt`rds, àtdak`s tbhpdkd puè` aiamuab e` jbs dtrbs frbkkida`s qu` ? gbk`rjd 5 kdatàmba ?, tbj`s kdhd yb qu` tdebs `stbs frbkkida`s puèà d ? ? ? ? ? s`r kdav`rtiebs à bj hujti hujtipjikbr pjikbrjbs jbs pdr bjmd. Sdr `l`hpjd `l`hpjd,, 8 2 ? 2 2 ? ? ? . Sdr dtrb pbrt`, ? 8 d, si s` hujtipj hujtipjikb ikba a `j auh` auh`rbe rbedr dr y `j ? 2 2 5 .(Eò` r`kdrebrs` qu` ? kubjqui`rb qu` eàdhiabed eàdh iabed pdr ?, `s imubj b ? s`b hujt hujtipjikb ipjikbeb eb pdr ? `s imubj b ?). Kdhd jb ? ad puè` fdrhbr pbrt` e` uab rbzµdda a qu` puèb `xpr`sbrs` sdjbhàt` à àt`rds, s` kdasie`rb qu` `s ua ‚aµuh`rd uh`rd irrbkidabj‘. Sdr

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70 dtrd jbed, jds aµu uh`rds h`rds qu` puèà fdrhbr pbrt` e` uab rbzµdda a qu` sdjbhàt` puèb `xpr`sbrs` à àt`rds, sda ‚aµ u uh`rds h`rds rbkidabj`s‘. Udeds jds àt`rds y frbkkida`s, s`ba pdsitivds d a`mbtivds, sda aµu uh`rds h`rds rbkidabj`s. [ukè` qu` kbsi td tdebs ebs jbs rb rbµµĲk`s kuberbebs sda irrbkidabj`s. @a r`bjiebe, jds u uaikds µaikds aµ u uh`rds h`rds qu` tià` rbµĲĲk`s k`s kuberbebs rbkidabj`s sda jds àt`rds qu` fdrhba fdrhb a jb s`ri s`ri`` e` kuber kuberbeds beds hàki hàkidabeds dabeds bj priak priakipid ipid e`j kbp kbpµµĲtujd 6 (jjbhbeds ‚kuber ‚kuberbeds beds p`rf`k p`rf`ktds‘, tds‘, eòie eòiedd b qu` jbs rbµ rbµĲk`s kuber kuberbebs bebs r`suj r`sujtba tba s`r àt`rds `xbktds), d frbkkida`s kdhpu`stbs e` `sds aµu uh`rds. h`rds. Sdr `l`hpjd, 26 = = 8 `s ua aµ u uh`rd h`rd rbkidabj, pdrqu` 2 9 9 qu` `s uab frbkkiµddaa kdhpu`s29 5 tb e` kuberbe kuberbeds ds p p`rf`ktds. `rf`ktds. Jb rbµĲĲzz kub kuberbeb erbeb e` 2 =9 `s, pdr jd tbatd, (siàed ; 5 jb rb rbµµĲz kuberbeb e` 26 y ; jb rb rbµµĲz kuberbeb e` 9), d s`b 2 2; . @j g`kgd e e`` qu` jb hbydr hbydrµµĲb e` jjbs bs rb rbµµĲk`s kuberbe kuberbebs bs s`ba irrbkida irrbkidabj`s bj`s , pdr p dr

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aiamµ u ua a fµ hdtivd gbk` utij`s. tij`s. Kda fr`kuàkib `s puè` a`k`sbrid usbr jb ?pdr à bjmuab ddrhujb rhujb jbs kiàt kiàtµ µĲflkbiaµ du e` iamài`r iamài`rµ µĲb. [u vbjdr e`sbrrdjjbrs`, hèid e`j hµ``tded tde d e` e`skritd skritd bj p priakipi riakipidd e` `st` kbp kbpµµĲtujd, gbstb tbatds puatds e`kihbj`s kdhd tàmb uad jb pbkiàkib e` kbjkujbr. Sdr `l`hpjd, jb ? puè` kbjkujbrs` gbstb sìs puatds e`kihbj`s y àkdatrbr qu` `s imubj b 2.525?25. @stb kifrb ad `stµb huy bj`lbeb e`j vbjdr v`rebe`rd, pu`std qu` 2.525?25 2.525?25 8 ?.333332?;==96. @st` r`sujtbed eifl`r` e` ? sdjbhàt` pdr s`r ua pdkd hµbs bs bjtd à uab hijjdaµ``sihb sihb pbrt`. Sdr `sb rbzµdda, a, 2.525?25 `s uab kifrb suflkiàt`hàt` ouàb pbrb kbsi kubjqui`r prdoj`hb à qu` `stµ ` iavdjuk iavdjukrbeb rbeb jb ?. Y bua si ad jd `stuvi`rb, ua p`qu`˚ a add `sfu`rzd beikidabj prdpdrkidabrµb ua hbydr aµu uh`rd h`rd e` e`kihbj`s... à r`bjiebe, tbatds kdhd uad qui`rb.

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[ia `hobrmd, si s` e`s`b, t`rkbhàt`, iasistir à dotà`r jb rbµ rbµĲz kuberbeb (s` jjbhb ‚`xtrb`r jb rb rbµµĲz‘) gbstb qu` s` bkboà jds e`kihbj`s, s` àkdatrbrµb qu` `sd ad `s pdsi pdsioj` oj`.. Ad gby gby tµ ``rhiad rhiad.. Ad ihpd ihpdrtb rtb kuµ bbatds atds e`kihbj`s s` kbjkujà pbrb jb rb rbµµĲz kuberbeb e` ?, si`hpr` gborµb hµbbs. s. @std puè` sdrpràe`r bj j`ktdr , kdhd jd gizd jb situbkiµdda a dkurrieb kda jb 2 v`rsiµdda a e`kihbj e` , jb kubj, kdhd s` r`kdrebrµbb,, `s imubj b 3.;;;;;;;;.... y bsµĲ ; 2 suk`sivbhàt`, suk`siv bhàt`, sia tà`r fla> d jb v`rsiµ dda a e`kihbj e` qu` `s 3.25?70=25?70=25?70=.... = sia tà`r tbhpdkd fla y sia qu` s`b kdhpj`tb. @sds e`kihbj`s siafµĲĲa a tiàà ua `quiv `quivbjàt` bjàt` frbkkidabrid, hiàtrbs qu` `j e`kihbj qu` r`pr`sàtb b jb ? ad jd ttià`. ià`. ´Sdr quµ``44 O Ouàd, uàd, jds `quivbjàt`s

∜

76

KBS µ IUPJD IUP JD =. OP[KBA OP[KBAED ED _Bµ IK@[

2 2 e`kihbj`s e` y e` , y ua aµuh`rd uh`rd iae`flaied e` dtrbs frbkkida`s e` jb his; = deikbs . Jds e`kihbj` hb `sp`ki` sda frbkkida`s e`kihbj`s p`riµ e`kihbj`ss siaf siafµµĲa puèà kdav`rtirs` b frbkkida`s sdjbhàt`  si si sda e`kihbj`s p`riµddeikds. eikds. deikd. @j puatd `s qu` `j e`kihbj uad, qu` dp`rb pbrb jb ?, ad `s p`riµ [ia ihpdrtbr kuµba ba j`lds s` jj`mu`, ad s` àkdatrbrµb aiamuab adrhb, yb s`b à jb ?, d à `j `quivbjàt` e` kubjqui`r aµu uh`rd h`rd irrbkidabj. [i s` àkdatrbrb kubjqui`r adrhb, pdr p`qu`˚a abb qu` fu`r fu`rb, b, pder pderµµĲb gbk` gbk`rs` rs` uab frbkk frbkkiµ iµdda a e`j e`kihbj. [i `xisti`rba ua hijjµda da e` eµĲmitds ad p`ri p`riµddeikds µeikds y sia adrhb bjmuab, y e`sp e`spuµ uµ``ss `s` hij hijjµ jµdda a s` r`piti`rb uab y dtrb v`z, p pderµ derµĲĲbb fdr fdrhbrs` hbrs` u uab ab frbkk frbkkiµ iµdda. a. D si `xisti`rb ua hijjµdda a e` eµĲmi Ĳmitds tds p`r p`riµ iµddeikds eikds (d ua oijjµdda a d ua trijjµdda) a) y, e` prdatd, uab iaflaitb s`ri` e` ...000 ...000000000... 000000... d kubjqui`rb dtrb, àtdak`s pd pder erµµĲb fdrhbrs` uab frbkkiµdda. a. [ia `hobrmd, à jds aµ u uh`rds h`rds irrbkidabj`s jds `quivbjàt`s e`kihbj`s ad r`pità kda p`rideikiebe y ai siqui`rb bpbr`k` uab jim`rb adrhb, ai puè` bpbr`k`r. (Sdr supu`std, pderµĲb Ĳb `j j`ktdr pr`muatbr kµddhd hd puè` sbo`rs` jd qu` sukè` e`spuµ `s `s e` gbo`r jj`mbe jj`mbedd bj trijj trijjba`si ba`sihd hd pua puatd td e`ki e`kihbj hbj si abei` jd gb iatàtbed. Jb kbràkib e` adrhbs e` jds e`kihbj`s irrbkidabj`s gb sied eèukiebb b trb eukie trbvµ vµ `s `s e` ki`rt ki`rtbs bs pr`hi pr`hisbs sbs fuaeb fuaebhà hàtbj`s tbj`s e` jbs hbt`h hbt`hµµbbtikbs. tikbs. [i bpbr`ki`rb bpbr` ki`rb kubjq kubjqui`r ui`r adrhb à tbj`s e`ki e`kihbj`s, hbj`s, bua jb hbs jim`r jim`rb, b, tàer tàerµµĲb qu` r`visbrs` tdeb jb `strukturb e` jbs hbt`hµbbtikbs. tikbs. [ia `hobrmd, ad `s e` `sp`rbrs` qu` `std bkdat`zkb).

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Jb `xistàkib e` jbs frbkkida`s Vbhds b kdasie`rbr bgdrb uab `xpr`siµdda a qu`, eimbhds, s`b (?5 )? . @std simaiflkb qu` jb `xpr`siµdda a ?5 tià` qu` `j`vbrs` b jb kubrtb pdtàkib. @j vbjdr e` ?5 `s ? ? ? ?, d s`b 26. @j kuberbed e` 26 `s 26 26, d s`b ?06, e` edae` (?5 )? 8 ?06. ? 06. S`rd ?06 `s tbhoiµ`à a ? ? ? ? ? ? ? ?,d ?7 . Sdr jd tbatd, (?5 )? 8 ?7. [i uad pruòb `stds prdoj`hbs vbribs v`k`s kda eif`ràt`s aµu uh`rds h`rds y `xpdaàt`s pbr`k`rµb, b, à mà`rbj, qu` bj `j`vbr ua aµu uh`rd h`rd `xpdaàkibj b ki`rtb pdtàkib jb r`spu`stb iakjuirµb jb hujtipjikbkiµdda a e` e eds ds `xpd `xpdaàt`s. aàt`s. Bs BsµµĲ, sia a`0 ? 23 k`siebe e` gbk`r aiamµu ua a kµbbjkujd, jkujd, pd pde`hds e`hds sbo`r qu` (; ) 8 ; y (=7 )= 8 =06. [i µ``s` s` `s `j kbsd, ´pdr quµ ` ad pde`hds ie`br ua auh auh`rd `rd qu` puèb s`r 5 r`pr`sàtbed, eimbhds, kdhd (? ) 4 [imuiàed `j sist`hb e` hujtipjikbr eds

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2 ?

7= 2

`xpdaàt`s, à tbj`s kdaeikida`s pbr`k`r pbr`k`rµµĲb qu` (?5 ) `s imubj b ?? . [i bsµĲ `s, kdhd ?5 8 26 y ?? 8 5, jd qu` `stbhds eiki` eikiàed aed `s qu` 26 8 5. Sbr Sbr`k`r `k`rµµĲb ?

àtdak`s qu` ua `xpdaàt` e` 2 `s `quiv `quivbjàt` bjàt` b `xtrb`r jb rbµ rbµĲz kuberbeb. ? @a dtrbs pbjborbs 26 8 26. Hµbbss bµ ua, ua, simuiàed `j hishd rbzdabhiàtd, puè` e`hdstrbrs` qu` 26 `s imubj b 26 (`s e`ki e`kir, r, jjbb rbµĲĲzz kkµ u uoikb µoikb e` 26)> 26 8 26 y bsµĲ suk` suk`sivbh` sivbhàt`. at`. BquµµĲ tà`hds jb iatrdeuk Bqu iatrdeukkiµ kiµdda a b jds `xpdaàt`s frbkkidabrids qu` prdh`t prdh`tµµĲ bj flabj e`j kbpituj kbpitujdd 6. Dosµ``rv`s`, rv`s`, tbhoiµ`à, a, kdhd kbsd kuridsd kuridsd,, qu` jb ? qu` ad puè`   `xpr`sbrs` kdhd frbkkiµ da, da, puè`  `xpr`sbrs` pdr hèid e` ua aµ u uh`rd h`rd sàkijjd iakjuyb ua `xpdaàt` frbkkidabrid, b sbo`r, ?. Ubhoi bhoiµµ`à a `s pdsioj` dotà dotà`r `r `j simaifl simaiflkbed kbed e` ua aµ uh`rd uh`rd kdhd 26 e` jb simuiàt` hba`rb. @j aµu uh`rd h`rd 26 puè` kdav`rtirs` b (26; ) pu`std qu` 2 ; ; 8 . Sdr jd tbatd, 26 siàed imubj b (26; ) `s imubj b 26; . @a mà`rbj, ? ? kubjqui`r `xpdaàt` frbkkidabrid `j`vb `j aµu uh`rd h`rd obs` b jb pdtàkib iaeikbeb pdr `j auh`rbedr y `xtrb` jb rb rbµµĲz bj auh`rd iaeikbed pdr `j eàdhiabedr. BsµĲ ? s`r s` rµĲb jb ?;0bvb. rb rbµµĲz e` jb 06=bvb. pdtà pdtàkib kib e` ?, pueiµ`àeds` aeds` sihodjizbr kdhd ?06=. Abturbjhàt`, tbj`s `xpdaàt`s frbkkidabrids sda eif eifµµĲkij`s e` hba`lbr. 2 Û@WUD FBJUBAU@X Eò` eµbbrs`j` rs`j` `sp`kibj simaiflkbed b kubjqui`r `xpdaàt` e`kihbj. Sdr `l`h0.2=0 pjd, ? `quivbj` b e`kir ? pu`std qu` 0.2=0 `s imubj b ?3= . B su v`z, ? 53 `s uab kifrb dotàieb pdr `j`vbkiµdda a e`j ? b jb ?3=bvb. pdtàkib y `xtrbkkiµda da e` jb kuberbmµ``sihb sihb pdt` pdtàkib akib ( d `xtrbkkiµdda ae e`` jb kuberbmµ``sihb sihb pdt` pdtàkib akib e` ? `j`vbaed `j r`sujtbed b jb ?3=bvb. pdtàkib, sia ihpdrtbr `j dreà à qu` s` jj`và b kbod kbo d jbs eds dp dp`rbkida`s. `rbkida`s. Jd puè` uad v`r pdr sµĲ hishd si jd pruòb 2 ?

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2

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2

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5

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5

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2

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06= ?;0

∜

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?3=

?3=

53

53

;

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kda 5. Jb rb rbµµĲz kuberbeb e` 5 `s imubj b jb 65 65 qu` `s 7. @j kuod e` 5 `s ?; , qu` tbhoiµ`à a `s 7). uh`rd h`rd obs` `j`vbed b kubjqui`r pd@j vbjdr e` ? (d `j e` kubjqui`r aµu tàkib, àt`rd, frbkkidabrid d e`kihbj, pdsitivd d a`mbtivd) puè` dotà`rs` bprdxihbebhàt` bprdxihbebhà t` pdr hèid e` tµ``kaikbs kaikbs hbt`hµbbtikbs tikbs bprdpibebs. @stbs ad kdhpràeà, à r`bjiebe, jb hujtipjikbkiµdda a e` ?3= eds`s, d àkdatrbr uab kuberbmµ``sihb sihb rb rbµµĲz e` kubjqui kubjqui`r `r kkdsb. dsb. @j vbjdr jj jj`mb `mb b kkbsi bsi ;6.2?6 ;6.2?6.. @j vbjdr `s bprdxihbed, pdrqu` ? `s irrbkidabj (kdhd jd sda kbsi tdeds jds aµ uh`rds uh`rds qu` kdatiàà ua `xpdaàt` frbkkidabrid d e`kihbj). Sdr jd tbatd, `j `quivbjàt` e`kihbj `s iaflaitd, y ad p`rideikd, p`rd puèà bprdxihbrs` ?3= 53

?3= 53

2

A. e`j Urbaskriptdr< bj trbaskrioir s` v` gbk` fbjtb ua pµbrrbfd brrbfd e`j jiord trbaskritd

77

KBS µ IUPJD IUP JD =. OP[KBA OP[KBAED ED _Bµ IK@[

tbatds puatds kdhd s`b a`k`sbrid b jb utijizbkiµdda a r`qu`rieb. Psbaed kubjqui`r aµu uh`rd h`rd kdhd obs`, pde`hds `xpr`sbr kubjqui`r dtrd

Ý

aµu uh`rd h`rd à fdrhb `xpdaàkibj. Sdr `l`hpjd, yd h` fllµ` `j prdoj`hb e` = 2= bj flabj e`j kbp kbpµµĲtujd 6. Bgdrb oià, = puè` `xpr`sb `xpr`sbrs` rs` kdhd ? ?.72, d ;2.==, d 02.?2, (`xistiàed fdrhbs e` kbjkujbr `stds `quivbjàt`s `xpdaàkibj`s) y 2= `s ?5.37, d ;?.07, d 02.=6. @j prdoj`hb `s e` = 2= puè` àtdak`s `xpr`sbrs` kdhd ? ?.72 ?5.37 8 ?6.79> d ;2.== ;?.07 8 ;5.;0> d 02.?2 02.=6 8 0?.9=. Jds aµ u uh`rds h`rds ?6.79, ;5.;0 y 0?.9= sda kbsi imubj`s àtr` sµĲĲ,, kdhd jd sda b 229, qu` `s `j r`sujtbed. @s e` supda`r, pdr priakipid, qu` pbr`zkb hµbbss sàki sàkijjd jjd hujtipji hujtipjikbr kbr = 2= qu` trboblbr pbrb àkdatrbr à e`kihbj`s jds `quivbjàt`s `xpdaàkibj`s (jds kubj`s, e` kubjqui`r hba`rb, sda sdjbhàt` bprdxihbeds< dosµ``rv`s` rv`s` qu` jds r`sujtbeds sda kbsi  imubj`s b 229). [ia `hobrmd, vbhds b s`muir be`jbat` pbrb v`r eµddae` ae` t`rhi t`rhiabhds. abhds. S`r S`rdd

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Ý

uab kdsb. Bj `skdm`r ua aµu uh`rd h`rd obs`, ´pdr quµ` `skdm`r `j ?, `j ; d `j 04 ´Sdr quµ` ad `skdm`r `j 23, qu` `s `j aµu uh`rd h`rd kjbv` à au`strd sist`hb aµ u uh`rikd4 h`rikd4

KbpµĲtujd 7 Jds huy mrbae`s y huy p`qu`˚ ads ads Pab au`vb kdavàiàkib e`j ei`z Pab e` jbs rbzda rbzda`s `s qu` dojimbrda b jds kià kiàttµĲflkds b usbr jds `xpdaà `xpdaàt`s t`s fu` jb fr`kuàkib kda qu` jds àkdatrbrda a`k`sbrids pbrb trboblbr kda kifrbs huy mrbae`s y huy p`qu`˚ a abs. bs. Jb hbsb e`j mjdod t`rrµbbqu`d qu`d tià` bjrèèdr e` 6333333333333333333333333333 e` mrbhds. Sdr dtrd jbed, `j µbbtdhd tdhd e` gierµddmàd màd tià` bjrèèdr e` 3.33333333333333333333333266 mrbhds. Kdhd puè` dos`rvbrs`, `s huy fµbbkij kij p`re p`re`rs `rs`` à jds k`r k`rds. ds. Urbt rbtbae baedd e` sihpji sihpjiflkbr flkbr su tbr`b tbr`b,, jds kià kiàttµĲflkds usba uab fdrhb e` `xpr` `xpr`siµ siµ dda a e` jds aµuh`rds uh`rds qu` `s, à pbrt`, jb dreiabrib y, à pbrt`, jb `xpdaàkibj. Y jb pbrt` `xpdaàkibj `xpda` akibj kdhprà kdhpràe` e` bj 23 kdhd obs` (uab iasiaub iasiaubkiµ kiµ dda a qu` s` gizd bj flabj e`j Jbs kbpµĲĲtujd tu jd bat`r bat`ridr) idr). pdtàkibs e` .23 p`rhità `xpr`sbr fµbbkijhàt` kijhàt` jbs kifrbs mrbae`s d jbs p`qu`˚ a abs, bs, à fdrhb tbj qu` r`bjhàt` s` blustba b au`strd sist`hb e` auh`rbkiµ dda, a, kuyb obs` `s `j ei`z. Suè` v`riflkbrs` jd bat`ridr, à jb tbojb simuiàt`, `f`ktubaed jbs hujtipjikbkida`s be`kubebs< Abturbjhàt`, `stb tbojb puè` bhpjibrs`, gbkib brriob d gbkib bobld, tbatd kdhd jd e`s``hds. 2333333 8 236 233333 8 230 23333 8 235 79

