6. Multiplicación de La Vid

8/3/2019 6. Multiplicacin de La Vid

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NULTMYLMH@HMGKDE L@ SMD


2/24

2

Cl`smg Hglunel` ser+ 8+9

Ygrt`d`1 Det`lle de dmcerekhm`hmk ek yrghexg de brgt`hmk de nermxtengx enbebmdgxek n`der` smea`, &gbxersexe, ek hglgr or`k`te, l` mkduhhmk de mkmhmg de r`hex `ere`x-+

El tevtg de exte n`ku`l ext tgn`dg, `d`yt`dg } xmnylmcmh`dg ` y`rtmr de l`

ngkgor`c` del nmxng kgnbre exhrmt` ygr el yrgcexgr Smhekte Xgtex Rumq+

&EUMT@ N`drmd-+


3/24

8

NULTMYLMH@HMGK DE L@ SMD

0+ OEKER@LMD@DEX

Yrgyò`hmk, nultmylmh`hmk g reyrgduhhmk ex l` gbtekhmk de mk.dmsmdugx de uk`x h`r`hterxtmh`x d`d`x g dexe`d`x ` y`rtmr de uk n`term`levmxtekte+

F`} dgx tmygx de yrgyò`hmk1

. xevu`l g xenmll`

. `xevu`l g seoet`tms`

Oeker`lnekte el trnmkg reyrgduhhmk xe xuele `ylmh`r ` nultmylm.h`hmk ygr xenmll`, nmektr`x pue el de yrgyò`hmk } el de nultmylmh`.hmk xe recmerek ` l` gbtekhmk de mkdmsmdugx ygr s` seoet`tms`+

Ek l` reyrgduhhmk xevu`l xe utmlmq` l` xenmll`, yrgduhmd` dexyuxde re`lmq`rxe lgx yrghexgx de clgr`hmk, yglmkmq`hmk } cehukd`hmk,f`bmekdg tekmdg luo`r l` cuxmk de dgx hlul`x pue xucrmergk l` nemgxmx }oeker`lnekte gh`xmgk` uk` xeoreo`hmk de h`r`hterex+

Ek l` yrgyò`hmk `xevu`l xe `yrgsehf` l` c`hult`d pue hmert`x y`r.tex de l` yl`kt` tmekek y`r` enmtmr brgtex g r`hex+ Ek exte h`xg, `l kg yrg.duhmrxe cehukd`hmk, xmkg xlg dmsmxmgkex nmttmh`x, l`x hlul`x fma`x xgkmdktmh`x ` xux n`drex } lgx mkdmsmdugx gbtekmdgx tmekek l`x nmxn`x h`.r`hterxtmh`x oektmh`x pue l` yl`kt` mkmhm`l, yudmekdg h`nbm`r xu `xyehtgevterkg debmdg ` l` mkcluekhm` del nedmg ek pue xe hultmsek+

L`x s`rmed`dex hultms`d`x tmekek uk mkterx pue r`dmh` ek uk` xermede h`r`hterxtmh`x bmek decmkmd`x } hgkxt`ktex pue yrexekt`k t`lex mkdmsm.dugx } pue decmkek ex` s`rmed`d+ El n`ktekmnmektg de extgx c`htgrex c`.sgr`blex xlg yuede re`lmq`rxe ygr nultmylmh`hmk seoet`tms`+ @x nmxng,or`hm`x ` l` nultmylmh`hmk seoet`tms`, xe yuedek gbteker } n`kteker lgxhlgkex+


4/24

4

2+ REYRGDUHHMGK XEVU@L

H`r`hterxtmh`xL` smd ex uk` yl`kt` `lo`n` hgk `ltg or`dg de feterghmogxmx, ek l`

pue ex nu} dmchml hgkxeoumr lke`x yur`x+

L` cehukd`hmk yuede xer hruq`d` g kg, yerg ek hu`lpumer h`xg lgxmkdmsmdugx yrghedektex de l`x xenmll`x xgk fetergokegx } yrexekt`kuk`x h`r`hterxtmh`x s`rmet`lex pue kg reyrgduhek l`x de kmkoukg de lgxy`rekt`lex+ Ek l` yrhtmh` exte tmyg de nultmylmh`hmk kg xe utmlmq` kukh`y`r` el ext`blehmnmektg de smedgx, yrehmx`nekte ygr ext` c`lt` de fgng.oekemd`d ek l`x yl`kt`x gbtekmd`x+

@ yex`r de extgx mkhgksekmektex l` reyrgduhhmk ygr xenmll` f`

dexenye`dg uk y`yel nu} mnygrt`kte ygrpue f` yernmtmdg1

. ` l`x ygbl`hmgkex x`ls`aex mkxt`l`rxe ek uk` qgk`, n`ktekerxe }enmor`r ` gtrgx luo`rex+

. l` mktrgduhhmk de kues`x exyehmex ek `louk`x reomgkex+

. l` gbtekhmk de kuesgx mkdmsmdugx mkterex`ktex1 Ygr xu utmlmq`hmkbmek hgng ygrt`mkaertgx, hgng yrgduhtgrex dmrehtgx &FYD- g mkhlu.

xg hgng s`rmed`dex yrgduhtgr`x de us`+

Ek decmkmtms`, exte ntgdg de nultmylmh`hmk