93

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

2333 8 23; 233 8 23? 23 8 232 2 8 233 3.2 8 23∟2 3.32 8 23∟? 3.332 8 23∟; 3.3332 8 23∟5 3.33332 8 23∟0 3.333332 8 23∟6 Sbrb e`hdstrbr kµddhd hd jb obs` e`j 23 àkblb eàtrd e` au`strd sist`hb auhµ``rikd, rikd, kdasie`r kdasie`r`hds `hds `j aµu uh`rd h`rd 5;=?.605. G`hds eivieied `stb kifrb à 5 ‚hijjbr`s‘, ‚hijjb r`s‘, ; ‚k ‚kàtàbs‘, àtàbs‘, = ‚‚e`kàbs‘ e`kàbs‘,, ? ‚uaieb ‚uaiebe`s‘, e`s‘, 6 ‚‚eµ eµ``kihds‘, kihds‘ , 0 ‚kàtµ``sisihds‘ y 5 ‚hijµ``sihds‘. sihds‘. Ubhoiµ `à a pdeµĲĲbhds bhds `skrioirjb kdhd 5333 + ;33 + =3 + ? + 3 .6 + 3.30 + 3.335. [ia `hobrmd, si usbhds jbs pdtàkibs e` 23, r`kdrebaed qu` 2333 `s 23; , 233 `s 23? y bs bsµµĲ suk`siv suk`sivbhàt`, bhàt`, p pde`hds de`hds `skrioir 5;=?.605 kdhd< (5 23; ) + (; 23? ) + (= 232 ) + (? 233 ) + (6 23∟2 ) + (0 23∟? ) + (5 23∟; ). @std, `f`ktivbhàt`, pda` à pbp`j jd qu` `j µbbobkd obkd gb `stbed gbkiàed eurbat` hijjbr`s e` b˚ a ads. ds. [i `j obrrdt` e` jbs ‚uaiebe`s‘ `s kjbsiflkbed kdhd ‚k`rd‘ y jds qu` simuà gbkib brriob sda kjbsiflkbeds kdhd 2, ?, ; y bsµ bsµĲ suk`sivbhàt`, y jds simuiàt`s gbkib bobld kdhd -2, -?, -;, r`bjhàt` `stµba siàed

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Ý

44brithµ kjbsiflkbeds pdr jb pdtàkib e`gbk` ei`z à qu`jb`s` obrrdt` r`pr`sàtb ). Bua kubaed `std auakb s` `sku`jb prihbrib, tdeb(jb ``tikb tikb qu` bpràe`hds, usbaed aµ u uh`rds h`rds brµbboimds, oimds, puè` s`r `xpjikbeb `stueibaed `j kdhpdrt kdhpdrtbhià bhiàtd td e` `stbs pdtàkibs. S`rd bgdrb eèikbr` eèikbr`hds hds bqu bquµµĲ bjm bjmµ u ua µa ti`hpd b `stueibr `stds `xpdaàkibj`s y v`r quµ` vàtblbs dfr`kà. Srih`rd kdasie`rbr`hds jbs pdtàkibs pdsitivbs e` 23. Dosµ``rv`s` rv`s` qu`, à `jjbs, `j `xpdaàt` `s imubj bj aµu uh`rd h`rd e` k`rds à `j aµu uh`rd h` rd dre dreiab iabrid. rid. Bsµ BsµĲ, siàed 6 `j auh`rd e` k`rds à 2333333, à fdrhb `xpdaàkibj pdr jd tbatd, 236. Bgdrb oià, b fla e` kdav`rtir ua aµu uh`rd h`rd fdrhbed pdr eµĲmitds qu` ad s`ba 2 y 3, b ua `xpdaàkibj kda obs` e` 23, `s hµbs fµbbkij kij kdav`rtir `s` aµu uh`rd h`rd à

92

µ Fimurbb 7.2< Bobkd Fimur kjbsiflkbed pdr pdtàkibs e` 23 uab `xpr`siµdda a qu` iakjuyb uab pdtàkib imubj e` 23. Sdr `l`hpjd, jb hbsb e`j

Ý

mjdod t`rrµ qu`d puè` `skrioirs` kdhd 2333333333333333333333333333 mrbhds. @jbbqu`d aµu uh`rd h`rd mrbae` `s bgdrb uab6 pdtàkib `xbktb e` 23, pu`std qu` kdasist` à ua 2 s`muied e` ua aµ u uh`rd h`rd e` k`rds, kdhd `s prdpid e` tdebs jbs pdtàkibs e` 23. Kdhd `j aµu uh`rd h`rd e` k`rds à `stb pdtàkib pbrtikujbrhàt` mrbae` `s e` ?=, puè` `xpr`sbrs` `xpdaàkibjhàt` kdhd 23?=. Jb hbsb e` jb Ui`rrb `s 6 v`k`s `st` aµ uh`rd, uh`rd, d 6 23?= mrbhds. @xistà eds vàtblbs dovibs à `stb fdrhb `xpdaàkibj e` aµu uh`rds h`rds mrbae`s. Srih`rb, s` dkupb hàds `spbkid. [`muaeb, `s hµbbss kjbrb pu`std qu` ad tà`hds qu` kdatbr jds k`rds kbeb v`z qu` ads àkdatrbhds kda `jjds. Jds aµ u uh`rds h`rds huy p`qu`˚a ads ds kdhpràeà jbs pdtàkibs a`mbtivbs e` 23. @stbs pdtàkibs a`mbtivbs, kdhd puè` v`rs` pdr jb tbojb bat`ridr, fdrhba aµuh`rds uh`rds dreiabrids qu` kdasistà à uab s`ri` e` k`rds b jb e`r`kgb e`j pua-

Ý

9?

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

td e`ki e`kihbj hbj y t`rh t`rhiaba iaba kda ua 2. @j vbjdr auh auhµµ``rikd rikd e`j `xpda` `xpdaàt` at` a`mbtiv a`mbtivdd `s imubj b uad hµbs bs qu` `j aµ uh`rd uh`rd e` k`rds à jb e`r`kgb e`j puatd e`kihbj. Sdr `l`hpjd, `j aµ u uh`rd h`rd 3.333332, tià` kiakd k`rds b jb e`r`kgb e`j puatd e`kihbj y, `xpdaàkibjhàt`, `s 23∟6 Jb hbsb e`j µbbtdhd tdhd e` gierµddmàd màd puè` `xpr`sbrs` kdhd 2.66 3.333333333333333333333332. ([i s` `f`ktµ u ubb jb hujtipjikbkiµ dda, a, ki`rtbhàt` s` dotàerµb `j aµu uh`rd h`rd ebed drimiabjhàt`). @j aµu uh`rd h`rd p`qu`˚ a add `s uab pdtàkib e` 23, kdasistàt` à k`rds y ua 2. Uià` ?; k`rds b jb e`r`kgb e`j puatd e`kihbj, pdr jd kubj `s 23 ∟?5. Jb hbsb e`j µbtdhd e` gierµddmàd btdhd màd `s 2.66 v`k`s `std y, pdr jd tbatd, puè` r`pr`sàtbrs` ∟?5 kdhd 2.66 23 mrbhds.

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Dtrds aµ uh`rds uh`rds bpbrt` e`j ei`z Pab v` v`zz qu` g`hds kdh` kdhàzbed azbed b kdjdkbr au` au`strd strd sist` sist`hb hb auh auhµµ``rikd rikd à fdrhb `xpdaàkibj kda obs` bj 23, s` gbk` hµbbss fµbbkij kij àtàe`r ua sist`hb obsbed à dtrd aµu uh`rd. h`rd. Sdr `l`hpjd, hàkidaµ` bj priakipid e` `st` jiord jb kdavàiàkib e` usbr `j 2? kdhd obs`, à v`z e`j 23, eòied b qu` `j 2? tià` hbs fbktdr`s qu` `j 23.2 Sbrb usbr `j 2? kdhd obs` a`k`sitbr a`k`sitbrµµĲbhds edk` eµĲmitds s`pbrbeds (iakjuyàed `j k`rd). @stds s`riba 3, 2, ?, ;, 5, 0, 6, =, 7, 9, C, y #. @stdy usbaed jds sµĲhodjds C y # pbrb r`pr`sà r`pr`sàtbr tbr jds aµu uh`rds h`rds qu`, à `j sist`hb obsbed à `j 23, sda r`pr`sàtbeds kdhd 23 y 22. Bgdrb oià, ua aµ uh`rd uh`rd kdhd `j ???, à `j sist`hb obsbed à `j 23, `s ? 2 (? 23 )+(? 23 )+(? 233 ). @j hishd aµuh`rd, uh`rd, à `j sist`hb obsbed, à `j 2?, pd pder erµµĲb kdav`rtirs` bj sist`hb obsbed à `j ei`z, `skrioiµ`àedjd, aedjd , baµ b aµbbjdmbhàt`, jdmbhàt`, ? 2 3 kdhd (? 2? )+(? 2? )+(? 2? ), qu` ads ads eb kdhd r`s r`sujt ujtbed bed ?77+ ?5+ ?5 + ?, d s`b ;25. @a dtrbs pbjborbs, ??? à `j sist`hb obsbed à `j 2? (eude`kihbj), `s imubj b ;25 à `j sist`hb obsbed à `j 23 (e`kihbj). SderrµĲb tà Sde tà`rs `rs`` tb tbhoiµ hoiµ`à a ua aµu uh`rd h`rd kdhd ;#5 à `j sist`hb eude`kihbj. ? @st`` s`r @st s `rµµĲb `qui `quivbjàt` vbjàt` b ((;; 2? ) + (# 2?2 ) + (5 2?3 ). Kdhd g` fllbed `j

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@j edk` tià` dtrbs và vàtblbs, tblbs, be`hµ bbss e`j aµ u uh`rd h`rd e` sus fbktdr`s. Jds batimuds hèiba `j ti`hpd pdr jb Juab. Kbeb ?3 d ;3 eµĲbs gbo gboµµĲb uab Juab au`vb au`vb y kdas`kuàt`hàt`, kdas`kuàt`hàt`, ua ‚h`s au`vd‘. [ukèiµd qu` `xist `xistµµĲba lustbhà lustbhàt` t` 2? e` tbj`s h`s`s (à r`bjiebe r`bjiebe 2? h`s`s y 22 eµĲbs) Ĳbs) à `j kikjd e` jbs `stbkida` `stbkida`s, s, `std `s, à `j jbpsd e` prihbv`r prihbv`rbb b prihbv`r prihbv`rb. b. @std eid bj 2? uab `sp`ki` e` vbjdr hµbbmikd. mikd. [` gizd ihpdrtbat` pbrb `j gdhor` prihitivd. Eòied b `jjd, `st` ie`µd 2? simads e`j zdeibkd, p`rhba`kiàed `j [dj ua h`s à kbeb simad su bpbràt` bpbràt` r`kdrr r`kdrried ied bjrèèdr e` jb Ui`rrb. Ui`rrb. Jds edk` simads e`j zdeibkd, zdeibkd, prdoboj`h` prdoboj`hàt` at`,, s` r`﬏`lbrba à kdsbs hµbbss t`rràbj`s, kdhd jbs edk` trious e` Isrb`j y jds edk` bpµdstdj`s. dstdj`s.

9; sµĲhodjd # kdhd imubj bj 22 à `j sist` sist`hb hb e`ki e`kihbj, hbj, jb `xpr `xpr`siµ `siµ dda a bat`ridr eb kdhd r`sujtbed 5;? + 2;? + 5, d s`b 067, à `s` µu ujtihd jtihd sist`hb. Sderµ Sd erµĲĲbb usb usbrs` rs` tb tbhoiµ hoiµ`à a ua au uh`rd µh`rd hàdr qu` `j 23 pbrb obsbr ua sist` sist`hb< hb< tbj v`z `j =. @atdak`s s` a`k`sitbr a`k`sitbrµµĲb sdjbhàt` = sµĲhodjds< 3, 2, ?, ;, 5, 0 y 6. @j aµ u uh`rd h`rd 5;0 à `j sist`hb obsbed à `j = puè` s`r kbhoibed b (5 =? ) + (; =2 ) + (0 =3 ), qu` eba kdhd r`sujtbed 296 + ?2 + 0, d s`b ??? à `j sist`hb e`kihbj. E` `st` hd hded, ed, `j sµĲhodjd pbrb ua aµu uh`rd h`rd à kubjq kubjqui`r ui`r sist`hb puè` s`r kdav`rtied bj sµĲhodjd `quivbjàt` p pbrb brb `s` aµuh`rd uh`rd à kubjqui`r dtrd sist`hb. Bua jds e`kihbj`s puèà s`r hba`lbeds bs bsµµĲ. @a `j sist`hb eude`kihbj puè` r`pr`sàtbrs` jb `xpr`siµdda a 3.20 kdhd (2 2?∟2 ) + ( 0 2?∟? ), d à `j sist`hb 2 0 2= e`kihbj kdhd + , qu` ads eba kdhd r`sujtbed . 2? 255 255 @a `j sist`hb obsbed à `j si`t`, jb hishb `xpr`siµda da s`r s` rµĲb (2 =∟? )+(0

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=∟?) d, à `j sist`hb e`kihbj, 2 + 0 , d s`b 2? . = 59 59 [ia `hobrmd, ´kµddhd hd pde`hds e`kir kuµbatds batds sµĲhodjd Ĳhodjdss s`pbrbe s`pbrbeds ds s` a`k`sitba pbrb kubjqui`r sist`hb à pbrtikujbr4 V`bhds, @j prih`r aµu uh`rd h`rd qu` (à kubjqui`r sist`hb) r`qui`r` eds sµĲhodjds `s `j 223. 3. Udeds jds aµu uh`rds h`rds hàdr`s e` 23 r`qui`rà ua sµĲhodjd eistiatd y s`pbrbed. Udeds jds aµu uh`rds h`rds hbydr`s e` 23 puèà fdrhbrs` e` kdhoiabkida`s e` jds sµĲhodjds e` jds a aµ u uh`rds µh`rds hàdr`s e` 23. @std `s v`rebe`rbhàt` ki`rtd à `j sist`hb e`kihbj, kda `j kubj yb `stbhds fbhijibrizbeds fbhijibrizbeds,, y eò` `sp`rbrs` qu` dtrds sist`h sist`hbs bs s` kdhpdrtà à fdrhb s`h`lbat` (y (y,, bj `xbhiabrjds, à r`bjiebe s` gb àkdatrbed qu` bsµ bsµĲ jd gbkà). @atdak`s, pdr `l`hpjd, ´kuµbbjj `s `j simaiflkbed e` jb `xpr`siµdda a 23 à `j 2

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sist`hb e`kihbj. eude`kihbj4 @s (2 2? à ) +`j(3 2? ),obsbed d 2? + 2?, à da sist`hb Baµbbjdmbhàt`, jdmbhàt`, sist`hb à3,`j d=,s`b jb `xpr`siµ d`ja 23 `s (2 =2) + (3 =3), d = + 3, d s`b =. [i s` v`riflkb `j bsuatd à dtrds sist`hbs, rµbbpiebhàt` piebhàt` s` kdavàk`rµb uad e` qu`, à ua sist`hb obsbed à ua aµ u uh`rd h`rd pbrtik pbrtikujbr, ujbr, jb `xpr` `xpr`siµ siµ dda a 23 `s imubj b `s` aµu uh`rd. h`rd. (@a `j sist`hb e`kihbj, abturbjhàt` 23 `s imubj b 23.) @a `j sist`hb eude`kihbj, àtdak`s, a`k`sitbhds s`pbrbr eµĲmitds pbrb kbeb aµ u uh`rd h`rd hàdr e` 2?. @std simaifl simaiflkb kb 2? eµĲmitds s`pbr s`pbrbeds, beds, iakju iakjuyà yàed ed `j k`rd. @a `j sist`hb obsbed à `j si`t`, a`k`sitbhds s`pbrbr e eµµĲmitds pbrb kbeb aµu uh`rd h`rd hàdr e` =, d s`b = e eµµĲmitds s`pbrbeds, iakjuyàed `j k`rd. @std dp dp``rbrµĲĲbb à fdrh fdrhbb baµbbjdmb jdmb pbrb kubjqui`r dtrd sist`hb. Pa sist`hb obsbed à `j aµu uh`rd h`rd ?7 r`qu`rirµĲĲbb ?7 eµĲmitds s`pbrbe s`pbrbeds, ds, iakjuyàed `j k`rd.

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95

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

Sbrb ebr uab ie`b hbs kjbrb e` jd qu` `std simaiflkb b kdatiaubkiµdda a bpbr`kà jds j ds sµĲĲhodjds hodjd s pbr pbrbb jds prih`rds trìatb a aµ u uh`rds µ h`rds e` jds sist`hbs obsbeds à `j edk`, à `j si`t` y à au`strd yb fbhijibr ei`z.

[ist`h [is t`hbb obs obsbed bed `à a `j si si`t` `t` [is [ist`h t`hbb obs obsbed bed `à a `j ei ei`z `z [is [ist` t`hb hb ob obsbe sbedd à `j edk` 2 2 2 ? ? ? ; ; ; 5 5 5 0 0 0 6 23 22 2? 2;

6 = 7 9 23

6 = 7 9 C

[ist`h [is t`hbb obs obsbed bed `à a `j si si`t` `t` [is [ist`h t`hbb obs obsbed bed `à a `j ei ei`z `z [is [ist` t`hb hb ob obsbe sbedd à `j edk` 25 22 # 20 2? 23 ?236 ?2 ?? ?; ?5 ?0 ?6

225; 20 26 2= 27 29 ?3

22?2 2; 25 20 26 2= 27

90 [ist`h [is t`hbb obs obsbed bed `à a `j si si`t `t`` [is [ist`h t`hbb obs obsbed bed `à a `j ei ei`z `z [is [ist`h t`hbb obs obsbed bed `à a `j edk edk`` ;3 ?2 29 ;2 ;? ;; ;5 ;0 ;6 53 52 5?

?? ?; ?5 ?0 ?6 ?= ?7 ?9 ;3

2C 2# ?3 ?2 ?? ?; ?5 ?0 ?6

Kdatbaed à uads y k`rds Sbrb kbeb sist`hb obsbed à ua aµ u uh`rd h`rd pbrtikujbr s` puè` kdaf`kkidabr tbojbs pbrb jb suhb, jb hujtipjikbkiµdda a y dtrbs dp`rbkida`s. @a `j sist`hb eude`kihbj 0 + 7 8 22 y ; 5 8 23. @a `j sist`hb obsbed à `j si`t` ; + 6 8 2?, hiàtrbs qu` 0 ; 8 ?2. @std puè` pbr`k`rads rbrd pdrqu` ad `stbhds bkdstuhorbeds b tbj`s sist`hbs. [ia `hobrmd, si tuviµ``rbhds rbhds qu` `stbr eàtrd e` u uad ad u dt dtrd rd ssist`hb, ist`hb, àkdatrbrµĲĲbhds bhds qu` s`rvir s`rvirµµĲba b au`s au`strbs trbs flabjiebe`s à fdrhb sbtisfbktdrib. G`hds bedptbed `j sist`hb obsbed à `j ei`z pdrqu` tà`hds tà`hds ei`z eèds, ad pdrqu` s`b hµbbss jµddmikd mikd qu` kubjqui`r dtrd.

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[ia `hobrmd, pbrb ua fla pbrtikujbr, ua sist`hb ua pbrtikujbr puè` r`µ qu` dtrd, bua kubaed ad s`b hµb sujtbr hbs utij bss jµddmikd. mikd. @std `s ki`rtd, pdr `l`hpjd, à `j sist`hb oiabrid, `j kubj `s ua sist`hb kdhpu`std e` eds mubrishds. @a ua sist`hb oiabrid, jb `xpr`siµdda a 23 `s imubj bj ? à `j sist`hb e`kihbj. @a kdas`kuàkib, sµddjd jd `xistà eds eµĲmitds à tbj sist`hb, sist`hb , 3 y 22.. J Jds ds sµĲhodjd Ĳhodjdss pbrb jds prih`rds trìatb aµuh`rds uh`rds e` `s` sist`hb bpbr`kà b kdatiaubkiµdda, a, bs bsµµĲ kdhd jds `quiv `quivbjàt`s bjàt`s à `j sist`hb e`kihbj.

96

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

[ist`h [is t`hbb oi oiabr abrid id [is [ist`h t`hbb e` e`kih kihbj bj 2 2 23 22 233 232 223 222 2333 2332 2323

? ; 5 0 6 = 7 9 23

[ist`h [is t`hbb oi oiabr abrid id [is [ist`h t`hbb e` e`kih kihbj bj 2322 22 2233 2232 2223 2222 23333 23332 23323 23322 23233

2? 2; 25 20 26 2= 27 29 ?3

[ist`h [is t`hbb oi oiabr abrid id [is [ist`h t`hbb e` e`kih kihbj bj 23232 ?2 23223 23222 22333 22332 22323 22322 22233 22232 22223

?? ?; ?5 ?0 ?6 ?= ?7 ?9 ;3

@s obstbat` fµbbkij kij kdav`rtir uab `xpr`siµdda a oiabrib b jb `xpr`siµdda a `quivbjàt` à `j sist`hb e`kihbj. Sdr `l`hpjd, kdasie`r`hds jb `xpr`siµdda a oiabrib 22332.

9= @stb `s `quivbjàt` b< (2 ?5 ) + (2 ?; ) + (3 ?? ) + (3 ?2) + (2 ?3 ) d, 26 + 7 + 3 + 3 + 2, d ?0, qu` `s `f`ktivbhàt` `j `quivbjàt` hdstrbed à jb

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tbojb. @st` prdkèihiàtd puè` sihpjiflkbrs` bj adtbr qu` jbs pdtàkibs e` ? sda hujtipjikbebs yb s`b p pdr dr 3 d p pdr dr 2. Ad `xistà dtrds e eµµĲmitds à `j sist`hb oiabrid qu` puèba s`r usbeds. [i jb pdtàkib `s hujtipjikbeb pdr 3 `s k`rd y puè` imadrbrs` à jb suhb. [i `s hujtipjikbeb pdr 2, `s jb pdtàkib hishb. Sdr jd tbatd, pde`hds kdatbr jds jumbr`s, à ua aµu uh`rd h`rd kdhd 22332, e` e` 2 2 3 3 2 . r`kgb r`k gb b izqui izqui`reb, `reb, `hp`zbae `hp`zbaedd kda k`rd, à jb fdrhb simuià simuiàt`< t`< 5 ; ? 2 3 Kbeb aµ u uh`rd h`rd p`qu`˚ a add r`pr`sàtb jb pdtàkib pbrtikujbr e` ?, simaiflkbeb pdr `sb pdsikiµda da à `j auh`rd. [djbhàt` a`k`sitba kdasie`rbrs` bqu`jjbs pdtàkibs à jbs kubj`s bpbr`k` ua 2. Psbaed `st` sist`hb, 22332 `s ? 5 +?; +?3 , dµ 26 + 7 + 22,, µd ?0. Pa aµ u uh`rd h`rd hµbbss mrbae`, kdhd 2223323233332332 puè` kbhoibrs` à jb hishb fdrhb. Kdhd `xist `xistà à uads  à jbs pdsikida`s 3, ;,7 23, 2;, 25 y 20, jds 20 aµu uh`rds h`rds r`pr`sàtba ? + ?25 + ?2; + ?23 + ?7 + ?; + ?3 , d s`b 07, 6;;. Kbhoibr à jb dtrb fdrhb `s hbs tèidsd, p p`rd `rd huy eif eifµµĲkij. [up [updambhds dambhds qu` g`hds kdasie`rbed `j `xpr`sbed 206? à sist`hb e`kihbj. Suè` kbhoibrs` bj sist`hb oiabrid kdhd simu`< Jb pdtàkib hµbbss mrbae` e` ? qu` `s hµbs bs p`qu`˚ a abb qu` 206?, `s ?23 (qu` `s imubj b 23?5). [i s` eèuk` `stb kifrb e` 206? quèba 0;7. Jb pdtàkib hµbbss mrbae` e` ? qu` `s hµbbss p`qu`˚ a abb qu` `sb kbatiebe, `s ?9 (qu` `s imubj b 02? quèbaed ?6). @atdak`s tà`hds ? 5 (qu` `s imubj b 26 y quèba 23), ?; (qu` `s imubj b 7 y quèba ?) y ? 2 (qu` `s imubj b ? y ad quèb abeb). @a pdtàkibs e` ?, pdr jd tbatd 206? 8 ?23 + ?9 + ?5 + ?; + ?2 . sdjbhàt` `s a`k`sbrid auh`rbr e` e`r`kgb eb, Bgdrb `hp`zbaed e`j k`rd. Kdjdkbhds jds 2 jds à `spbkids jbs pdsikida`s 2, ;, b5,izqui`r9 y 23 (kdrr`spdaeiàt`s b jbs pdtàkibs e` ? qu` fdrhba `j aµu uh`rd) h`rd) y jds 3 à jbs e`hµbbs. s. Gbkiàed `std, tà`hds 22333322323. @st` `s `j `quivbjàt` oiabrid e` 206? à `j sist`hb e`kihbj. @a `j sist`hb oiabrid tà`hds uab tbojb huy sàkijjb e` suhbs y hujtipjikbkida`s< @stb `s jb jistb kdhpj`tb. 3+383 3

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3+282 3

Ý283

2 + 2 8 23 2

Ý282

97

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

BsµµĲ, à `j sist`hb oiabrid Bs oiabrid `std, kdhd s` bkbob e` iaeikbr, ad kbhoibrµb `j vbjdr tdtbj e`j aµuh`rd. uh`rd. Hujtipjikbaed ;.? pdr 23, abturbjhàt` dotà`hds ;?. Eivieiàed àtr` 23 (à r`bjiebe 232, gbojbaed `xpdaàkibjhàt`) dotà`hds 23? pu`std qu` bj eivieir r`stbhds `xpdaàt`s. @j aµ u uh`rd, h`rd, pdr jd tbatd, s` kdavi`rt` à ;? 23? sia aiamµ u ua a kbhoid à vbjdr. Sde`hds gbk`rjd à jb dtrb fdrhb y eivieir ;.? àtr` 23 (dotàiàed 3.;?) y hujtipjikbaed be`hµbs bs pdr 23 (dotàiàed 235 ). @j aµ u uh`rd h`r d s` kdav`rt kdav`rtir irµµĲb 5 à 3.;? 23 , p`rhba`kiàed sia kbhoid à vbjdr. S`rd kdhd ;? 23? , ;.? 23; y 3.;? 235 sda `j hishd aµ u uh`rd, h`rd, ´`xist`

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bjmuab dda a pbrbpdr qu`hdtivds eòbhds uab fdrhb u dtrb4 Hbt`hµ bbtikbhàt`, tikbhàt`, ad. [ia rbzµ `hobrmd, e` `skdm`r kdavàiàkib, s` bkdstuhorb `skdm`r jb fdrhb e`j aµ uh`rd uh`rd à jb kubj jb pbrt` ad `xpdaàkibj s` àkuàtrb àtr` 2 y 23. @a ;? 23?, jb pbrt` ad `xpdaàkibj (;?) `s hbydr e` 23 hiàtrbs qu` à 3.;? 235 jb pbrt` ad `xpdaàkibj (3.;?) `s hàdr e` 2. Sdr jd tbatd, ; .? 23; , kda ;.? àtr` `j 2 y `j 23, `s jb fdrhb e` aµu uh`rds h`rds dreiabribhàt` usbeb. @xk`ptd pdr `j e`tbjj` e`j `xpdaàt` a`mbtivd, tded `std dp`rb jd hishd pbrb jds aµu uh`rds h`rds hàdr`s e` 2. Udh`hds, pdr `l`hpjd, 3.3305. Suè` `xpr`sbrs` kdhd 05 3.3332, d kdhd 0.5 3.332 d kdhd 3.05 3.32. Kbeb uab e` `stbs hujtipjikbkida`s ebrµb 3.3305. @xpd @xpdaàkibj aàkibjhàt`, hàt`, µ``stbs stbs puèà ∟5 ∟; ∟? `skrioirs` kdhd< 05 23 , 0.5 23 ,d 3.05 23 . Au`vbhàt` `stds aµu uh`rds h`rds sda `quivbjàt`s. @hp`zbaed kda 0.5 23∟;

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233

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

puè` hujtipjikbrs` 0.5 pdr 23 y eivieir 23 ∟; àtr` 23. Sdr supu`std, 0.5 23 8 a kdhpràe` jb 05> y 23∟; 232 `s 23∟5 pu`std qu`, prdpibhàt`, jb eivisiµdda

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∟ ∟

ý

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r`stb e`∟5`xpdaàt`s y ; 2 `s 5 pdr jd qu` `j aµu uh`rd h`rd `s kbhoibed b 05 23 sia aiamµ u ua a kbhoid à vbjdr. @a jb hishb fdrhb, eivieiàed 0.5 àtr` 23 y hujtipjikbaed 23 ∟; pdr 23 dotàer`hds 3.05 23∟? sia kbhoid à vbjdr. Y, au`vbhàt`, jb fdrhb pr`f`rieb `s 0.5 23 ∟;, pdrqu` `s à jb kubj jb pbrt` ad `xpdaàkibj e`j aµu uh`rd h`rd `stµb àtr` 2 y 23.

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Hµ bs bs lu`mds hbjbobr`s kda jds `xpdaàkibj`s `xpda àkibj`s Udebs jbs dp`rbkida`s b jbs kubj`s `stµbba a sul`tds jds aµu uh`rds h`rds dreiabrids puèà s`r bpjikbebs tbhoiµ `à a b jds aµu uh`rds h`rds `xpdaàkibj`s. @a jb suhb y jb r`stb sdjbhàt` `stµbba a kdhpràeiebs jbs pbrt`s ad `xpdaàkibj`s e`j aµ u uh`rd. h`rd. Sdr `l`hpjd, jb suhb e` ?.; 235 y 5.? 235 `s 6.0 235 . (Suè` `j j`ktdr kdhprdobr `std kdavirtiàed jds `xpdaàkibj`s qu` `stµba ba siàed suhbeds b aµ u uh`rds h`rds dreiabrids ‛?;, 333 y 5?, 333‛ suhµbbaedjds aedjds pbrb dotà`r 60, 333 y kbhoiµbaedjds baedjds au`vbhàt` b ua `xpdaàkibj. Suè` gbk`r jd hishd à tdeds jds `l`hpjds qu` edy. J` ebrµb prµbbktikb ktikb y j` gbrµb ad kr``r a`k`sbribhàt` à jd qu` s` j` eik`, bua kubaed `stµ` ihpr`sd). Jb suhb e` 7.= 235 y ;.9 235 r`sujtb s`r 2?.6 235 . Eikgd r`sujtbed puè` e`lbrs` à `stb fdrhb. [ia `hobrmd, jb pbrt` ad `xpdaàkibj `s hbydr e` 23. Sdr hèid e` jb bprdpibeb eivisiµdda-hujtipjikbkiµ a-hujtipjikbkiµ dda a eiskutieb à jb s`kkiµda da 0 bat`ridr, s` puè` kdav`rtir `j aµu uh`rd h`rd b 2.?6 23 , qu` `s uab r`spu`stb kdrr`ktb y `stµb à jb fdrhb pr`f`rioj`. [ia `hobrmd, ´quµ` sukè` si jbs pb pbrt`s rt`s `xpdaàkibj`s e` eds aµu uh`rds h`rds qu` 5 `stµba ba siàed suhbeds ad sda imubj`s4 ´Kuµbbjj `s jb suhb e` 2.7= 23 y 9 23? 4 B fla e` dotà`r jb suhb `s a`k`sbrid imubjbr jbs pbrt`s `xpdaàkibj`s e` jds aµu uh`rds. h`rds. Sdr `l`hpjd, 2.7= 235 puè` kdav kdav`rtirs`, `rtirs`, pdr `j hµ``tded tded usubj, b ? 27= 23 . Pab v`z g`kgd `std, jb suhb s` kdavi`rt` à (27= 23? )+(9 23? ) y `j r`sujtbed, `vieàt`hàt`, `s 296 23? . Jb dtrb pdsioijiebe `s qu` 9 23? puèb kdav`rtirs` b 3.39 235 (yb bgdrb s` `sp`rb qu` ad a`k`sit` `j j`ktdr jds e`tbjj`s) y jb suhb r`sujtb (2.7= 235 )+ (3.39 235 ) 8 2.96 235 . BsµµĲ s` tiàà eds r`spu`stbs pdsioj`s< 296 23? y 2.96 235 . S`rd siàed Bs µ`stbs `stbs ieµàtikbs àtikbs à vbjdr, `s jb s`muaeb fdrhb jb pr`f`rioj pr`f`rioj`. `.

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232 Jbs dp`rbkida`s pbrb jb r`stb sda imubj`s qu` pbrb jb suhb, `xk`ptd qu` trboblba bj kda kdatrbri trbrid. d. @a r`bjiebe, jds `xpdaàkibj`s rbrbhàt` sda usbeds pbrb jbs suhbs y jbs r`stbs. Ad `s hµbbss fµbbkij kij hba`lbr µ``stds stds à `stbs dp dp`rbkida`s `rbkida`s qu` jds aµu uh`rds h`rds dreiabrids. @a r`bjiebe, sda hµbs bs eifµĲĲkij`s kij `s pu`s pu`std td qu` gby q qu` u` iirr kdavir k davirtiµ tiµ`àedjds aed jds e` uab pdtàkib b dtrb. Jb hujtipjikbkiµdda a y jb eivisiµdda a sda dtrd bsuatd. [updambhds qu` s` e`s`b hujtipjikbr 63333 pdr 3 .337. Sbr`k` obstbat` fµbbkij kij y puè` gbk`rs` kdhd simu`<

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Jb uaikb u µ aikb eiflkujtbe kdasist` à qu` gby qu` jj`vbr jb kuàtb e` jds k`rds y bs`murbrs` uad e` qu` tià` `j puatd e`kihbj à `j jumbr kdrr`ktd. (@s fµbbkij kij `quivdkbrs` b `st` r`sp`ktd.) Vbhds kdav`rtir bgdrb bhods aµ u uh`rds h`rds b `xpdaàkibj`s. Srih`rd, 63333 `s 6 235 , hiàtrbs qu` 3.337 `s 7 23∟; . Hujtipjikbaed sda< 6 235 7 23∟; . Kdhd ad ihpdrtb `j dreà à qu` s`ba hujtipji hujtipjikbeds kbeds jds aµu uh`rds, h`rds, `std puè` ∟; 5 `skrioirs` `skr ioirs` kdhd 6 7 23 23 . Bgdrb oià, 6 7 `vieàt`hàt` sda 57, 5 ∟; hiàtrbs qu` 23 23 r`sujtb 232 . [` suhba jds `xpdaàt`s, y 5+( ;) 8 2. @j r`sujtbed `s 57 232 , d `skrioiµ`àedjd aedjd à jb fdrhb pr`f`rioj pr`f`rioj`, `, 5 .7 23? . @a kubjqui`r kbsd, `j `xpdaàt` tià` `j vbjdr e` 573 à aµu uh`rds h`rds dreiabrids qu`, pdr supu`std, `s kdrr`ktd. S`rd jb dp`rbkiµdda a `xpdaàkibj s` gb àkbrmbed e` jds k`rds y e` jb pdsikiµda da

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e`j puatd e`kihbj.s`Be`hµ s, bbs jbs tbr`b e`kdafdrh` jj`vbr jb kuàtb e` jds e` jds puatds e`kihbj`s gbk`bbs, hµ tèidsb jbs kifrbs sdak`rds hµbbss y mrbae`s d hµbbss p`qu`˚ a abs, bs, kdsb qu` jds `xpdaàkibj`s puèà hba`lbr kda sia imubj fbkijiebe à kifrbs e` kubjqui`r hbmaitue. [updambhds, pdr `l`hpjd, qu` s` pr`muatbrb bj j`ktdr jd simuiàt`. [i jb Ui`rrb `stuvi`rb kdhpu`stb `xkjusivbhàt` `xkjusivbhàt` e` µbbtdhds tdhds e` gierµddmàd, màd, ´kuµbbatds atds e` µ`std `s tdss kd kdat` atàe aerrµĲb Ĳb44 Jb hbsb e`j mjdod t`rrµbbqu`d, qu`d, kdhd yb g`hds vistd, tià` 6333333333333333333333333333 mrbhds y jb hbsb e` µbbtdhds tdhds e` gierµddmàd màd 3.33333333333333333333333266 mrbhds. Sbrb dotà`r jb r`spu`stb b jb pr`muatb bat`ridr gby qu` eivieir jb hbsb e` jb Ui`rrb àtr` jb hbsb e` µbbtdhds tdhds e` gierµddmàd> màd> `s e`kir, 6333333333333333333333333333 3.3333333 33333333333333 33333333333333 333333333266. 33266. Suè` gbk`r gbk`rs` s` si s` e`s`b e`s`b..

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23?

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

[ia `hobrmd, si s` utijizba `xpdaàkibj`s, `j prdoj`hb s` kdavi`rt` sihpj`hàt` à (6 23?=) (2.66 23∟?5). Kdhd à jb hujt hujtipjik ipjikbkiµ bkiµ dda, a, puè` ustè

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kbhoibr `std b uab eivisiµda da e` uab pbrt` ad `xpdaàkibj àtr` jb dtrb y e` uab pbrt` `xpdaàkibj àtr` jb dtrb. @j kdkiàt` e` 6 2.66 `s ;.6 (ad `xbktbhàt`, p`rd obstbat` bprdxihbed)> hiàtrbs qu` 23 ?= 23∟?5 8 2302. @j aµ u uh`rd h`rd e` µbtdhds e` gierµddmàd btdhds màd à jjbb U Ui`rrb i`rrb s`r s`rµµĲb, pdr jd tbatd, (si µ``stb stb `stuvi`rb kdh02 pu`stb tdtbjhàt` e` gierµddmàd) màd) imubj b ;.6 23 , qu` à aµ u uh`rds h`rds dreiabrids jj`mbrµµĲb b ;633333333333333333333333333333333333333333333333333, jj`mbr ;6333333333333333333333333333333333333333 33333333333, kifrb qu` gborµĲb Ĳb ustè dotàied si guoi`rb jj`vbed b kbod jb eivisiµdda a hàkidabeb à `j pµbbrrbfd rrbfd bat`ridr. Jbs pdtàk pdtàkibs ibs y jbs rb rbµµĲk`s e` jds aµu uh`rds h`rds `xpdaàkibj`s ad pr`sàtba ? 5 ? pbrtikujbr pbrtiku jbr eiflkujtb eiflkujtbe. e. Bs BsµµĲ (9 23 ) `s imubj b 9? (235 ) , qu` r`sujtb s`r 72 237 , d pr`f`rioj`hàt` 7.2 239 . Sdr dtrd jbed, jb 9 235 `s imubj b 9 235 , d ; 23? .

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Jds `xpdaàt`s s` gbkà kbrmd tdtbjhàt` e`j bsuatd Udebv debvµµĲb gby ju jumbr mbr pb pbrb rb ihpb ihpbkiàtbrs` kiàtbrs` kd kda a `j us usdd e` jjds ds `xp `xpdaàkib daàkibj`s j`s à jjbb hujtipjikbkiµ dda a y jb eivisiµdda. a. Kdhd g`hds vistd, fuakidaba p`rf`ktbhàt` kda jds aµ uh`rds uh`rds kda hukgds k`rds, p`rd supdambhds qu` e`s`bhds hujtipjikbr 67;= W 27??. [i kdav`rtihds `stbs kifrbs b `xpdaàkibj`s, s` trbasfdrhba à 6.7;= 23; y 2.7?? 23; . Jb hujtipjikbkiµdda a e` jbs pbrt`s `xpdaàkibj`s `s r`jbtivbhàt` r`jbtiv bhàt` fµbbkij, kij, p p`rd, `rd, ´quµ` e`kir e` jbs ad `xp `xpdaàkibj`s, daàkibj`s, kdhd sda 6 .7;=

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y 2 7??4 @sdda abebtdhbr hµbbss qu` buhàtbed kda jb kdhpjikbkiµ aau`vbhàt` e` ua puatdau`strd e`kihbjprdoj`hb, qu` gby qu` à kuàtb. Jd qu` qu`r`hds, à dtrbs pbjborbs, `s ua aµu uh`rd h`rd obsbed à `j ei`z y à `j kubj jb fµbbkijhàt kijhàt`` hba`l hba`lboj` boj` pbrt` `xpdaà `xpdaàkibj kibj trbobl` tba tbatd td kdhd s`b pdsioj` y jb ihpdrtuab pbrt` ad `xpdaàkibj jd hàds pdsioj`. Jd ie`bj s`rib kdatbr kda ua aµuh`rd uh`rd obsbed à `j ei`z qu` fu`rb tdtbjhàt` `xpdaàkibj y qu` `jihiabrb kdhpj`tbhàt` jb pbrt` ad `xpdaàkibj. @std ads kdaeuk` bj bsuatd e` jds `xpdaàt`s e`kihbj`s, e` jds qu` s` gbojµd bj flabj e`j kbp kbpµµĲtujd =. V`bhds kda hµbs bs e`tbjj` jds `xpdaàkibj`s obsbeds à ei`z. Sbrb `hp`zbr, 3 2 23 8 2 y 23 8 23. ´Ruµ` gby à kubatd b jds `xpdaàt`s àtr`  3 y 24 Sdr `l`hpjd, 233.0 `s

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imubj b 23 , d b 23, siàed bprdxihbebhàt` ;.26??=7. @a imubj fdrhb (p`rd hµbbss kdhpj kdhpjikbebh ikbebhà àt`), t`), puèà dotà`r dotà`rs` s` jds vbjdr` bjdr`ss e` 23 b dtrbs pdtàkibs ?

e`kihbj`s. @xistà tbojbs e`tbjjbebs e` tbj`s vbjdr`s à eiv`rsds jiords e` kdasujtb. kdasujt b. Ebrµ` b kdatiaubkiµdda a uab p`qu`˚ a abb tbojb e` vbjdr`s e` jbs pdtàkibs e`kihbj`s e` 23. S`qu`˚ a abb tbojb batijdmbrithikb `xpdaà `xpd aàkib kibjj jdmb jdmbrit rithd hd ba batij tijdmbr dmbrith ithdd 3.3 23 3.3 2.333 3.2 23 3.2 2.?09 3.? 23 3.? 2.070 3.; 23 3.; 2.990 3.5 23 3.5 ?.02? 3.0 23 3.0 ;.26? 3.6 23 3.6 ;.972 233.= 3.= 0.32? 3.7 23 3.7 6.;23 3.9 23 3.9 =.95; 232.3 2.3 23.333 Kdhd s` pr`supda` jb obs` e` 23, mà`rbjhàt` bpbr`k` sdjbhàt` `j `xpdaàt` à eikgbs tbojbs. @j `xpdaàt` qu` eò`hds `j`vbr `s jjbhbed ‚jdmbrithd‘. @j vbjdr e`j `xpdaàkibj à kifrbs dreiabribs `s `j ‚batijdmbrithd‘. Kdhd `l`hpjd, à jb `xpr`siµdda a 23? 8 233, `j ? `s `j jdmbrithd e` 233 y `j 233 `j batijdmbrithd e` ?. Pab tbojb kdhd jb bat`ridr qu`, prdpdrkidab jds batijdmbrithds pbrb uab s`ri` r`mujbr e` jdmbrithds, `s jjbhbeb ‚tbojb batijdmbrithikb.‘ Sdr supu`st supu`std, d, jdsq bat batijdmbr ijdmbrithds ithds flmurba à jb tbojb sda ia`x ia`xbktds, bktds, ad pueiµ `àeds` aeds` `s `sp`rbr p`rbr qu` u` s`ba `xb `xbktds ktds qu` pu`s pu`std td qu`, `xk`p `xk`ptd td jds vbjdr`s imubj`s e` 3.3 2.3 jbs pdtàkibs e`j ei`z, kdhd 23 y 23 tdebs jbs pdtàkibs sda irrbkidabj`s (`s e`kir, ad puèà `xpr`sbrs` kdhd frbkkida`s `xbktbs). [ia `hobrmd, jds batijdmbrithds puèà jj`vbrs` b tbatds puatds e`kihbj`s kdhd s`b u utij µ tij y a`k`sbrid. [i tdhbhds `std bj kdatrbrid, pde`hds `xpr`sbr kubjqui`r aµu uh`rd h`rd àtr` 2 y 23 kdhd bjmuab pdtàkib e`kihbj e` ei`z. @a dtrbs pbjborbs, `s pdsioj` e`sbrrdjjbr `j jdmbrithd `quivbjàt` e` kbeb aµu uh`rd h`rd pdr hèid e` kµbbjkujds jkujds qu` ad a`k`sitbhds hàkidabr à `st` jiord. BquµµĲ, pdr `l`h Bqu `l`hpjd, pjd, s` pr`s` pr`sàtb atb uab p`qu`˚ a abb tbojb e` jdmbrithds pbrb uab s`ri` r`mujbr e` aµ u uh`rds h`rds dreiab dreiabrids. rids. Pabs tbojbs huk hukgd gd hµbs bs e`tbjjbebs,

235

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

qu` eba jds vbjdr`s e` hukgds aµ uh`rds uh`rds hµbbs, s, pderµbba a àkdatrbrs` à eiv`rsds jiords e` kdasujtb. Kdhd r`sujtbed e` jd bat`ridr, yb abei` kbjkujb jds jdmbrithds. @j trbobld yb gb sied g`kgd. Bgdrb sdjbhàt` `s a`k`sbrid ouskbr à jbs tbojbs. Sdr `l`hpjd, usbaed jb p`qu`˚a abb tb tbojb ojb jjdmbr dmbrµµĲthikb qu` s` pr`sàtb b kdatiaubk kdatiaubkiµ iµdda, a, s` puè` v`r qu` `j jdmbrithd e` ;.? (pbrb tdhbr ua aµu uh`rd h`rd bj bzbr) `s 3.0302, hiàtrbs qu` `j jdmbrithd e` ?.5 (tdhµbbaedjd aedjd imubjhàt` bj bzbr) `s 3.;73?. (@stds vbjdr`s sda, pdr supu`std, sdjbhàt` bprdxihbeds, kdhd sda tdeds jds jdmbrithds, `xk`ptd bqu`jjds e` jbs pdtàkibs iat`mrbj`s e`j ei`z.) Bgd rb,, bquµ Bgdrb b quµĲ ``st stµbµ `j vbjdr e` jds jdmbrithds. [iàed r`bjhàt` `xpdaàt`s, bj hujtipjikbr jds aµu uh`rds h`rds u uaikbhàt` µaikbhàt` a`k`sitbhds suhbr jds `xpdaàt`s. Sbrbb hujt Sbr hujtipjikb ipjikbrr ;.? y ?.5, sµddjd jd a`k a`k`si `sitbh tbhds ds suh suhbr br 3.0302 y 3.;73? pbrb dotà`r 3.770;. @stb kifrb bµua ua `s ua `xpdaàt`, pdr supu`std, y sdjbhàt` ads eik` qu` eò`hds àkdatrbr `j vbjdr e` 23 3.777; . Sbrb tbj vbjdr, eò`hds ouskbr uab u ab tboj tbojbb batijdm batijdmbr brµµĲthikb ssuflkiàt`hàt` uflkiàt`hàt` e`t e`tbjjbeb bjjbeb b fla e` à àkdatrbr kdatrbr qu` `j batijdmbrithd e` 3.770; `s =.67. @s` `s `j r`sujtbed e` ;.? ?.5.

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aµu uh`r h`rdd jdmbr jdmbrit ithd hd 2.6 3.?352 2.= 3.?;35 2.7 3.?00; 2.9 3.?=77 ?.3 3.;323

aµu uh`r h`rdd jdmb jdmbri rith thdd ?.2 3.;??? ?.? 3.;5?5 ?.; 3.;62= ?.5 3.;73? ?.0 3.;9=9

aµu uh`r h`rdd jdmb jdmbri rith thdd ?.6 3.5203 ?.= 3.5;25 ?.7 3.55=? ?.9 3.56?5 ;.3 3.5==2

aµu uh`r jd3mbrit ;h.2`rdd jdmbr .59ithd 2hd 5 ;.? 3.0302 ;.; 3.0270 ;.5 3.0;20 ;.0 3.0552 aµu uh`r h`rdd jdmbr jdmbrit ithd hd 5.6 3.66?7 5.= 3.6=?2 5.7 3.672? 5.9 3.693? 0.3 3.6993

aµu uh`r jd3mbri ;h.6`rdd jdmb .0rith 0thd 6; d ;.= 3.067? ;.7 3.0=97 ;.9 3.0922 5.3 3.63?2 aµu uh`r h`rdd jdmb jdmbri rith thdd 0.2 3.=3=6 0.? 3.=263 0.; 3.=?5; 0.5 3.=;?5 0.0 3.=535

aµu uh`r jd3mbri 5h.2`rdd jdmb .6rith 2thd ?7 d 5.? 3.6?;? 5.; 3.6;;0 5.5 3.65;0 5.0 3.60;? aµu uh`r h`rdd jdmb jdmbri rith thdd 0.6 3.=57? 0.= 3.=009 0.7 3.=6;5 0.9 3.==39 6.3 3.==7?

230 aµu uh`r h`rdd jd jdmb mbri rith thdd 6.2 3.=70;

aµu uh`r h`rdd jd jdmbr mbrit ithd hd 6.6 3.7290

aµu uh`r h`rdd jdmb jdmbri rith thdd =.2 3.702;

=.? 3.70=; 6.= 3.7?62 6.? 3.=9?5 =.; 3.76;; 6.7 3.7;?0 6.; 3.=99; =.5 3.769? 6.9 3.7;77 6.5 3.736? =.0 3.7=02 =.3 3.7502 6.0 3.72?9 aµu uh`r h`rdd jd jdmb mbri rith thdd aµu uh`r h`rdd jd jdmbr mbrit ithd hd aµu uh`r h`rdd jdmb jdmbri rith thdd =.6 3.7737 7.2 3.9370 7.6 3.9;50 =.= 3.7760 7.? 3.92;7 7.= 3.9;90 =.7 3.79?2 7.; 3.9292 7.7 3.9550 =.9 3.79=6 7.5 3.9?5; 7.9 3.9595 7.3 3.93;2 7.0 3.9?95 9.3 3.905? aµu uh`r h`rdd jdmb jdmbri rith thdd aµu uh`r h`rdd jdmb jdmbri rith thdd 9.2 3.9093 9.6 3.97?; 9.? 3.96;7 9.= 3.9767 9.; 3.9670 9.7 3.992? 9.5 3.9=;2 9.9 3.9906 9.0 3.9=== 23.3 2.3333 Sdr dtrd jbed, pbrb eivieir ;.? àtr` ?5, µu uaikbhàt` aikbhàt` a`k`sitbhds r`stbr 3.;73? e` 3.0302 y dotà`r 3.2?59. @j batijdmbrithd e` `stb µu ujtihb jtihb kifrb `s 2.;;; qu` `s jb r`spu`stb. Bgdrb vdjvbhds bj prdoj`hb kda `j kubj `hp`zbhds `stb s`kkiµdda< a< 667; 7;== + 27??. Kdavirtiàed `stds aµu uh`rds h`rds b `xpdaàkibj`s, tà`hds 6.7;= 23; y 2.7?? 23; . @j jdmbrithd e` 23; `s sàkijjbhàt` ;, pu`std qu` ua jdmbrithd `s tba sµdjd djd `j `xpdaàt` e` 23 qu` eb `j vbjdr e` ua aµu uh`rd h`rd e`t`rhiabed.

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@j `xpdaàt` e` 23 qu` eb `j vbjdr e` 23 `s `vieàt`hàt` ;. @a jb hishb fdrhb, `j jdmbrithd e` 232? `s 2? y `j jdmbrithd e` 23∟25 `s 25. @j jdmbrithd e` 6.7;= eò` ouskbrs` à uab tbojb hµbbss e`tbjjbeb qu` jb qu` s` pr`sàtb à `st` jiord y r`sujtb s`r 3 .7;57=. @j jdmbrithd e` 6.7;= 23; `s, pdr jd tbatd, 3.7;57= + ; (r`kuµ ``re`s` re`s` qu` kubaed hujtipjikbhds jds aµu uh`rds h`rds suhbhds jds jdmbrithds) d ;.7;57=. @a jb hishb fdrhb, `j jdmbrithd e` 2.7?? r`sujtb s`r 3.?6300, pdr jd qu` `j jdmbrithd e` 2.7?? 23; `s 3.?630 ?63000 + ;;,, d s`b ;.?6300. Jb hujtipjikbkiµda da e` 67;= y 27?? quèb àtdak`s g`kgb bj suhbr jds jdmbrithds ;.7;57= + ;.?6300 dotàiàed ua r`sujtbed e` =.3905?. Vbhds b kbhoibr `sd b 3.3905? + = y b àkdatrbr` batijdmbrithd. (Jb pbrt` e`kihbj e` ua jdmbrithd ^3.3905? à `st` kbsdX `s jjbhbeb ‚hbatisb‘, `j àt`rd ^= à

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236

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

`st` kbsdX jb ‚kbrbkt ‚kbrbkt`r `rµµĲstikb‘ Ĳstikb‘). ). @j batdjdmbrithd e` ua aµuh`rd uh`rd àt`rd `s sihpj`hàt` jb pdtàkib e` 23 b `s` aµ u uh`rd. h`rd. Bs BsµµĲ, `j batijdmbrithd e` = `s 23= . @j batijdmbrithd e` 3.3905? `s 2.?56, (s`mµ u ua a bpbr`k` à uab tbojb). @j bat batijdmbr ijdmbrithd ithd e` =.3905? `s, pdr jd = tbatd, 2.?56 23 ( bj ir e` jdmbrithd b batijdmbrithd, abturbjhàt` jb suhb eò` r`kdav`rtirs` b hujtipjikbkiµdda) a) d, à aµ u uh`rds h`rds dreiab dreiabrids, rids, 2? , 563, 333. [i s` `f`ktµ u ubb jb hujtipjikbkiµdda a e` 67;= pdr 27?? dotàerµb kdhd r`sujtbed 2?, 50=, 325. [ia `hobrmd, gby qu` r`kdrebr qu` jds jdmbrithds ad sda `xbktds, pdr jd kubj su r`sujtbed sdjbhàt` puè` s`r bprdxihbed. Jb eivisiµdda a e` 67;= àtr` 29?? simaiflkb qu` gby qu` r`stbr jds jdmbrithds. Kdhd ;.7;57= ;.?6300 8 3.0=5;? pbrb kuyb kifrb `j batijdmbrithd `s ;.=0?, µ`stb `stb `s su r`spu`st r`spu`stb. b. Jb eivisiµdda a `f`ktivb ebrµb b ust` ustè e ua r`suj r`sujtbed tbed e` ;.=029? p`rd, r`p`ti r`p`tihds hds au` au`v vbhà bhàt`, t`, jds jdmbri jdmbrithds thds tba sµddjd jd bprdxihbeds. @j j`ktdr puè` àkd àkdatrbrs` atrbrs` ihp ihpbkiàt` bkiàt` kda u ua a hµ``tded tded qu` sdjb sdjbhàt` hàt` ria-

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e` r`sujtbeds bprdxihbeds y qu` r`qui`r` kdasujtbr eds tbojbs eif`ràt`s, tded `std à jumbr e` jbs dp`rbkida`s e` hujtipjikbkiµdda a y eivisiµdda a qu` ads sda tba fbhijibr`s. S`rd gby qu` kdasie`rbr qu` jds prdoj`hbs kiàt kiàtµµĲflkds y e` iamài`r iamài`rµµĲb fr`kuàt fr`kuàt`hàt` `hàt` kdhpràeà uab s`ri` kdatia kdatiaub ub e` hujtipjikbkida`s y eivisida`s. [` puè` kdafrdatbr jb r`sdjukiµdda a e` ua prdoj`hb kdhd ? (295.=67 3.350 29.? ) (2.007 ;0.5) . Udhbr dhbrµµĲb mrba kbatiebe e` ti`hp ti`hpdd `j r`sdjv`rjd pdr jds hµ``tdeds tdeds dreiabrids e` hujtipjikbkiµ dda a y eivisiµdda a hiàtrbs qu` uab p`rsdab gµbboij oij à `j hba`ld e` jds jdmbrithds, tded jd qu` tià` qu` gbk`r `s v`r uabs kubatbs kifrbs à uab tbojb (y kda prµbktikb, `std puè` gbk`rs` rµbbpiebhàt`) piebhàt`) suhbr, r`stbr, v`r dtrb kifrb y dotà`r jb r`spu`stb. Hµbbss bµ u ua, a, si uad `stµb sbtisf`kgd kda jbs r`spu`stbs qu` sda kbsi, p`rd ad tdtbjhàt` `xbktbs ‛y hukgbs v`k`s tbj`s r`spu`stbs ‚kbsi `xbktbs‘ puèà

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s`r huy u utij`s‛ µtij`s‛ àkdatrbrµb à `st `st`` hµ``tded tded uab mb mbabakib abakib beikid beikidabj abj à ti`hp ti`hpd. d.

@j au`vd au` vd µ bobk bo bkd d Jb mu muµµĲb bj hµ`tded `tded bor`vib bor`vibed ed puè` àkdatrbrs àkdatrbrs`` vi viàed àed e`tàieb e`tàiebhàt` hàt` jjbb fdrhb à qu` jds jdmbrithds buhàtba. Dosµ``rv`s` rv`s` qu` `j jdmbrithd e` 2.3 `s 3.333, qu` ` e` ?.3 `s 3.;323 y qu` `j e` ;.3 `s 3.5==2. Bj pbsbr e` 2 b ? `j vbjdr e`j jdmbrithd `s buhàtbed à 3.;323> bj pbsbr e` ? b ;, buhàtb beikidabjhàt`, p`rd sµddjd jd à 3.2=62. Kdatiaubaed, `j jdmbrithd e` 5.3 `s 3.63?3, qu` simaiflkb sdjbhàt` ua buhàtd beikidabj e` 3.2?59. Y bj pbsbr e` 9.3 b 23.3, `j jdmbrithd kbhoib

23= e` 3.905? b 2.3333, ua buhàtd e` 3.3507, hiàtrbs qu` bj pbsbr e` 29.3 b ?3.3, `j jdmbrithd kbhoib e` 2.?=77 b 2.;323, ua buhàtd sdjbhàt` e` 3.3???. [i uad fu`rb b trbzbr uab `skbjb à uab r`mjb pdaiàed ihpbrkibjhàt` `spbkibeds `spbk ibeds jds vbjdr` bjdr`ss jdmbr jdmbrµµĲthik Ĳthikds, ds, y à jb pbrt` dpu`stb s` kdjdkbr kdjdkbrba ba jds aµu uh`rds h`rds dreiabrids (batijdmbrithds) à jb pdsikiµdda a kdrr`spdaeiàt` b jds jdmbrithds, jds aµ u uh`rds h`rds dreiabrids pbr`k`r pbr`k`rµµĲba `str`kgbrs` hutubhàt` à `j `xtr`hd hµbs bs bjtd e` jb hishb. [` puè` v`r `std à jb ijustrbkiµdda a (44). @stb `s uab r`﬏`xiµdda a e`j g`kgd e` qu`, kda aµu uh`rds h`rds hbs bjtds, oblb jb prdpdrkiµda da e` buhàtd à jds jdmbrithds.

Fimurb 7.?< Batijdmbrithds y Jdmbritthds B tbj `skbjb e` bpr`tulbeds aµu uh`rds, h`rds, kdrr`spdaeiàt`s b uab s`ri` imubj e` jdmbrithds, s` j` jjbhb ‚`skbjb jdmbr jdmbrµµĲthikb‘. Hbt`hµ bbtikbhàt`, tikbhàt`, jb `skbjb jdmbrµĲĲthikb thikb kdastit kdastituy` uy` ua iastruh` iastruhàtd atd huy u utij. µ tij. Suèà utijizbrs` eds `skbjbs jdmbrµĲĲthikbs thikbs bj hishd ti`hpd, à fdrhb tbj qu` uab puèb hdv` hdv`rs` rs` kda r`sp`ktd b jb dtrb kdhd s` dos`rvb à jb ijustrbkiµda da (44).

Fimurb 7.;< @skbjb jdmbrµĲĲthikb thikb hdvioj` Hdviàed jbs eds `skbjbs pde`hds suhbr aµu uh`rds h`rds qu`, à jb fdrhb à qu` `stµbba a brr`mjbeds à uab `skbjb jdmbrµĲĲthikb, thikb, `quivbj` à r`bjiebe b suhbr jds jdmbrithds e` jds aµ uh`rds uh`rds iaeikbeds. @std b su v`z simaiflkb qu` `stbhds hujtipjikbaed jds hishds aµu uh`rds. h`rds. Sdr `l`hpjd, pbrb hujtipjikbr ? y ;, kdjµddqu`s` qu`s` `j 2 e` jb `skbjb hdvioj` sdor` `j ? e` jb `skbjb `stbkidabrib. [i s` kdrr` jb vistb bj aµu uh`rd h`rd ; e` jb `skbjb hdvioj`, s` suhb butdhµbbtikbhàt` tikbhàt` `j jdmbrithd e` ? bj e` ; y, kda

237

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

bosdjutb s`muriebe, s` àkdatrbrµbb,, eòbld e`j ; e` jb `skbjb hdvioj`, `j 6 à jb `skbjb `stbkidabrib. Y, kdhd puè` v`rs`, ? ; 8 6. Ubj`s pi`zb pi`zbss e` hbe`r hbe`rb, b, rby rbybebs bebs kda jia`b jia`bss (e` bqu bquµµĲ qu` s` jjbhà ‚r`mjbs‘), uab e` jbs kubj`s puè` e`sjizbrs` kda r`sp`ktd b jb dtrb, sda kdadkiebs kdhd ‚r`mjbs e` kµbbjkujd‘. jkujd‘. Pab r`mjb e` kµbbjkujd jkujd `jbodrbeb kdhd jb qu` s` hu`strb à `j eiould ( 44), tià` ua aµ u uh`rd h`rd eif`ràt` e` `skbjbs ‛tdebs sutij y `xbktbhàt` eivieiebs‛ jbs kubj`s puèà usbrs` pbrb jb r`sdjukiµdda a e` uab vbrièbe e` prdoj`hbs, tba fµbbkijhàt` kijhàt` kdhd `j e` ? ;.

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Fimurb 7.5< _`mjb e` kµbbjkujd jkujd Kdhd uab r`mjb e` kµbbjkujd jkujd `s, à rr`bjiebe `bjiebe,, uab tbojb jdmbrµĲĲthikb thikb fllbhàt` kdjdkbeb à ua pèbzd e` hbe`rb, puè` r`sdjv`r kubjqui`r prdoj`hb qu` jds jdmbrithds puèba r`sdjv`r. [u `xbktitue e`pàe` e` kuµbba a `str`kgbh` `str`kgbhàt` at` puèba j``rs` jbs eivisida`s qu` kdatià` jb hishb. Kbeb mbabakib mbab akib tià` su pµ``reieb reieb kdhp kdhpàsbtdrib. àsbtdrib. Pab r`mjb e` kµbbjkujd jkujd gbk` iaa`k`sbribs jbs tbojbs jdmbr jdmbrµµĲthikbs, sµĲ, p`rd uab e` jbs virtue`s e`j hba`ld e`j jdmbrithd à jb fdrhb dreiabrib `s qu` butdhµbbtikbhàt` tikbhàt` s` àkbrmb e`j puatd e`kihbj. Jb r`mjb e` kµbbjkujd jkujd ad jd gbk`. Sbrb jdkbjizbr `j puatd e`kihbj, trboblbaed kda jb r`mjb e` kµbjkujd bjkujd,, `s prih`rbhàt` a`k`sbrid gbk`r ua kbjkujd rµbbpied pied e` jb r`sdjukiµdda a e` su prdoj`hb. Sdr `l`hpjd, `l` hpjd, kdasieµ``r`s` r`s` `j prdoj`h prdoj`hbb q qu` u` pr`sàtµ` à uab e` jbs pµbbmiabs miabs bat`ridr`s< (295.=67 3.350 29.?? ) (2.007 ;0.5)

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Kbhoibaed jbs kifrbs b jds hbs k`rkbads ‚aµu uh`rds h`rds rèdaeds‘ `st` puè` 2 2 trbasfdrhbrs` b ?33 ?3? ;0 , r`suj r`sujtbaed tbaed 5333 03 8 73. 2 ? ?3 @j r`sujtbed, bsµĲ jd sbo`hds, eò` `stbr hµbbss k`rkb e` 73 qu` e` 7 d e` 733, y `sd fllb `j puatd e`kihbj (d, kdhd e`kihds bjmuabs v`k`s, fllb `j ‚dreà e` hbmaitue‘). Bgdrb oià, usbaed hi prdpib r`mjb e` kµbbjkujd, jkujd, puèd hba`lbr jb `skbjb hdvioj` e` uab pbrt` b dtrb, suhbaed y r`stbaed jdmbrithds, y àkdatrbr qu` jb kifrb e` jb r`spu`stb `s 07=. Kdhd sµ` qu` jb r`spu`stb eò` `stbr hµbs bs

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239 k`rkb e` 73 qu` e` 7 u 733, kdjdkd `j puatd e`kihbj pbrb gbk`r qu` `j aµu uh`rd h`rd s`b j` j`µµĲed kdhd 07.=. Jb `l`kukiµ dda a e`j kµbbjkujd jkujd tdtbj h` tdhµd ;0 s`muaeds, y eòd hbaif`stbr qu` jd gik` e`spbkid, kda dol`td e` ad `quivdkbrh`. Jb r`sdjukiµdda a e`j prdoj`hb pdr hèids brithµ``tikds tikds kdhua`s h` ebr ebrµµĲb uab r`spu`stb hµbbss `xbkt `xbktb. b. Jb jj`v jj`vµµ` b kbod pbrb dotà dotà`r `r 07.6 p`rd eurba eurbat` t` `j prdk`sd e` jb dp`rbkiµdda a kdh`t kdh`tµµĲ eds `rrdr`s qu` guo` e` kdrr`mir. @a tdtbj dkupµ` ?3 hiautds, p`rd gik` trbhpbs trbhpbs.. @a prih`r jumbr, e`skuorµĲ qu` gbo gboµµĲb kdh`tied kdh`t ied ua `rrd `rrdr, r, pdrqu` hi r`suj r`sujtbed tbed brith brithµµ``tikd tikd ad kdakd kdakdrebob rebob kda `j r`sujtbed dotàied pdr hèid e` hi r`mjb e` kµbbjkujd jk ujd.. Y e`s e`spuµ puµ`s, `s, jdkbj jd kbjikµ ikµ` his e`hµbbss `rrdr `rrdr`s `s y `vit `vitµµ` tà` tà`rr qu` r`p`tir tded `j prdoj` prdoj`hb, hb, v`r v`riflkba iflkbaed ed kbeb pbsd iaeivieubj kda jb r`mjb e` kµbbjkujd. jkujd. Jbs kdhputbedrbs hde`rabs, pdr supu`std, dp`rba hµbbss `xbktb y rµbbpiebpiebhàt` qu` jbs r`mjbs e` kµbbjkujd, jkujd, p`rd ad sda pdrtµbbtij`s. tij`s. Sbsbrµb hukgd ti`hpd bat`s e` qu` `j iamài`rd d `j kiàt kiàtµµĲflkd s` siàtba b mmustd ustd sia tà`r b su bjkbak` uab r`mjb e` kµbbjkujd. jkujd. Bjmuabs v`k`s, piàsd qu` `j kiàt kiàtµµĲflkd hde`rad `s jb r`àkbrabkiµ da da e`j batimud bhbauàs`, y su r`mjb e` kµbbjkujd jkujd jb r`àkbrabkiµda da e`j batimud µbbobkd. obkd.

223

˚ KBS µ IUPJD IUP JD 7. JD[ HPY HPY M_BAE@[ M_BAE@[ Y HPY S@RP@ S@RP@ AD[

KbpµĲtujd 9 E` aµ uh`rds e` jµĲa`b uh`rds Ĳa`b b auh`rds u µ h`rds e` sup`rflki` @atrb i Gbstb bgdrb, à tdebs his `xpdsik `xpdsikida`s ida`s sdor` jbs rbµ rbµĲk`s kuberbebs g` `vitbed kuiebedsbhàt` hàkidabr jds aµ u uh`rds h`rds a`mbtivds. Sdr `l`hpjd, eimd qu` 5 `s imubj b ?, pdrqu` ? ? 8 5. S`rd, pdr jb hishb pruòb, 5 eò` tbhoiµ`à a s`r ‛? pu`std qu` ? ? 8 5. (_`kuµ ``re`s` re`s` qu` jb hujtipjikbkiµ da da e` aµ u uh`rds h`rds a`mbtivds pdr aµu uh`rds h`rds a`mbtivds eb kdhd r`sujtbed aµu uh`rds h`rds pdsitivds). E` jd bat`ridr s` e`spràe` qu` à r`bjiebe `xistà eds rbµ rbµĲk`s p pdsioj`s dsioj`s e` 5 y, gbojbaed bprdpibebhàt`, `std `s jd qu` eò eòµµĲb `skrioirs`< 5 8 ?. @j

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sµĲho hod djd eò` iat`rpr`tbrs` s µĲz d hàds‘. Bgdrb oià, si ? y ? sda kdhd bhods‚hµ jb bbs rb rbµ kuberbeb e` +5, ´kuµbj bj `s jb rbµĲz Ĳz kuberbeb e` 54 ´Ru` auh` auh`rd rd huj hujtipjik tipjikbed bed pdr sµĲ hishd `s imubj b 54 @stbhds s`murds e` qu` +? ? 8 5, p`rd +? y ? ad sda `j hishd aµu uh`rd, h`rd, bs bsµµĲ qu`, q u`, bj hujtipjikb hujtipjikbrjds, rjds, aiamua aiamuadd hujtipjikb pdr sµĲ h hishd. ishd. Ubj pbr`k` kdhd si à aiamuab pbrt` `xisti`rb ua aµuh`rd, uh`rd, pdsitivd d a`mbtivd, qu` puei`rb s`r tbj v`z jb rb rbµµĲz kuberbeb e` 5 d e` kubjqui`r aµu uh`rd h`rd a`mbtivd. [ia `hobrmd, pdr t`rquèbe, vbhds b kdasie`rbr ua pdkd `j prdoj`hb. Srih`rd qu` tded, gbmbhds µ``st` st` tba sàkijjd kdhd pdebhds. Pa a aµ u uh`rd µh`rd kdhd 65 puè` frbkkidabrs` à fbktdr`s, e` hba`rb qu` puèb `skrioirs` 26 5. B ssu u v`z, µ``st` st` puè` trbasfdr trbasfdrhbrs` hbrs` à 26 5. [u s`pbrbkiµdda a à

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µ @_D[ µ @_D[ 22?KBS µ IUP IU PJD 9. E@ E@ A PH PH@_ D[ E@ Jµ IA@B B A PH PH@_ D[ E@ [PS@_F [PS@_FIKI IKI@ @

∜ ∜ ∜ Ý 5 eds rb rbµµĲk`s kuberbebs ad bf`ktb su vbjdr, pu`std qu` jb 65 `s 7 y 26

`s 5 ?, qu` `s 7. Jb r`p`tikiµdda a e` pruòbs e`hdstrbrµb qu` si ua aµu uh`rd h`rd `s frbkkidabed à fbkt fbktdr`s, dr`s, jb rb rbµµĲz kuber kuberbeb beb e`j hishd `s imubj bj prdeukt prdeuktdd e` jbs rbµ rbµĲk`s kuberbebs kuber bebs e` jds fbktd fbktdr`s. r`s. @std tbh tbhoiµ oiµ à à puè` s`r bpjikb bpjikbed ed b jds aµu uh`rds h`rds irrbkidabj`s. Sdr `l`hpjd, kdasie`r`hds jb 20 y jb e` 0 ;. [` puèà ouskbr à jbs tbojbs jbs rb rbµµĲk`s kuberbebs bprdpibebs, y àkda àkdatrbr trbr qu` jb e` 20, `s ;.7=?97;, jb e` 0 `s ?.?;6367 y jb e` ; `s 2.=;?302 (kifrbs abturbjhàt` bprdxihbebs). Kdhd ?.?;6367 2.=;?302 r`sujtb s`r imubj b ;.7=?97;, ;. àtdak`s, 20 8 0 Bgdrb oià. Kubjqui`r aµ u uh`rd h`rd a`mbtivd puè` kdasie`rbrs` kdhd `j prdeuktd e`j aµ u uh`rd h`rd pdsitivd kdrr`spdaeiàt` 2. @a dtrbs pbjborbs, 65 8 65 2> ?=6 8 ?=6 2> 2.97 8 2.97 2 y bsµ bsµĲ suk` suk`sivbh` sivbhàt`.) at`.) Jb rb rbµµĲz kuberbeb e` kubjqui`r aµu uh`rd h`rd a`mbtivd, eimbhds 2=?, puè`, pdr jd tbatd, frbkkidabrs` kdhd simu`<

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Sbrb bor`vibr, si sdjbhàt` supiµ``rbhds rbhds jb rbµ rbµĲz kuberbeb e` hàds uad ∜ ( ∟2) pd pder erµµĲbhds e`sbrrd e`sbrrdjjbr jjbr jb rb rbµµĲz kuberbe kuberbebb e` kubjqui kubjqui`r `r aµu uh`rd h`rd a`mbtivd. S`rd bquµĲ ads àkdatrbhd àkdatrbhdss bat` `j eij eij`hb `hb h hàkidab àkidabed< ed< 2 Ý 2 8 2 y ∟2 Ý ∟2 8 2. Aiamuad e` `stds aµu uh`rds, h`rds, hujtipjikb hujtipjikbeds eds àtr` sµĲ, ebrµba ba ∟2. Jb u uaikb µ aikb kdsb qu` pde`hds gbk`r `s iavàtbr   uab r`sp r`spu`st u`stb. b. Sd Sder erµµĲbh Ĳbhds, ds, pdr `l`hpjd, fllbr ua simad kdhd # y e`kir qu` # Ý # eb uab r`spu`stb a`mbtivb. @atdak`s s`r s`rµµĲb #2 Ý #2 8 ∟ 2. @sd s`r s`rµµĲb ki`rtd kdhd e`flaikiµ da da

y, si ad s` dpusi`rb kda kubjqui`r dtrb kdsb yb `stboj`kieb à `j `squ`hb

hbt`hµbbtikd, tikd, ad gbor gborµµĲb rbzµdda a pbrb qu` ad ads sbji sbjiµµ``rbhds rbhds kda jb au` au`strb strb à `st` bsp`ktd. Sdr supu`std, ua aµ u uh`rd h`rd kdhd µ``s` s` tàer tàerµµĲb qu` s`r ihbmia ihbmiabrid. brid. [bo` [bo`hds hds jd qu` `s +$2 y $2. @j prih`rd `s ua eµddjbr jbr (d kubjqui`r dtrb uaiebe hda`tbrib) e` kbpitbj, y `j µu ujtihd jtihd ua eµddjbr jbr (d kubjqui`r dtrb uaiebe hda`tbrib) e` beùed.. ´Ru` kdsb ss`r beùed `rµµĲb #$2 #$244 Jds prih`rds hbt`hµbbtikds tikds qu` trboblbrda kda jbs rb rbµµĲk`s kuberbe kuberbebs bs e` jjds ds aµu uh`rds h`rds a`mbtivds jds jjbhbrda ‚aµu uh`rds h`rds ihbmiabrids‘, y bua à jb bktubjiebe p`rsist` `s` adhor`. B eif`ràkib e` µ``stds, stds, jds aµuh`rds uh`rds dreiabrids, yb s`ba pdsitivds d a`mbtivds, rbkidabj`s d irrbkidabj`s, sda jjbhbeds ‚aµ u uh`rds h`rds r`bj`s‘. [ia `hobrmd, gbstb jb f`kgb, jds hbt`hµbbtikds tikds ad gba kdaf`kkidabed ua au`vd simad kdrr`spdaeiàt` b + d . (Y (Ydd e`s`br e`s`brµµĲb qu` jd guoi`rba g`kgd,

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22; s` s`rµ rµĲb Ĳb hµbs bs abturbj). @a kbhoid, ie`brda `j sµĲhodjd i (pbrb ‚ihbmiabrid‘) y p`rhiti`rda qu` s` tdhbrb kdhd jb rb rbµµĲz kuberbeb e` 2. @a dtrbs pbjborbs, 2 8 i , be`hµbbs, s, y pdr supu`std, `std simaiflkb qu` i i 8 i ? 8 2. [ia eueb, i i `s tbhoiµ`à a imu imubj bj b i? d 2 y, pdr `stb rbzµdda, a, r`bjhàt`, eò`hds i 8 i? 8 ( 2) 8 2. `skrioir 2 8 i. Y, pbrb rèdae`br `j bsuatd, i Y bgdrb yb tà` tà`hds hds jb r`spu r`spu`stb `stb pbrb tdebs jbs rbµ rbµĲk`s kuberbebs e` jds aµu uh`rds h`rds a`mbtivds. @j vbjdr e` 5, `s imubj b 5 2, d b ? i, qu` puè` sihpj`hàt` `skrioirs` ?i, d hµbbss be`kubebhàt`, ?i. @a jb hishb fdrhb, `j vbjdr e` 65 `s 7i, hiàtrbs qu` `j vbjdr e` 20i, 20 `s d, bprdxihbebhàt`, ;.7=?97;i.

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Aµ uh`rds uh`rds y puatds e` jb orµ ulujb ulujb [ia `hobrmd, yb pbrb bgdrb puè` `j j`ktdr `stbr rboibaed e` iaeimabkiµdda. a. Suè` `stbr pàsbaed E`l`hds `stb pjµbbtikb tikb r`sp`ktd b i. ´R ´Ruµ uµ` sima si maifl iflkb kb i4 @a r`bjiebe, puè` simaiflkbr jd qu` b `j`kkiµ dda a au`strb e`s``hds qu` simaiflqu`. Eò` r`kdrebrs` qu` jds aµu uh`rds h`rds sda sdjbhàt` kr`bkiµdda a e`j gdhor` pbrb byuebrjd b àtàe`r bj uaiv`rsd qu` jd rde`b. Sde`hds gbk`r jd qu` qu`rbhds kda au`strbs prdpibs kr`bkida`s à tbatd ads byueà b àtàe`r. Sdr `l`hpjd, ` l`hpjd, jds batimuds mri`md mri`mdss ad pd pde eµĲba v`r quµ` simaiflkb s imaiflkbob ob 2< j`s `rb hist`ridsd hist`rid sd kdhd jd gbor gborµµĲb sied jb 2. Urbtµ` e` `xpjikbr à `j kbpitujd ? jd qu` `s 2 usbaed uab jia`b v`rtikbj à jb qu` kdjd kdjdquµ quµ` jds aµu uh`rds h`rds pdsitivds àkihb e`j k`rd y jds aµu uh`rds h`rds a`mbtivds bobld e`j k`rd. Kdhd `std pbr`k` fuakidabr, vdjvbhds b `jjd. [updambhds qu` trbzbhds dtrb jµĲa`b gdrizd gdrizdatbj atbj b trbvµ`s `s e`j puatd k`rd e` `sb prih`rb jµĲa`b, hbrkbae hbrkbaedd

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jb jµĲĲa`b a`b gdrizdatb gdrizdatbjj b `spbkids imubj`s b jd jdss yb u usbeds sbeds à jb jµĲa`b v`rtikbj. B jb e`r`kgb hbrkbr`hds jds `spbkids kdhd +i, +? i, +; i, +5 i y bsµĲ suk` s uk`si sivbh` vbhàt`. at`. B jb izqui`reb, kdhd i, ?i, ;i, 5i, `tk. Jb ijustrbkiµdda a ( 44) hu`strb `std. Bgdrb pde`hds v`r jd qu` simaiflkb `j tµ``rhiad rhiad i. Sdr `l`hpjd, `stuvihds e` bku`red à qu` +2 simaiflkb ua nijµddh`trd h`trd bj adrt` e` uab pdojbkiµdda a (pdr `l`hpjd) y 2 ua nijµddh`trd h`trd bj sur e` uab pdojbkiµdda. a. @atdak`s +i simaiflkb ua nijµddh`trd h`trd `st` e` uab pdojbkiµdda a y i, ua nijµddh`trd h`trd bj d`st` e` uab pdojbkiµdda. a. (@std ad `s jd qu` ‚r`bjhàt`‘ simaiflkba jds aµ uh`rds. Ad tiàà ua uh`rds. simaiflkbed ‚r`bj‘. @s sdjbhàt` uab ijustrbkiµdda a pbrb byuebrads b kbptbr jb r`jbkiµ dda a bostrbktb à tµ``rhiads rhiads e` bjmd kdakr`td.) Sdrr `l` Sd `l`hpj hpjd, d, à v` v`zz e` usb usbrr sµĲh Ĳhodjds odjds kdh kdhdd + y , pd pder erµµĲbhds gbo` gbo`rr e`kieied hbrkbr jds aµu uh`rds h`rds kda jbs iaikibj`s e` jds puatds e` jb orµu ulujb. lujb.

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µ @_D[ µ @_D[ 225KBS µ IUP IU PJD 9. E@ E@ A PH PH@_ D[ E@ Jµ IA@B B A PH PH@_ D[ E@ [PS@_F [PS@_FIKI IKI@ @

Fimurb 9.2< JµĲa`bs pbrb aµu uh`rds h`rds r`bj` r`bj`ss ` ihbmia ihbmiabrids brids

SderµĲbhds gbo`r e`kieied jjbhbr b jds aµu Sderµ uh`rds h`rds r`bj`s pdsitivds, ‚aµu uh`rds h`rds A‘ (pdr `j adrt`), b jds aµ u uh`rds h`rds r`bj`s a`mbtivds ‚aµu uh`rds h`rds [‘ (pdr `j sur), jds aµ u uh`rds h`rds ihbmiabrids pdsitivds ‚aµ u uh`rds h`rds @‘ (pdr `j `st`), y b jds aµ u uh`rds h`rds ihbmiabrids a`mbtivds ‚aµ u uh`rds h`rds D‘ (pdr `j d`st`). [dor` `stbs obs`s, ad `xistir `xistirµµĲb rbzµdda a pbrb jjbhbr ‚r`bj`s‘ b jds aµ u uh`rds h`rds A y [, ` ‚ihbmiabrids‘ b jds aµu uh`rds h`rds @ y D. Jds kubtrd puatds e` jb orµu ulujb lujb sda tdeds imubjhàt` r`bj`s. [ia `hobrmd, `j adhor` ‚ihbmiabrid‘ surmiµd pdr rbzda`s gistµddrikbs rikbs y bktubjhàt` `s yb huy tbre` pbrb kbhoibrjd. Psbaed `stds ‚aµu uh`rds h`rds e` orµ u ulujb‘ lujb‘ pbrb e`sbrrdjjbr uabs hbt`hµbbtikbs tikbs butdkdasistàt`s (qu` `s jd qu` jds hbt`hµbbtikds tikds à r`bjiebe `stµbba a trbtbaed e` gbk`r) pderµĲĲbb s`r a`k`sbrid e`kir qu`<

220

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@st` dreà (uab `sp`ki` e` tbojb e` hujtipjikbr pbrb simads) `s kda `j qu` `f`ktiv `f`ktivbhàt` bhàt` trbob trboblbhds lbhds à jbs hbt`hµbbtikbs tikbs qu` gba sied iav` iavàtbebs atbebs pdr jb guhbaiebe. Suè` pbr`k`r rbrb `stb tbojb e` hujtipjikbr, p`rd tià` uab sàkijjb simaiflkbkiµdda a m`dhµ``trikb trikb qu` `j j`ktdr e`skuorirµ b bj `stueibr jb ijustrbkiµ dda a (44) @std ad qui`r` simaiflkbr qu` jbs au`strbs s`ba jbs µu uaikbs aikbs ‚v`rebe`rbs‘ hbt`hµbbtikbs tikbs d qu` bjmµ u ua a dtrd dreà puei`rb ad s`r imubjhàt` ‚v`rebe`rd‘. [ukè` qu` µ``st` st` gb sied `j dreà qu` ads gb e`hdstrbed s`r u utij µ tij y gb sied `j qu` g`hds usbed.

Hµ bs bs puatds e` jb orµ orulujb u µ lujb Yb qu` `s `stbhds tbhds gbojba gbojbaed ed e` a adrt`, drt`, su sur, r, `st` y d`s d`st`, t`, ´quµ` gby à ku kubatd batd b uab eir`kkiµdda a kdhd adre`st` y sue`st`4 Gbstb bgdrb, à au`strbs hujtipjikbkida`s e` aµu uh`rds h`rds r`bj`s ` ihbmiabrids, jbs r`spu`stbs gba r`sujtbed si`hpr` à uab e` jbs eds jia`bs, adrt` sur d`st` dd`st`. `st`. [ia `hobrmd, ´quµ` e`kir e` jb suhb4 ´Kuµbbjj `s jb suhb e` 2 hµbbss i4 Jds hbt`hµbbtikds tikds ad gba àkdatrbed a`k`sbrid iav` iavàtbr atbr sµĲĲhodjds hodjds qu` r`pr`sàtà uab suhb tbj y jb gba e`lbed sihpj`hàt` kdhd 2+ i. Ad dostbat`, puè` gbk`rs` qu` `stb suhb tàmb simaiflkbed à au`strd sist`hb auhµ auhµ``rikd rikd eir`kkida bj. Vbhds b trb eir`kkidabj. trbzbr zbr au`vbhàt` jbs jµĲa`bs adrt`-su adrt`-surr y `st`-d `st`-d`st` `st` y, à v`z e` pda` pda`rr ad adtbs tbs auh`rbebs à jjbs bs jia`bs, trbk`hds au`vbs jjµµĲa`bs b ttrbvµ rbvµ`s `s e` kbeb adtb. [updambhds qu`, b trbv trbvµµ`s `s e` kbeb adtb à jb jµĲa`b v`rtikbj e` aµu uh`rds h`rds r`bj`s, trbzbhds uab j`v` ji jia`b a`b gdrizdatbj gdrizdatbj.. Jb jµĲa`b qu` p pbsb bsb b trbvµ t rbvµ``ss e` +2 `s +2 à tdeb su jdamitue à kbeb puatd. Jb jia`b gdrizdatbj qu` pbsb b trbv trbvµµ`s `s e` +? `s +? à td tdeb eb su jdamitue, hiàtrbs qu` jb qu` pbsb b trbv trbvµµ``ss e` ; `s ; b jd jbrmd e` tdeb jb jµĲa`b, y bs bsµµĲ suk`sivbhàt`. Gbmbhds jd hishd kda jb jµĲa`b gdrizdatbj e` aµu uh`rds h`rds ihbmiabrids, trbzbaed j`v`s jµĲa`bs v`rtikbj` v`rtikbj`ss b trbv trbvµµ`s `s e` kbeb adtb. Au`v Au`vbhàt`, bhàt`, jb jµĲa`b v`rtikbj b trbvµ`s `s e` +?i s`rµb +?i à tdeb su jdamitue, jb qu` trbv trbvµµ`s `s e` 0i s`rµb 0i à tdeb td eb ssu u jjdamitu damitue, e, y bs bsµµĲ suk`sivbhàt`. s uk`sivbhàt`.

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µ @_D[ µ @_D[ 226KBS µ IUP IU PJD 9. E@ E@ A PH PH@_ D[ E@ Jµ IA@B B A PH PH@_ D[ E@ [PS@_F [PS@_FIKI IKI@ @

(b)

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Fimurb 9.?< Fimurb A8Aµ u uh`rds h`rds r`bj`s pdsitivds, tbj`s kdhd +2 @8Aµ u uh`rds h`rds ihbmiabrids pdsitivds, tbj`s kdhd + i [8Aµ u uh`rds h`rds r`bj`s a`mbtivds, tbj`s kdhd 2

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38Auh`rds 38A uh`rds ihbmiabrids a`mbtivds, tbj`s kdhd

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22= Kubaed `std gbyb sied g`kgd, kdhd s` hu`strb à jb ijustrbkiµdda( a(44), tàer` tà er`hds hds ua hde` hde`jd jd e` tboj`r tboj`rdd e` ebh ebhbs bs kda jb jµĲa` Ĳa`bb e` kbe kbebb aµ u uh`rd h`rd ihbmiabrid kruzbaed jb jµĲa`b e` kbeb aµu uh`rd h`rd r`bj y vik`v`rsb.

Fimurb 9.; Bgdrb pde`hds dotà`r jb r`spu`stb bj bsuatd r`f`ràt` b jb suhb e` 2 + i. Pa aµu uh`rd h`rd kdh kdhdd 2 + i r`pr`sàtbr r`pr`sàtbrµµĲb ua puatd qu qu`` `s b jb v`z +2 y +i y `s` puatd sµddjd jd puè` `xistir edae` jbs jia`bs s` kruzba à `j tboj`rd hde`jd. Kdhd jds `spbkids sda imubj`s à bhobs jµĲĲa`bs, a`bs, 2 + i r`pr`sàtb ua aµ u uh`rd h`rd à eir`kkiµdda a adr adre`st e`st`. `. Bs BsµµĲ dk dkurr` urr`,, tbhoiµ`à a à ? + ?i, ; + ;i, 5 + 5i, `tk. Pa aµ u uh`rd h`rd kdhd 2 i puè puè`` `skr `skrioi ioirs` rs` kdh kdhdd +2 + ( i) y `j puatd qu` jd r`pr`sàtb `stbr `stbrµµĲb edae` kruzba jbs jµĲa`bs e` +2 y i uoikbed à eir`kkiµda da adrd`st`. [`h`lbat`h [`h`lbat`hàt` àt` 2+ i `st `stbr brµµĲb à `j sue sue`st` `st` y 2 i, à `j sued`st`. Dtrbs eir`kkida`s puèà s`r r`pr`sàtbebs pdr aµu uh`rds h`rds tbj`s kdhd 0+?i, = ;i, `tk. @a r`bjiebe, kbeb pdsikiµdda a à `j tboj`rd hde`jd (qu` puè` ihbmiabrs` s` `xtiàe` iae`flaiebhàt` à kbeb eir`kkiµdda) a) puè` kdasie`rbrs` kdhd ua puatd qu` r`pr`sàtb bjmµu ua a aµ u uh`rd h`rd qu` `s jb suhb e` aµu uh`rd h`rd r`bj y uad ihbmiabrid. @s hµbbs, s, jb pdsikiµda da puè` tà`r vbjdr`s e`kihbj`s, d bua irrbkidabj`s, y `stbr situbeb àtr` jbs jµĲa`bs e`j tboj`rd kdhd 9 .0 5 + 3.320i, d ? = 0 ?i. Jds aµ uh`rds uh`rds e` `st` tipd, qu` iakjuyà uab pbrt` r`bj y dtrb ihbmiabrib,

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µ @_D[ µ @_D[ 227KBS µ IUP IU PJD 9. E@ E@ A PH PH@_ D[ E@ Jµ IA@B B A PH PH@_ D[ E@ [PS@_F [PS@_FIKI IKI@ @

sda jjbhbeds ‚aµ u uh`rds h`rds kdhpj`lds‘. @a r`bjiebe, jds aµu uh`rds h`rds qu` sda àt`rbhàt` r`bj`s d àt`rbhàt` ihbmiabrids miabri ds tbhoiµ`à a pd pder erµµĲb kdasie`r kdasie`rbrs` brs` kdhd aµu uh`rds h`rds kdhpj`lds. @j aµu uh`rd h`rd 5?, pdr `l`hpjd, puè` `skrioirs` kdhd 5? + 3 i, hiàtrbs qu` `j aµu uh`rd h`rd 0i puè` `skrioirs` kdhd 3 0i. Jb ihpdrtbakib e` jds aµu uh`rds h`rds kdhpj`lds à jds bsuatds prµbbktikds ktikds `stµb à qu` µ``stds stds ad sdjbhàt` eàdtba jb hbmaitue kdhd aµu uh`rds h`rds dreia dreiabrids, brids, siad tbhoiµ`à a jb eir eir`kki `kkiµµdda. a. Sbrb ebr ua `l`hpjd, à fµĲsikb ` iamài`r iamài`rµµĲb jds ‚`sfu`rzds‘ lu`mba ua pbp`j ihpdrtbat`. Jds `sfu`rzds sda `hpul`s y trbkkida`s. @xistà `sfu`rzds r`sistàt`s, iat`rhèids y pdkd r`sistàt`s. Jds `sfu`rzds puèà s`r hèieds sdjbhàt` à aµ u uh`rds h`rds r`bj`s, kda `hpul`s pdsitivds, pdr `l`hpjd, y trbkkida`s a`mbtivbs. S`rd jds `sfu`rzds tbhoiµ `à a tiàà eir`kkiµ dda. a. Pa `hpul` d uab trbkkiµda da

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puè` s`r sup`ridr, iaf`ridr, jbt`rbj d iat`rhèid. Y pbrb bhods, aµu uh`rd h`rd y eir`kkiµ dda, a, s` a`k`sitba jds aµuh`rds uh`rds kdhpj`lds. @a `stb fdrhb, `j aµu uh`rd h`rd i, qu` pbrb hukgbs p`rsdabs blàbs b jbs hbt`hµbbtikbs tikbs pbr`k` kdhpj`tbhàt` hist`ridsd ` iaµu utij, tij, `s pdsitivbhàt` µu utij. tij. @a `j`ktrµddaikb, aikb, pdr `l`hpjd, ad `xistirµĲĲbb p pdsioij dsioijiebe iebe e` trb trbtbhiàtd tbhiàtd hbt`hµbbtikd tikd sia kdatbr kda i . Jbs kdrriàt`s bjt`rabs kbhoiba kdastbat`hàt`, tbatd à kbatiebe kdhd à eir`kkiµda da y, pbrb `xpr`sbr `stds kbhoids hbt`hµbbtikbhàt` tikbhàt` à bhods bsp`ktds, eòà usbrs` jds aµu uh`rds h`rds kdhpj`lds. Jds aµ u uh`rds h`rds kdhpj`lds puèà suhbrs` y r`stbrs` kdhd jds aµu uh`rds h`rds dreiabrids, eiabri ds, suhµbbaeds` aeds` y r`stµbaeds` baeds` s`pbrbebhàt` jbs pbrt`s r`bj`s y jbs ihbmiabribs. miabri bs. Bs BsµµĲ, si +? 5i s` suhba b 0 + =i, `j r`sujtbed `s ; + ;i, p`rd si s` r`stba, `j r`sujtbed `s = + 22i. (@std puè` e`hdstrbrs` à `j tboj`rd

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hde`jd, kdhd puè` puèà a e`hds e`hdstrbrs trbrs`` jb suhb y jb r`st r`stbb dreia dreiabribs bribs à jb jµĲa`b adrt`-sur p`rd tbj v`z yb b `stbs bjturbs puèb `j j`ktdr hishd gbk`rjd.) Jb hujtipjikbkiµdda a `s hµbbss kdhpjikbeb, p`rd à r`bjiebe ad hµbbss kdhpjikbeb qu` jb e` aµu uh`rds h`rds dreiabrids. Bj hujtipjikbr ;0 ?7, kdhd s` `xpjikµd à `j kbpµµĲtujd ;, s` frbkkidab `j a kbp aµ u uh`r µ h`rdd ;0 à ;3 + 0 y `j ?7 à ??33 + 7. Kb Kbeb eb pb pbrt rt`` e` ua aµ u uh`rd h`rd `s àtdak`s hujtipjikbed pdr kbeb pbrt` e`j dtrd, suhµbbaeds` aeds` jds suohµ u ujtipjds. jtipjds. @xbktbhàt` imubj s` prdkè` kda jds aµu uh`rds h`rds kdhpj`lds. Sbrb hujtipjikbr ; + 0 i 6 + i, s` pjbat`b jb dp`rbkiµdda a à jb fdrhb simuiàt` (44) [imuiàed jbs ﬏`kgbs y hujtipjikbaed kbeb kifrb sup`ridr pdr kbeb kifrb iaf`ridr v`hds qu`< ; 6 8 27> ; i 8 ;i> 0i 6 8 ;3i> y 0i i 8 0i? . Kdhd i? 8 2, 0i? 8 0.

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Fimurb 9.5

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Eds e` jds suohµujtipjds ujtipjds sda r`bj`s. @stds, 27 y 0, si s` suhba gbkà 2;. Jds dtrds eds, ;3i y ;i, sda ihbmiabrids y si s` suhba gbkà ;; i. @atdak`s `j r`sujtbed e` au`strb hujtipjikbkiµdda a e` aµ u uh`rds h`rds kdhpj`lds `s 2; + ;;i. Dtrds tipds e` dp`rbkida`s brithµ``tikbs tikbs puèà e`hdstrbrs` tbhoiµ `à a à fdrhb s`h`lbat`. Sbrb bor`vibr, jds aµu uh`rds h`rds kdhpj`lds puèà hba`lbrs` pdr jds hishds hèids qu` jds aµu uh`rds h`rds dreiabrids y, pdr jd tbatd, abeb tiàà e` hist`ridsds.

Ouskb Ou skbae aed d rbµ rbµĲk`s Ĳk` s hµ b bss prdf pr dfua uaeb ebh` hàt` at` @j kbhpd e` jds aµ u uh`rds h`rds kdhpj`lds bor` jb pdsioijiebe e` kdasie`rbr bjmuads g`kgds r`sp`ktd b jbs rb rbµµĲk`s hbs `j`v `j`vbebs. bebs. Yb h` g` p`rhiti p`rhitied ed `xpji `xpjikbr kbr qu` jb rb rbµµĲz kuber kuberbeb beb e` kubjq kubjqui`r ui`r a aµ u uh`rd µh`rd tià` eds sdjukida`s, pdr jd qu` jb +2 `s d oià +2 d 2 y jb 2 `s d oià +i d i. ´Ruµ` e`kir e` jb +24 @vieàt`, +2 +2 +2 +2 8 +2, pdr jd qu` +2 `s uab e` jbs kubtrd rb rbµµĲk`s e` +2. @atdak`s, sup supdambhds dambhds kdasie`rbr 2 2 2 2. Jds eds prih`rd prih`rdss tµ``rhiads rhiads hujtipjikb hujtipjikbeds eds àtr` sµĲ eba +2, y jd hishd jds eds u ujtihds> µ jtihds> p pdr dr jd tba tbatd, td, 2 2 2 2 `s jd hishd qu` +2 +2. @a kdas`kuàkib, 2 `s uab s`muaeb kubrtb rbµĲz Ĳz e` +2. S`rdd ad g`hds t`rh S`r t`rhiabed. iabed. ´Ru ´Ruµµ` gby r`sp`k r`sp`ktd td b +i +i +i + i4 @j prdeuktd e` +i y +i `s 2, pdr jd kubj jb hujtipjikbkiµdda a kuµbberupj` erupj` `s `quivbjàt` b 2 2, qu` `s +2. @a kdas`kuàkib, +i `s uab kubr kubrtb tb rbµ rbµĲz2 e` +2. Sdr `j hishd rbzdabhiàtd puè` e`hdstrbrs` qu` i `s uab kubrtb rbµĲĲzz e` +2 +2.. Jb kdakjusiµdda a `s qu` jb +2 tià` kubtrd r`spu`stbs< +2, 2, +i y i. 2 tià` kubtrd r`spu`stbs< + +i, [ihijbrhàt` s` puè` e`hdstrbr qu` jb +i, + i y i. SderµĲb Ĳb `j j`kt j`ktdr dr pr`m pr`muatbr uatbr quµ` `s +i. B `sd jb r`spu`stb `s< `s ua aµuh`rd uh`rd qu` bj hujtipjikbrs` pdr sµĲ hishd eb kdhd r`sujtbed

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5

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A. e`j Urbaskriptdr< @a `j sàtied e` s`r uab ‚rb ‚rbµµĲz kubrtikb‘

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µ @_D[ µ @_D[ 2?3KBS µ IUP IU PJD 9. E@ E@ A PH PH@_ D[ E@ Jµ IA@B B A PH PH@_ D[ E@ [PS@_F [PS@_FIKI IKI@ @

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i. Sdr `sb rbzµ dda, a, + +i + +i `s imubj b +i y + +i + +i + +i + +i puè` trbasfdrhbrs` à +i +i, qu` b su v`z `s imubj b 2. E` bquµĲ qu` + +i s`b uab e` jbs kubrtbs rb rbµµĲk`s e` 2 y, pdr imubj rbzdabhiàtd, jd sda +i, + tbho hoiµ iµà i y i.

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@a jb hishb fdrhb puè` e`hdstrbrs` qu` tdeds jds aµu uh`rds, h`rds, tià tiàà à kubtrd kub trd kubr kubrtbs tbs rb rbµµĲk`s Ĳk`s.. S`rd si tdeds jds aµ u uh`rds h`rds tiàà eds rbµ rbµĲk`s kuberbebs y kubtrd kubrtbs rbµµĲk`s rb Ĳk`s,, ´tà ´tàerµ erµb tbhoiµ`à a tr`s rb rbµµĲk`s kµu uoikbs, oikbs, kiakd quiatbs rbµ rbµĲk`s, kubràtb y eds kuberbmµ``sihbs sihbs rb rbµµĲk`s, y bs bsµµĲ suk`sivbhàt`4 @a r`bjieb r`bjiebe, e, bsµ bsµĲ `s. S`rd s` r`qu` r`qu`rir rirµµĲb `j usd e` hbt`h hbt`hµµbbtikbs tikbs hµbbss `j`v `j`vbebs bebs pbrb e`hdstrbr e`hdstrbrjd, jd, kdsb ihpdsioj` à `st` jiord. [ia `hobrmd, pd pde`hds e`hds kdhprdobr e` kubjqui`r hba`rb jb t`dr t`drµµĲb sdor` jbs rbµĲk Ĳk`s `s ku uoikbs. µoikbs. ´Ruµ` `s `s.. p pdr dr ``l`hpjd, l`hpjd, jb +24 Sbrb `hp`zbr, +2 +2 +2 8

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;

+2>S`rd, pdr tbatd, +2`stµ `sbba uab jb jbss rbµ ĲĲk`s k`s kkµ uoikbs µ oikbs éµddae` ae` a jbse`dtrbs eds4 [iu vbhdse`b+2. jds a`mbt a`mbtivd ivds, s, pder pderµµĲbhds àsbybr 2 2 2. [ia `hobrmd, 2 2 8 +2> pdr jd tbatd, 2 2 2 puè` trbasfdrhbrs trbasfdrhbrs`` b +2 2 y `sd `s imubj b 2. @a kdas`kuàkib, 2 ad `s uab e` jbs rb rbµµĲk`s kµu uoikbs oikbs e` +2. Be`hµbbs, s, kubjqui`r aµu uh`rd h`rd r`bj eistiatd b +2 d 2, d kubjqui`r aµu uh`rd h`rd ihbmiabrid eistiatd b +i d i, ad ebrµĲb Ĳb jb kbatiebe e` 2 (e` kubjqui`r simad) si s` `j`vb bj kuod. Éµddae` ae` `stµbba, a, àtdak`s, jbs eds rb rbµµĲk`s kµu uoikbs oikbs fbjtbat`s e` +24 ´@xistà4 [i, `xistà> p`rd à `j kbhpd e` jds aµu uh`rds h`rds kdhpj`lds. [ia e`skrioir `j hµ``tded tded pbrb àkdatrbrjbs, sihpj`hàt` dfr`k`r dfr`k`rµµ` jds aµu uh`rds h`rds kdhpj`lds 2 2 2 2 ;i y + ;i, kdhd kd hd jjbs bs rrbbµĲk`s fbjtbat` fbj tbat`s. s. @ @std std tbhoiµ`à a jjdd p pde de`hd `hdss ? ? ? ? kdhprdobr. ; 2 2 2 2 + ;i ;i, `s uab e` jjbs bs rrbbµĲk`s kµu uoikbs oikbs e` +2, àtdak`s + [i ? ? ? ? 2 2 2 2 2 2 + ;i eòà s`r imubj`s b +2. + ;i + ;i d ? ? ? ? ? ? Sde`hds `f`ktubr jb prih`rb hujtipjikbkiµdda a à jb fdrhb e`skritb à pµbbmiabs miabs bat`ridr`s (44) 2 2 ;i. ;i, suhba, pd supu`std, Jds eds suohµu ujtipjds jtipjds ihbmiabrids, ? 5 ; @a jd r`f`ràt` b + i?, `s `f`ktivbhàt` ua auh`rd r`bj, pu`std qu` i? 8 5 ; ? ; 2 2 2, pdr jd qu` + i 8 . @s` y `j dtrd suohµu ujtipjd jtipjd r`bj, + , suhba . 5 5 5 ?

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2?2

Fimurb 9.0 @j prdeuktd e` `stb hujtipjikbkiµdda a r`sujtb s`r<

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? ? @a s`muieb `s a`k`sbrid hujtipjikbr uab v`z hµbbs, s, pdr fdrhb qu` s` e`tbjjb (44).

∟ 2? + 2? ∜ ;i, à jb

Fimurb 9.6

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2 2 Jds eds suohµu ujtipjds jtipjds ihbmiabrids, + ;i, y ;i, suhba 3, pdr jd 5 5 ; ? i `s `f`ktivbhàt` r`bj y, pu`std qu` qu` puèà djviebrs`. @j aµ u uh`rd h`rd 5 ; 2 i? 8 2, r`sujtb s`r . [µ u uh`s` h`s` `std bj suohµujtipjd ujtipjd r`stbat` + , y `j mrba 5 5 tdtbj e` jb kuoikbkiµdda a r`sujtb s`r +2. 2 2 ;i, y @j hishd prdkèihiàtd puè` jj`v jj`vbrs` brs` b kbod b trbv trbvµµ`s `s e` ? ? 2 2 bhods y + ;i sda, pdr jd tbatd, rb rbµµĲk`s kµu uoikbs oikbs e` +2. ? ?

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µ @_D[ µ @_D[ 2??KBS µ IUP IU PJD 9. E@ E@ A PH PH@_ D[ E@ Jµ IA@B B A PH PH@_ D[ E@ [PS@_F [PS@_FIKI IKI@ @

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Ubhoiµ`à a puè` puè ` e e`hd `hdstrb strbrs` rs` qu` 2 tià t ià`` tr tr`s `s rbµ rbµĲk`s kµu uoikbs, oikbs, kdhd bsihishd jbs tiàà i y i. @a kbeb kbsd, eds e` jbs rbµĲĲk`s k`s sda aµu uh`rds h`rds kdhpj`lds.

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Hµ bs bjjµ b e` i Ubhoiµ `à a `s pdsioj` ie`br uab t`rk`rb jµĲa`b à au`strd tboj`rd hde`jd, b fla e` tà`r ad sdjbhàt` adrt`, sur, `st` y d`st`, siad, bsihishd, pdrhàdr`s hiaukidsd hiau kidsds. s. @std ads p`rhit p`rhitirµ irµ b tà`r uab kifrb sµddjieb jieb y, à v`z e` fdrhbr ua tboj`rd e` kuberbeds, pde`hds fdrhbr ua hdsbikd e` kuods. @j t`rk`r `l` kdastb e` ‚gip`rihbmiabrids‘, qu` sda sihodjizbeds kdhd l . @a dtrbs pbjborbs, jds aµuh`rds uh`rds b jd jbrmd e`j t`rk`r `l` sda< +2 l , +? l , +; l , bs bsµµĲ suk`sivbhàt` à uab eir`kkiµdda, a, y, 2 l , ? l , ; l , y bs suk`sivbhàt`, à jb dtrb.

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àtdak`s pdsioj` jdkbjizbr kubjqui`r aµ u uh`rd h`rd à`st`-d`st` `j `spbkid,ypdr e`j @s puatd e` kruk` e` ua pjbad adrt`-sur, ua pjbad ua hèid pjbad e` pdrhàdr`s hiaukidsds. (Jds kuods s` fdrhba e` jbs iat`rs`kkida`s e` jds pjbads,, bs pjbads bsµµĲ kdhd jd jdss kuberb kuberbeds eds e`j tb tboj`rd oj`rd s` fd fdrhbrda rhbrda e e`` jbs iat`rs`kki iat`rs`kkida`s da`s e` jbs jia`bs.) Pa puatd à `j `spbkid p pder derµµĲb s`r r`pr`sà r`pr`sàtbed tbed kdhd +0 +0+ + ?i 5 l , siàed `st` aµ u uh`rd h`rd ua ‚aµ u uh`rd h`rd gip`rkdhpj`ld‘. Bua kubaed ad hµbbss e` tr`s `l`s puèà s`r visubjizbeds à `j huaed qu` kdadk`hds, pu`std qu` `xistà sdjbhàt` tr`s eihàsida`s à au`strb fdrhb dreiabrib e` pàsbr, jds hbt`hµbbtikds tikds puèà ie`br hµbbss eihàsida`s. @s hµbs, bs, puèà trboblbr kda sist`hbs e` `l`s edae` `j aµuh`rd uh`r d `xbk `xbktd td ad `stµ` `sp`kiflkbed. E`spuµ``ss ads gbojba e` ‚`spbkid eihàsidabj a‘, fllbaed a pbrb ‚kubjqui`r ‚kubjq ui`r aµu uh`rd.‘ h`rd.‘

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KbpµĲtujd 23 S`rp`tuiebe

@j ‚ad aµ uh`rd‘ uh`rd‘ Udeb p`rsdab qu` s` eèikb b pàsbr sdor` jds aµu uh`rds h`rds jj`mb b jb kdakjusiµdda a e` qu` `xist` uab mrba kbatiebe e` `jjds y s` siàt` iakbpbz e` `xpr`sbr `xbktbhàt`` kuµbbatds. `xbktbhàt atds. @ @a ap pd`sµ d`sµĲĲbb pu puè` è` ua uadd usb usbrr bjm bjmµ u ua µa sµĲĲhij< hij < ‚tbatb ‚ tbatbss kd kdhd hd jjbs bs bràbs e`j hbr‘> ‚tba auh`rdsbs kdhd jbs `str`jjbs qu` orijjba y pbrpbe`ba à jb VµĲb Jµbbkt`b‘. kt`b‘. S`rd pbrb `j hbt`hµbbtikd, tikd, jds sµĲhij`s kbr`kà e` utijiebe. Sbrb µ``j, j, sdjbhàt` `xist` jb k`rt`zb e` qu` jds àt`rds s` fdrhba `hp`zbaed kda `j uad, `j kubj suhbed b dtrd uad fdrhb dtrd aµu uh`rd, h`rd, qu` b su v`z, si s` j` bmr`mb dtrd uad fdrhb tbhoiµ`à a dtrd aµu uh`rd, h`rd, bs bsµµĲ su suk`sivbhàt`. k`sivbhàt`. Kd Kdhd hd jjbs bs rr`mjbs `mjbs hbt`hµbbtikbs tikbs ad fllba aiamµ u ua a jµĲhit` b jb j b suhb s uhb (eds aµuh`rds uh`rds kubj`squi`rb puèà s`r Be`hµ suhbeds) adihpdrtbr puè` tà`r `st`s`b prdkèihiàtd. bbs, s, sia kuµbba aflajbrmd `j adhor` qu` s` bsima` b ua aµu uh`rd, h`rd, ai kuµbba a `j`v `j`vbed bed s`b µ``st` st` ‛buaqu` s` bjbrmu bjbrmu`` à uab jia`b iat iat`rhi `rhiaboj` aboj` e` p`qu`˚ a abs bs kifrbs e`se` bqu bquµµĲ gbstb jb `str`jjb hµbbss r`hdtb‛ si`hpr` `s pdsioj` e`kir ‚`s` aµ u uh`rd h`rd hµbbss uad‘ y dotà`r uab kifrb bµu ua a hµbbss `j`vbeb. Jb s`ri` e` àt`rds, si s` `skrioà à dreà, 2 , ?, ;... `s ‚iaflaitb‘, pbjborb qu` s` e`r e`riv ivbb e`j jbt jbtµµĲa y sim simaifl aiflkb kb ‚qu ‚qu`` ad ti` tià` a` ai pu` puè` e` tà tà`r `r fla. fla.‘‘ @a kdas`kuàkib, kubaed pde`hds `skrioir jb s`ri` e` aµ uh`rds uh`rds kdhd 2, ?, ;... jd qu` qu`r`hds hbaif`s hbaif`stbr tbr `s ‚2, ?, ; y bs bsµµĲ suk`sivbhàt` siaf siafµµĲa‘. @a imubj fdrhb, si kdasie`rbhds jds aµ u uh`rds h`rds a`mbti a`mbtivds vds,, 2, ?, ;, y bs bsµµĲ suk`siv suk`sivbhàt`, bhàt`, pde`hds v`r qu` tbhoi tbhoiµµ`à a sda iaflaitds y puèà `skrioirs` kdhd 2, ?, ;... [`h`lbat`hàt`, jbs s`ri`s e` ihbmiabrids pdsiti-

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2?;

2?5

KBS µ IUPJD 23. S@_S@UPIE S@_S@UPIEBE BE

vds puèà `skrioirs` kdhd +2i, +?i, +;i... y jds ihbmiabrids a`mbtivds kdhd 2i, ?i, ;i... Kdasie`r`hds bgdrb dtrbs s`ri`s e` àt`rds. Sàs`hds à jds aµu uh`rds h`rds pbr`s< ?, 5, 6, 7 y bs bsµµĲ suk`siv suk`sivbhàt`. bhàt`. ´Kuµbbatds atds aµ u uh`rds h`rds pbr`s `xistà4 Pab fdrhb e` eiskutir `st` bsuatd s`r s`rµµĲb e`kir< Ouàd, jds àt`rds puèà eivieirs` bjt`rabtivbhàt` à aµ u uh`rds h`rds ihpbr`s y à aµu uh`rds h`rds pbr`s, pdr jd tbatd, à jds prih`rds ei`z aµ uh`rds uh`rds `xistà kiakd ihpbr`s y kiakd pbr`s> à jds prih`rds kià aµu uh`rds, h`rds, kiaku kiakuàt àtbb ihpbr` ihpbr`ss y kiak kiakuà uàtb tb pbr`s pbr`s,, y bsµ bsµĲ suk`sivbhàt`. @stb kuàtb s` s`muir muirµµĲb ssia ia ih ihpdrt pdrtbr br kuµbbatds atds àt`rds s` tdhbrba. Kdas`kuàt`hàt`, `j tdtbj e` aµ u uh`rds h`rds pbr`s `s imubj b jb hitbe e`j tdtbj e` tdeds jds aµ u uh`rds h`rds àt`rds. S`rd ad `s bsµĲ. Ĳ. @j aµu uh`rd h`rd `s iaflai iaflaitd td y ad puè` uad gbojbr e` ‚uab hitbe e`j iaflaitd.‘ @a jumbr e` `sd, kdasie`r`hds jds pbr`s à `stb fdrhb< Ad `xist` ‚`j aµu uh`h`rd pbr hµbbss mrbae`‘, kdhd ad `xist` `j ‚aµ u uh`rd h`rd hµbbss mrbae`‘. Su`s buaqu` s` `skriob ua aµu uh`rd h`rd pbr kda kifrbs huy p p`quàbs `quàbs y jj`muà j j`muà µ``stbs stbs e`se` bqu bquµµĲ gbstb `j bstrd hµbbss r`hdtd, si`hpr` gborµb jb pdsioijiebe e` e`kir ‚`s` a aµ u uh`rd µh`rd hµbs bs eds‘. E` bqu bquµµĲ qu` jbs s`ri`s e` aµuh`rds uh`rds pbr`s eòba `skrioirs`< ?, 5, 6... @a imubj fdrhb, jds aµ u uh`rds h`rds ihpbr`s sda 2, ;, 0...> jbs s`ri`s e` aµu uh`rds, h`rds, kdatbaed e` kiakd à kiakd sda< 0, 23, 20...> y jbs s`ri`s e` aµu uh`rds h`rds kdatbaed pdr hijjda`s sda 2333333, ?333333, ;333333... Udebs `stbs s`ri`s ad tiàà fla y `s `sd jd qu` iaflaitd (d aµuh`rd uh`rd iaflaitd) simaiflkb.

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Kdatbaed sia kdatbr S`rd `std ad puè` sbtisfbk`r. [`murbhàt`, s` puè` pàsbr qu` bua kubaed jbs s`ri`s e` aµu uh`rds h`rds pbr`s sda iaflaitbs y jd hishd jbs s`ri`s e` tdeds jds àt`rds, suosist` `j g`kgd e` qu` sdjbhàt` `xist` jb hitbe e` aµuh`rds pbr`s qu` e` aµ u uh`rds h`rds àt`rds y qu` u uaikbhàt` µaikbh` at` `xis `xist` t` jb hijjdaµ``sihb sihb pbrt` e` aµ uh`rds uh`rds qu` r`pr`sàtà pbr`s e` hijjda`s. ¡@s jµddmikd! mikd! djd pdrqu` s`b jµ (Auakb kdaf kdafµµĲ` à ua rbzdabh rbzdabhiàtd iàtd sµ ddmikd. mikd. @s jµddmikd mikd qu` si ua gdhor` `stµb hirba hirbaed ed gbkib `j adrt` adrt`,, su `spbje `spbjebb `st `stµµ` gbkib `j sur. [ia `hobrmd, si `stµb pbrbed à `j Sdjd [ur y `stµb hirbaed gbkib `j adrt`, su `spbjeb `stµb tbhoiµ`à a gbkib `j adrt`.) U`rhi `rhia`hds, a`hds, à àtdak` tdak`s, s, `j bsua bsuatd, td, àkda àkdatrbae trbaedd kuµ batds batds aµ u uh`rds h`rds pbr` pbr`ss `xistà à kdhpbrbkiµda da kda tdeds jds àt`rds. ´Kµddhd, hd, si jb kbatiebe `s iaflaitb4 Oià, kdatbr`hds.

2?0 V`bhds, prih`rd, jd qu` qu`r`hds e`kir pdr kdatbr. @a `j simaiflkbed kdhµ u ua a e` jb pbjborb, kdatbhds jds dol`tds bsimaµbbaedj` aedj` b kbeb uad ua aµu uh`rd h`rd suk`sivd. @st` dol`td `s `j aµu uh`rd h`rd uad, bquµ`j `j `s `j aµu uh`rd h`rd eds, `j dtrd `j aµu uh`rd h`rd tr`s, y bs bsµµĲ suk`siv suk`sivbhàt`. bhàt`. Kubaed t`rhiabhds, si bj u ujtihd µ jtihd dol`td s` j` bsimaµd `j aµu uh`rd h`rd ei`z, `xistà ei`z dol`tds. S`rd, ´pde`hds kdatbr sia aµu uh`rds4 h`rds4 @stbhds tba bkdstuhorbeds b jds aµu uh`rds h`rds pbrb tbj do l`td qu` suàb kdhd si `stuviµ``rbhds rbhds pr`muatbaed< ´pde`hds kdatbr sia kdatbr4 Y, sia `hobrmd, jd pde`hds gbk`r. [updambhds qu` tià` `j j`ktdr uab kbatiebe e` kbrbh`jds (imadrb kuµbbatds), atds), y uab ruiedsb hukgèuhor` e` kgiquijjds qu` bag`jba s` jds r`pbrtb (ad sbo` kuµbatds). batds). Eistriouy` jds kbrbh`jds ebaed uad b kbeb kgiquijjd y kubaed gb t`rhiabed, tdeds jds kgiquijjds tiàà kbrbh`jds, p`rd bj j`ktdr j` sdorba bjmuads. @atdak`s, `vieàt`hà `vieàt`hàt`, t`, y bua sia kdatbr à jb fdrhb usubj, µ``jj sb sbo` o` qu` gbo gboµĲb hµbs bs kbrbh`jds qu` kgiquijjds. [i, pdr dtrd jbed, s` j` bkbobrba jds kbrbh`jdss y tdebvµĲĲbb guoi kbrbh`jd guoi`rb `rb kgiq kgiquijjds uijjds `sp`rb `sp`rbaed aed b qu` j`s e ei`rb, i`rb, sborµĲĲbb q qu` u` gb gboµ oµĲb Ĳb hµbs bs kgiquijjds qu` kbrbh`jds. S`rd, si bj t`rhiabr jb eistrioukiµdda, a, kbeb kgiquijjd tià` ua kbrbh`jd sia fbjtbrj` b aiamuad y bj j`ktdr ad j` sdorb aiamuad, µ``st` st` sborµbb,, fu`rb e` tdeb eueb y bua sia gbo`r kdatbed à jb fdrhb usubj, qu` `j aµu uh`rd h`rd e` kgiquijjds `rb imubj bj aµ u uh`rd h`rd e` kbrbh kbrbh`jds. `jds. @atdak`s, `stb fdrhb e` kdatbr, qu` kdasist` à bjia`br s`pbrbebhàt` eds s`ri`s (uab e` kgiquijjds y uab e` kbrbh`jds> d uab e` tdeds jds àt`rds y uab e` jds aµ u uh`rds h`rds pbr`s), iaeikbrµb bj j`ktdr si jbs eds s`ri`s sda imubj`s d e`simubj`s y, si sda e`simubj`s, kuµbbjj `s hbydr. Sdambhds Sdam bhds p pdr dr kbsd qu` fdrhb fdrhbhds hds eds jµĲa`bs kda jbs s`ri` s`ri`ss e` tdeds jds àt`rds y tdeds jds aµu uh`rds h`rds pbr`s, kdhd s` iaeikb b kdatiaubkiµdda< a< 2 ? ; 5

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Kdhd puè` v`rs`, `xist` ua aµu uh`rd h`rd pbr pdr kbeb àt`rd ihbmiaboj`, pueiµ`àeds` aeds` dotà`r `j aµuh`rd uh`rd pbr, sihpj`hàt`, eupjikbaed `j àt`rd. [ia ihpdrtbrr kuµ ihpdrtb bba a j`lds s` jj`mu` à `st` bsuatd, aiamµu ua a àt`rd a`k`sitb dhitirs` y ad fbjtbrµb aiamµ u ua a aµ u uh`rd h`rd pbr. (Kbeb kgiquijjd, à dtrbs pbjborbs, gb sied sbtisf`kgd kda ua kbrbh`jd). @a aiamµ u ua a kbsd, pdr hukgd qu` jb jistb s` prdjdamu`, s` àkdatrbrµb ua àt`rd pbrb `j kubj ad s` puèb `skrioir ua aµuh`rd uh`rd pbr, y kbeb àt`rd (kubjqui`rb) tià` su aµu uh`rd h`rd pbr eif`ràt`.

2?6

KBS µ IUPJD 23. S@_S@UPIE S@_S@UPIEBE BE

´[imaiflkb `std qu` `xistà à kdaluatd tbatds aµ u uh`rds h`rds pbr`s kdhd àt`rds4 @a r`bji r`bjiebe, ebe, jb `xpr` `xpr`siµ siµ da da ‚tbatds kdhd‘ ad tià`, à r`bjiebe, `j sàtied mà`rbj qu` j` bpjikbhds eibribhàt` kubaed `stbhds gbojbaed e` kdsbs qu` sda iaflaitbs. @s hµbs bs prdpid e`kir, à su jumbr, qu` jbs s`ri`s e` aµ u uh`rds h`rds pbr`s `stµba ba ‚kdrr`spdaeiàed pbrt` b pbrt`‘ kda jbs s`ri`s e` tdeds jds àt`rds> simaiflkbaed qu` jds aµ u uh`rds h`rds pbr`s puèà bjia`brs` sist`hµbbtikbhàt` tikbhàt` kda jds àt`rds, e` hba`rb qu` `xistb uab jµĲĲa`b a`b e` jb prih`rb s`ri` p pdr dr kbeb uab e` jb s`muaeb, y vik`v`rsb. Ubhoiµ`à ap puè` uè` uad fdrhbr uab s`ri` e` a aµ u uh`rds µh`rds kda kdatbaed tbaed pdr hijjda hijjda`s, `s, y kdhpbrbrjb, à imubj fdrhb, kda jbs s`ri`s e` tdeds jds àt`rds. Sdr kbeb àt`rd `xist` ua aµu uh`rd h`rd imubj à hijjda`s, dotàied bj hujtipjikbr µ``st` st` pdr ua hijjµdda. a. Sbrb 2, `xist` 2333333, pbrb 6, 6333333, pbrb ?7=;, ?7=;333333, y bs bsµµĲ suk` suk`sivbhàt`. sivbhàt`. Sdr jd tbatd, à tdtbj gby ‚tbatds ‚tbatds‘‘ a aµ u uh`rds µh`rds kdatbaed pdr hijjµda, da, kdhd gby àt`rds. D, kubaed hàds, jbs eds s`ri`s `stµba ‚kdrr ‚kdrr`spda`spdaeiàed pbrt` b pbrt`‘. Udebs jbs s`ri`s e` aµu uh`rds h`rds qu` s àkuàtrba à jb kitbeb situbkiµdda a kkda da jjds ds àt`rds, eµĲkàs` s`r ‚‚e`sauh`rbo e`sauh`rboj`s j`s2 ‘. Y, `j kdaluatd e` àt`rds, ` at`rds, tbhoiµ`à a `s jjbhbe jjbhbedd ‚‚e`sauh`rboj e`sauh`rboj`‘. `‘.

@j iaflaitd à su hµ b bss sihpj` `xpr`siµ da da @j ‚iaflaitd‘, ‚iaflaitd‘, pdr supu` supu`std, std, eb b uad jb adkiµdda a e` iahàsiebe y `t`raiebe. Bj j`ktdr puè` pbr`k`rj` qu` ad tià` utijiebe. [ia `hobrmd, bua kubaed sµddjd jd `stuviµ``rbhds rbhds iat`r`sbeds à jds aµu uh`rds h`rds p`qu`˚ a ads, ds, s` e`lbr e`lbrµµĲba v`r jbs adkida`ss e`j ‚iaflai adkida` ‚iaflaitd‘. td‘. 2 . _`kdrebaed jb Sdr `l`hpjd, supdambhds tà`r qu` eivieir 2 àtr` 23 2 r`mjb sdor` jds aµu uh`rds h`rds iav`rsds, `s imubj b 2 23 y pdr jd tbatd 2 8 23. 23 2 2 @a fdrhb s`h`lbat`, 2 8 233 y 2 8 2333. 233 2333 @a r`bjiebe, kubatd hµbbss p`qu`˚ a add s`b `j eivisdr, hµbbss mrbae` s`rµb `j kdkiàt`. @a v`rebe, kubaed s` eivie` 2 (d kubjqui`r aµ u uh`rd) h`rd) àtr` uab s`ri` e` aµu uh`rds h`rds qu` vba gbkiµ`àeds` aeds` kbeb v`z hµbbss p`qu`˚ a ads, ds, su kdkiàt` s` vb gbkiàed kbeb v`z hbs mrbae`, y kdafdrh` `j eivisdr s` gbk` iaflaitbhàt` p`qu`˚a ad, d, jb r`spu r`spu`stb `stb s` gbk` iaflaitbh` iaflaitbhàt` at` mrbae`.

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A. e`j Urbaskriptdr< Ubhoiµ à à s` j`s puè` jjbhbr auh`rboj`s

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2?= [ia `hobrmd, puè` uad pr`muatbrs`< ´quµ` s` qui`r` e`kir pdr ‚iaflaitbhàt` p`qu`˚ a ad‘4 d‘4 Iaeueboj`hàt` jb p`qu`˚a a`z `z t`rhiab à k`rd. ¡Bg! p`rd jb p`qu`˚ a a`z `z puè` `xpr`sbrs` à fdrhb e` frbkkiµdda. a. BsµĲ, Ĳ, `s ua aµu uh`rd h`rd p`qu`˚ a ad, d, `s hµbbss p`qu`˚ a ad, d, `s hµbbss p`qu`˚ a add bµ u ua> a> ad `xist `xistiàed iàed jµĲhit` b jb p`qu`˚ a a`z `z kdafdrh` buhàt` uad `j aµu uh`rd h`rd e` k`rds b .... yb qu`, sia ihpdrtbr kuµbbatds atds buhàt`, auakb jj`mbrµb tdtbjhàt` b k`rd. Sde`hds e`kir, pdr jd tbatd< kubaed s` eivie` 2 (d kubjqui`r aµu uh`rd) h`rd) àtr` uab s`ri` e` aµu uh`rds h`rds qu` vba gbkiµ`àeds` aeds` kbeb v`z hµbbss mrbae`s, su kdkiàt` vb gbkiµ`àeds` aeds` kbeb v`z hµbbss p`qu`˚ a ad, d, hiàtrbs qu` si `j eivisdr s` gbk` iaflaitbhàt` iaflaitbhàt` mrbae`, `j kdkià kdkiàt` t` s` gbk` iaflaitbh` iaflaitbhàt` at` p`qu`˚ a ad. d. [ia `hobrmd, dos`rv`s` qu` auakb pde`hds eivieir àtr` 3. @stb dp`rbkiµda da `stµb `xkjuieb e` jbs hbt`hµ bbtikbs. tikbs. Y pdr uab huy ouàb rbzµdda a ´R ´Ruµ uµ` au uh`rd, µh`rd, pdr `l`hpjd, s`r s`rµµĲb `j kdkiàt` si trbtµbbrbhds rbhds e` eivieir, eimbhds 6 àtr` 34 6 uh`rd, yb qu` tded aµu uh`rd h`rd hujtipjikbed Uàerµ Uà erµĲbhd Ĳb hdss 84 Ad `xist` `s` aµuh`rd, pdr 3 `s imubj3 b 3. Sdr jd tbatd, ad pde`hds eivieir aiamµu ua a aµ u uh`rd h`rd àtr` 3. @j `spbkid `xistàt` àtr` eds aµ u uh`rds h`rds kubj`squi`rb, eimbhds àtr` 2 y ?, puè` eivieirs` à kubjqui`r aµu uh`rd h`rd e` frbkkida`s, pdr kdav`rsiµdda a bh hij ijjd jdaµ aµ``sisi hds,, d trij hds trijjdaµ jdaµ``sihds sih ds y kbe kbebb trij trijjdaµ jdaµ``sihd sih d à trij trijjdaµ jdaµ``sihds, sih ds, y bsµĲ suk` suk`sivbh` sivbhàt`. at`. @std tbhoiµ t bhoiµ`à a pu` puè` e` gbk` gbk`rs` rs` pbr pbrbb kubj kubjqui`r qui`r `spbkid hµbs bs p`qu`˚ a ad, d, kdhd `j qu` 2 2 `xist` àtr` y , d àtr` 3.333333 y 3.333332. 5 ? Y, b p`sbr e` tded, jds hbt`hµbbtikds tikds gba prdkurbed e`hdstrbr qu` tdebs jbs frbkkida`s ihbmiaboj`s (`s e`kir, tdeds jds aµu uh`rds h`rds rbkidabj`s) puèà kdjdkbrs` à fdrhb tbj, qu` uab kdrr`spdaeàkib e` pbrt` b pbrt` puèb fl lbrs` kda jbs s`ri`s e` jds àt`rds. [ia `xk`pkiµdda, a, pbrb tded àt`rd gborµb uab frbkkiµ dda, a, y vik`v`rsb y, pdr jd tbatd, tdebs jbs s`ri`s pdsioj`s e` frbkkida`s sda e`sauh`rboj`s. Hµbbss k`rkb, hµbbss k`rkb y hµbbss k`rkb. 2 2 2 2 2 2 Kdasie`r`hds uab s`ri` e` frbkkida`s kdhd µ``stb< stb< , , , , , , ? 5 7 26 ;? 65 2 2 2 , , , y bs bsµµĲ su suk`sivbhàt`, k`sivbhàt`, g gbstb bstb `j iiaflaitd. aflaitd. 2?7 ?06 02? Dosµ``rv`s` rv`s ` qu qu`` kb kbeb eb frbk frbkkiµ kiµdda a `s lustbhàt` jb hitbe e` jb frbkkiµdda a bat`ridr, pu`std qu` `j eàdhiabedr s` eupjikb à kbeb dkbsiµda. da. (E`s (E`spuµ puµ``ss e e`` td tded, ed, si ss`` tdhb kubjqui`rb e` jbs frbkkida`s y s` eivie` àtr` ?, `quivbj` b hujtipjikbrjb 2 2 2 2 pdr . BsµĲ 8 , eupjikµbbaeds` aeds` à `stb fdrhb `j eàdhiabedr). ? 2?7 ? ?06 Bua kubaed jbs frbkkida`s s` gbkà kdatiaubhàt` hµbs bs p`qu`˚ a abs, bs, jbs s`ri`s puèà kdasie`rbrs` kdhd iaflaitbs, pu`std qu`, sia ihpdrtbr jd p`qu`˚a abs bs

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2?7

KBS µ IUPJD 23. S@_S@UPIE S@_S@UPIEBE BE

qu` s` gbmba jbs frbkkida`s, si`hpr` `s pdsioj` hujtipjikbr `j eàdhiabedr pdr ? y dotà`r uab frbkkiµdda a hµbbss p`qu`˚ a abb bµ u ua a y simuiàt` à jb s`ri`. Be`hµbbs, s, jbs frbkkida`s auakb jj`mba b k`rd, pdrqu` `j eàdhiabedr puè` gbk`rs` kbeb v`z hµbbss mrbae`, gbstb `j iaflaitd, y sdjbhàt` si s` puei`rb bjkbazbr ua fla (kdsb qu` ad puè` dkurrir) p pder derµµĲb jj`mbr b k`rd jb frbkkiµdda. a. @j bsuatd `s< ´kuµbbjj `s jb suhb e` tdebs `sbs frbkk frbkkida`s4 ida`s4 Sder SderµµĲb pbr`k pbr`k`r `r qu` jb suhb e` uab s`ri` iaflai iaflaitb tb e` a aµ u uh`rds µh`rds eòi`rb s`r iaflaitbhàt` mrbae` (‚`s jµddmikd‘) mikd‘) p`rd, e` kubjqui`r hba`rb, `hp`k`hds b suhbr. 2 2 ; 2 = 2 Srih`rd hµbbss sda . Bmr`mu`hds y tàer`hds bmr`mbaed jb ? 5 5 7 7 26 20 2 ;2 suhb s`rµb e` b jb kubj, si j` bmr`mbhds ads ebrµb , y bsµĲ suk suk`sivb`sivb26 ;? ;? ;2 2 6; hàt`, e` b jb kubj, si j` bmr`mbhds ads ebrµb , y bsµ bsµĲ suk suk`sivbh `sivbhàt`. àt`. ;? 65 65 Dosµ``rv`s` rv`s` qu`, uab v`z qu` gba sied suhbebs jbs eds prih`rbs frbkkida`s, b jb kbatiebe e` ; sdjbhàt` j` fbjtb pbrb 2 jj`mbr b 2. Bmr`mbaed jb t`rk`rb, 5 5 2 àkdatrbhds qu` sdjbhàt` fbjtb pbrb jj`mbr b 2. Jb simuiàt` frbkkiµdda a eb 7 2 2 2 uab kbatiebe b jb qu` sdjbhàt` j` fbjtb pbrb p brb jj`mbr b 2> e`spuµ`s `s , , 26 ;? 65 y bsµ bsµĲ suk suk`sivbh `sivbh`t`. `t`. @a dtrbs pbjborbs, kdafdrh` vb uad suhbaed hµbbss frbkkida`s e` `sb s`ri`, hµbbss s` bk`rkb b 2< tba k`rkb kdhd jd e`s``, gbstb ua hijjdaµ``sihd sihd e` uad, ua trijjdaµ``sihd sihd e` uad, ua trijjdaµ``sihd sihd e` trijdaµ``sihd sihd e` uad. [` bk`rkb kbeb v`z hµbbs, s, p`rd auakb jj`mb b 2. Jds hbt`hµbbtikds tikds `xpr`sba `std eikiàed qu` jb suhb e` jb s`ri` e` frbk 2 2 2 kida`s iaflaitbs ? , 5 , 7 , ... ‚s` bprdxihb b 2 kdhd jµĲhit`.‘ @st` `s ua `l`hpjd e` uab ‚s`ri` kdav`rmàt`‘, `s e`kir, uab s`ri` kda ua aµu uh`rd h`rd iaflaitd e` tµ``rhiads, rhiads, p p`rd `rd kuyb suhb tdtbj s` bprdxihb b ua a aµ u uh`rd µh`rd dreiabrid (ua aµ u uh`rd h`rd ‚flaitd ‚flaitd‘) ‘) kdhd jµĲhit`. Jds mri`m mri`mds ds e`sk e`skuori` uori`rda rda `sbs s`ri`s kda kdav` v`rmà rmàt`s, t`s, p p`rd `rd quèbrda tba ihpr`sidabeds pr`si dabeds kda jd iaflait iaflaitdd e` jds tµ ``rhiads rhiads e` jbs hishbs qu` ad s` ei`r ei`rda da kuàtb kuà tb e` qu` jb suhb pder pderµµĲb ad s`r iaflait iaflaitb. b. @a kdas` kdas`kuàk kuàkib, ib, ua mri`m mri`mdd jjbhbed \àµdda a `stboj`kiµd ua aµ u uh`rd h`rd e` prdoj`hbs jjbhbeds ‚pbrbedlbs‘ jds kubj`s pbr`k pbr`kµµĲba r`futbr kdsbs `vieàt` `vieàt`hàt` hàt` ki`rtbs. \àµda da ‚r`futbob‘, pdr `l`hpjd, qu` fu`rb pdsioj` `j hdvihiàtd. @stbs pbrbedlbs fu`rda fbhdsbs eurbat` hij`s e` b˚ ads, ads, p`rd tdebs s` e`svba`ki`rda tba prdatd kdhd s` kdaflrhµd jb v`rebe sdor` jbs s`ri`s kdav`rmàt`s.

2?9 Jb pbrbedlb hbs fbhdsb e` \àµdda a `s jb kdadkieb kdhd ‚Bquij`s y jb tdrtumb‘. tdrtumb ‘. Bq Bquij`s uij`s fu` ua gµ``rd` rd` gdhµ``rikd rikd ffbhdsd bhdsd pdr su rb rbpie`z, pie`z, y jb tdrtumb `s ua baihbj fbhdsd pdr su jàtitue. [ia `hobrmd, \àµdda a s` `hp`˚ a add à e`hdstrbr qu` à uab kbrr`rb à jb kubj s` j` ei`rb vàtblb b jb tdrtumb, Bquij`s auakb pd pder erµµĲb bjkbazb bjkbazbrjb. rjb. [updambhds, pdr `l`hpjd, qu` Bquij`s puè` kdrr`r ei`z v`k`s hµbbss v`jdzhàt` hà t` qu` jb tdrt tdrtumb umb y qu` b µ``stb stb s` j` kdak kdakè` è` uab v` vàtblb atblb e` kià h`trds bj iaikibrs` jb kbrr`rb. Bquij`s r`kdorb `sds kià h`trds e` vàtblb à uabs kubatbs zbakbebs, p`rd, kubaed jd gb jdmrbed, jb tdrtumb, kdrriàed b ua eµ`kihd `kihd e` jb v` v`jdkiebe jdkiebe e` Bquij Bquij`s `s (e`hb (e`hbsibed sibed bpris bprisbb pbrb uab tdrtu tdrtumb) mb) gb be`jbatbed ei`z h`trds. @a s`muieb, Bquij`s kuor` `sds ei`z h`trds> p`rd, eurbat` `s` ti`hpd, jb tdrtumb gb bvbazbed ua h`trd hµbbs. s. Bquij`s kuor` `s` h`trd, p`rd p `rd jb tdrtumb gb r`kdrried ei`z kàt kàtµµĲh`trds hµbbs. s. Bquij`s ... S`rd v`b `j j`ktdr kdhd `s `std, Bquij`s kdatiaub bvbazbaed, p`rd tbhoiµ`à a jd gbk` jb tdrtumb, y Bquij`s auakb s` `hpbr`lb. Hµbs b s bµ u ua, a, kdhd `j j`ktdr pderµ pd erµĲĲbb brm brm¨ u uir ¨ ir à jb hishb fdrhb sia ihpdrtbr kuµbba a p`qu`˚ a abb s`b jb vàtblb qu` tàmb jb tdrtu tdrtumb mb à jb sbjieb ‛ei`z d trìa trìatb tb kà kàttµĲh`t Ĳh`trds‛ rds‛ à obs` b jb pbrbedlb pbrbed lb e` \àµdda, a, Bq Bquij`s uij`s auakb p pderµ derµĲĲbb r` r`kup`rb kup`rbrr `sb vàtblb, pdr p`qu` p`qu`˚ ab a ˚b qu` fu`rb. Y `std simaiflkb qu` `j hdvihiàtd `s ihpdsioj`. pd deµĲb  bjkbazbr b jb tdrtumb y qu` Sdr supu`std, `j j`ktdr sbo` qu` Bquij`s p `j hdvihiàtd sµĲ  `s `s pdsioj`. Sdr jd tbatd, jb e`hdstrbkiµdda a e` \àµdda a `s uab pbrbedlb. Bgdrb oià, ´quµ` tià` e` `rrµda`b da`b jb e`hdstrbkiµdda a e` \àµdda4 a4 V`bhds. [updambhds [updambh ds qu` Bquij Bquij`s `s pde pdeµµĲb r`kd r`kdrr`r rr`r ei`z h`trd h`trdss pdr s`mua s`muaed. ed. @a `s` kbsd, Bquij`s r`kdrr` jb vàtblb e` kià h`trds ebeb b jb tdrtumb à ei`z s`muaeds, eurbat` kuyd ti`hpd jb tdrtumb r`kdrr` ei`z h`trds. Bquij`s r`kdrr` jds ei`z h`trds à 2 s`muaed, eurbat` kuyd ti`hpd jb tdrtumb r`kdrr` ua h`trd. Bquij`s b su v`z r`kdrr` `s` h`trd à 3.2 e` s`muaed, eurbat` kuyd ti`hpd jb tdrtumb r`kdrr` ua eµ``kihd kihd e` h`trd. @a dtrbs pbjborbs, `j ti`hpd r`qu`ried pdr Bquij`s pbrb kuorir kbeb uad e` jds bvbak`s suk`sivds e` jb tdrtumb, fdrhb uab s`ri` qu` tiàe` b s`r kdhd µ`stb< `stb< 23, 2, 3.2, 3.32, 3.332 3.332,, 3.3332, 3 .3332, 3.333 3.33332, 32, y bsµĲ suk`sivbhàt`. s uk`sivbhàt`. ´Kuµbbatd atd ti`hpd j` tdhb b Bquij`s r`kup`rbr tdebs jbs vàtblbs4 Kdhd gby ua aµu uh`rd h`rd iaflaitd e` tµ``rhiads rhiads à `stb s`ri` e` \àµdda, a, `j hishd supusd qu` jb suhb tdtbj `rb iaflaitb. Ad kdhpràeiµd qu` bjmuab bjmuabss s`ri`s e` tµ``rhiads rhiads e` aµ u uh`rds h`rds iaflait iaflaitds ds ‚kdav`rm ‚kdav`rmµµĲba‘ y tà tàµµĲba uab suhb flaitb. Sdr `l`hpjd, `l`hpjd, jb suhb e` jds eds prih` prih`rds rds tµ``rhiads rhiads à jb s`ri` e` \àµ da da `s 22> jb suhb e` jds prih`rds tr`s `s 22.2< e` jds prih`rds kubtrd 22.22> e`

2;3

KBS µ IUPJD 23. S@_S@UPIE S@_S@UPIEBE BE

jds prih`rds kiakd 22.222, y bs bsµµĲ suk`siv suk`sivbhàt`. bhàt`. Kdhd puè` dos`rvbrs`, si s` suhba tdebs td ebs jbs s`ri`s iaflaitbs e` tµ``rhiads, rhiads, s` dotià` ua e`kihbj iaflaitd, kdhd jd `s jb kbatiebe e` 22.2222... y bsµĲ suk`siv suk`sivbhàt`, bhàt`, gbstb `j iaflaitd. S`rd si s` dotià` `j e`kihbj `quivbjàt` e`j aµ u uh`rd h`rd 22, s` àkuàtrb qu` `s tbhoiµ `à a `j iaflaitbhàt` r`p`tied e`kihbj 22.222222... y bsµ bsµĲ iae`flaieb2 e ` s`m s`muaed uaedss y µ``s` s` hàt`. @s, pdr jd tbatd, jb suhb e` jbs s`ri`s \àµda da 22 9 e` `s `j ti`hpd à `j kubj Bquij`s bjkbazbrµb y pbsbrµb b jb tdrtumb, bua kubaed tàmb qu` qu ` kdhp kdhp`tir `tir b tr trbvµ bvµ``ss e` uab iaflaitb s`ri` e` kdatiaubs y hµbbss p`qu`˚ a abs bs b  b vàtblbs qu` kdas`rvb jb tdrtumb. B p`sbr e` tded, bjkbazbrµ b jb tdrtumb> `j hdvihiàtd `s  pdsioj` y tdeds pde`hds e`skbasbr. Kda jbs s`ri`s s` puèà bprdxihbr e`kihbj`s iaflaitds, qu` ad s`ba frbkkida`s e`kihbj`s p`riµdeikbs, deikbs, kdhd ua uabb suhb jµĲhit`. @s @sbs bs s`ri s`ri`s `s puèà gbk`rs` pbrb `xpr`sbr ua aµuh`rd uh`rd irrbkidabj, y pdr hèid e` ir suhbaed kbeb v`z hµbs bs tµ``rhiads rhiads e` tbj`s s`ri`s, pd pder erµµĲbhds bk`rkbrad bk`rkbradss kbeb v`z hµbbss bj vbjdr e`j aµu uh`rd h`rd irrbkidabj, bua kubaed, p pdr dr supu`std, auakb pd pder erµµĲbhds jj`mbr `xbktbhàt` bj hishd. @sbs s`ri`s kdav`rmàt`s sda utijizbebs pbrb e`t`rhiabr aµuh`rds uh`rds irrbkidabj`s kdhd, pdr `l`hpjd, jds jdmbrithds.

Hµ bs bs qu` qu ` iafla ia flait itd d S`rd, ´sda iaflaitbs tdebs jbs kdsbs ` iaflaitbs à jb hishb fdrhb4 ´Suè` uad ie`br uab s`ri` iaflaitb e` kubjqui`r µĲaedj`, qu` ad s`b e`sauh`rboj` kda jb s`ri` e` àt`rds4 @a r`bjiebe, sµĲ. Kdasie`r`hds jb jµĲa`b qu` g` vàied `xpdaiàed à eiv`rsbs s`kkida`s e` `st` jiord< jb e` jds aµu uh`rds h`rds badtbeds b `spbkids imubj`s. [updambhds qu` `j `spbkid àtr` kbeb aµu uh`rd h`rd fu`rb dkupbed kda tdebs jbs frbkkida`s ihbmiaboj`s. Udeds jds `spbkids àtr` àt`rds bpbr`k`r bpbr`k`rµµĲba tdtbjhàt` jjà jjàds ds kda t`rk t`rkids ids,, sµ``ptihds pti hds,, hij hijjdaµ jdaµ``sihds sih ds y ttrijj rijjdaµ daµ`sihds `si hds.. BsµµĲ y tded Bs tded,, gbor gborµµĲb puatds à jb jµĲa`b qu` ad tàer tàerµµĲba frbkkida frbkkida`s `s r`pr`sàtbtivbs, bua kubaed guoi`rb ua aµ u uh`rd h`rd iaflaitd e` frbkkida`s iakjuiebs. _`kuµ``reàs` re` as` jds aµu uh`rds h`rds irrbkidabj`s. Sdr `l`hpjd, jb ? ad ti` tià` a` ua pua puatd td qu` r`p r`pr`s r`sà àt` t` à jb ‚jµ ‚jµĲa` Ĳa`bb e` frbkkida`s‘, pdrqu` ad puè` `xpr`sbrs` kdhd frbkkiµda. da. [ia `hobrmd, `xist` à jb jµĲa`b. Ihbm IhbmµµĲa`s` ua ku kuberbed, berbed, kdhd `j qu` flmurb à jb ijustrb ijustrbkiµ kiµdda, a, qu` `xtàei`rb jbt`rbjhàt` e` ua àt`rd b dtrd. Jb eibmdabj e`j kuberbed `s ? y si `sb eibmdabj fu`rb hèieb à jb jµĲa`b e`se` `j puatd k`rd, t`rhiab t`rhiabrrµĲb à ua puatd kdrr`sp kdrr`spdaeiàt` daeiàt` b ? y, b

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2;2 p`sb r e` tded p`sbr tded,, ad gbor gborµµĲb u uab ab frb frbkkiµ kkiµdda a à `s` pu puatd. atd. Ad pd pder erµµĲb gb gbo`rjb o`rjb.. Uded dtrd aµ u uh`rd h`rd irrbkida irrbkidabj bj p pderµ derµĲĲbb ss`r `r bad badtbed tbed à jb jµĲa`b, à ua puatd e edae` dae` ad puei`rb puei` rb `xist `xistir ir uab frbk frbkkiµ kiµ dda. a. (2.2)

Fimurbb 23.2 Fimur Uded `std `quivbj` b jd simuiàt`< si jds puatds qu` r`pr`sàtba tdeds jds aµu uh`rds h`rds rbkida rbkidabj`s bj`s sda badtbe badtbeds ds à uab jµĲa`b, tdebv tdebvµµĲb `xistirµbba a puatds e`lbeds fu`rb< à r`bjiebe, ua aµu uh`rd h`rd iaflaitd e` puatds e`lbeds fu`rb. Jd qu` `s hµbbs, s, ad `xistirµbba a eds puatds beybkàt`s qu` r`pr`sàtà aµ u uh`rds h`rds rbkidabj`s. Jds hbt`hµbbtikds tikds gba e`hdstrbed qu`, àtr` eds aµu uh`rds h`rds rbkidabj`s kubj`squi`rb, ad ihpdrtb jd bprdxihbeds qu` `stµà à à vbjdr, si`hpr` eò` `xistir, kubaed hàds, ua aµu uh`rd h`rd irrbkidabj, y vik`v`rsb, àtr` eds aµu uh`rds h`rds irrbkidabj`s kubj`squi`rb, sia ihpdrtbr kuµbba a bprdxihbeds `stµ`à a à vbjdr, si`hpr` eò` `xistir, kubaed hàds, ua aµu uh`rd h`rd rbkidabj. [i, b p`sbr e` tded, `stµbba a badtbeds tdeds jds aµu uh`rds h`rds rbkidabj`s `   irrbkidabj`s kdhd puatds à jb jµĲa`b, `sd qui`r` e`kir qu` gba sied usbeds td tdeds eds jds puatds. Jb s`ri` e` aµu uh`rds h`rds r`bj`s, pdr jd tbatd, (qu` iakjuy` bhods, aµuh`rds uh`rds rbkidabj`s irrbkidabj`s) fdrhb ua ‚kdatiaud‘. S`rd, ´quµ` e`kir r`sp r`sp`ktd `ktd b tdeds jds aµu uh`rds h`rds r`bj`s4 ´[da e`sauh`rboj`s kda jbs s`ri`s e` àt`rds, kdhd jd sda tdeds jds aµ u uh`rds h`rds rbkidabj`s4 @a

2;?

KBS µ IUPJD 23. S@_S@UPIE S@_S@UPIEBE BE

r`bjiebe, ad jd sda. Gb quèbed e`hdstrbed qu`, sia ihpdrtbr jb fdrhb à qu` trbt` uad e` brr`mjbr jds aµu uh`rds h`rds r`bj`s pbrb qu` puèba imubjbrs` uad pdr uad kda jds àt`rds, si`hpr` eòà quèbr fu`rb ua aµu uh`rd h`rd iaflaitd e` aµu uh`rds h`rds r`bj`s. Jb p`rp`tuiebe e` jds aµu uh`rds h`rds r`bj` r`bj`ss `s simaifl simaiflkbeb kbeb pdr `j sµĲhodjd K (pdr ‚kdatiaud‘). K `s uab `sp`ki` e` p`rp`tuiebe hµbs bs bvbazbeb qu` jbs p`rp`tuiebe`s e`sauh`rboj`s, yb qu` ai bua jbs s`ri`s iaflaitbs e` àt`rds sda suflkiàt`s pbrb kdatbr jbs s`ri`s iaflaitbs e` aµuh`rds uh`rds r`bj`s. Suèà jj`vbrs` b kbod v`riflkbkida`s kda dtrbs kjbs`s e` s`ri`s iaflaitbs, à ouskb e` e`sauh`rboijiebe, kda jbs s`ri`s e` aµ u uh`rds h`rds r`bj`s. Sdr `l`hpjd, jbs s`ri`s e` tdeds jds aµ u uh`rds h`rds kdhpj`lds (qu` r`pr`sà r`pr`sàtbr tbrµµĲba tdeds jds puatds à ua pjbad, à v`z e` sdjbhàt` uab jµĲĲa`b) a`b) d jbs s`ri`s e` tdeds jds aµu uh`rds h`rds gip`rkdhpj`lds (qu` r`pr`sàtbr r`pr`sàtbrµµĲba tdeds jds puatds à ua sµddjied, jied, `s e`kir à `j uaiv`rsd àt`rd, kdasie`rbaed `j uaiv`rsd kdhd iaflaitd à tdebs eir`kkida`s pbrb `st` dol`td) sda e`sauh`rboj`s kda jbs s`ri`s e` aµu uh`rds h`rds r`bj`s.

S`rp`tuiebe iaflaitb @a 2796, `j hbt`hµbbtikd tikd M`drm Kbatdr e`sbrrdjjµd u uab ab t`drµĲb Ĳb e` ‚aµ ‚au uh`rds µh`rds trbasflaitds‘, e` bku`red kda jb kubj `xist` ua aµu uh`rd h`rd iaflaitd e` eif`ràt`s kjbs`ss e` p`rp`tui kjbs` p`rp`tuiebe`s. ebe`s. Kbatdr r`pr`sàtµd `stbs p`rp`tuiebe`s pdr hèid e` jb j`trb gòr`b ‚bj`pg‘ ( ), y jbs eif`rà eif`ràt`s t`s p`rp`tuiebe` p`rp`tuiebe`ss fu`r fu`rda da hbrkbebs p pdr dr ‚suoskrit ‚suoskritds‘ ds‘ (aµu uh`h`rds p`qu`˚ a ads ds `skritds bob bobld ld e` jb jµĲa`b) à jb fdrhb simuiàt`<

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Jb prih`rb à jb s`ri` `s jjbhbeb ‚bj`pg aujb‘, y kdrr`spdae` b jb p`rp`tuiebe e` jbs s`ri`s e` àt`rds. @std simaiflkb qu` jb iahàsb p`rp`tuiebe kda jb kubj iai iaikiµ kiµ` `s `st` t` kkbp bpµµĲtuj Ĳtujd, d, pu` puè` e` ss`r `r jjbb h hµµbbss p`qu`˚ a abb p`rp`tuiebe qu` tbj v`z puèb `xistir. Abei`, à dtrbs pbjborbs, gb e`skuoi`rtd uab s`ri` iaflaitb e` kubjqui`r kubjq ui`r µĲĲaedj` aedj` qu` ad s`b e`sa e`sauh`rb uh`rboj` oj` kda jbs s`ri` s`ri`ss e` àt àt`rds, `rds, pdrqu` bjmuads e` jds àt`rds s` e`lba fu`rb. [` piàsb qu` ‚bj`pg uad‘ ( 2 ) r`pr`sàtb b K, ‚`j iaflaitd e` jb kdsb kdatiaub,‘ p`rd gbstb bgdrb ad quèbed e`hdstrbed. Abei` gb àkdatrbed uab s`ri` iaflaitb e` kdsbs qu` s` àkuàtr` àtr` 3 y K, p`rd tbhpdkd abei` gb e`hdstrbed qu` `std s`b ihpdsioj`.

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2;; Jb p`rp`tuiebe e`j aµ u uh`rd h`rd e` kurvbs eif`ràt`s qu` puèà s`r trbzbebs à ua pjbad, puè` s`r ‚bj`pg eds‘ ( ? ). _`sp`ktd b jbs bj`pgs hµbbss `j`vbebs e` bj`pg eds, ad gba sied àkdatrbebs s`ri`ss e`flai s`ri` e`flaiebs ebs qu` j`s kdrr` kdrr`spdaeba. spdaeba. Ad dostbat`, suosist` `j kdak`ptd e` jbs vbrièbe`s iaflaitbs e` p`rp`tuiebe`s, `hp`zbaed kda `j ‚iaflaitd‘ dreiabrid kdhd jb vbrièbe hµbbss p`qu`˚ ab ab e` jbs hishbs. Y bs bsµµĲ, jb hàt` guhbab, qu` pàdsbhàt` trboblµd pbrb àkdatrbr jb eif`ràkib àtr` 2 y ?, s` gb `j`vbed gbstb edae`, sia t`hdr, trbtb e` àkdatrbr jb eif`ràkib àtr` jb vbrièbe e` p`rp`tuiebe`s. @atdak`s, à jiord qu` trbt` sdor` jbs r`bjizbkida`s e` jb hàt` guhbab, ad eò` uad `skr `skrioir ioir FIA, pu`std qu` µ``st` st` ad `xist `xist`. `. [djbh` [djbhàt` at` eò` uad `skrioir

ℰ

El Reino de Los Numeros Isaac Asimov

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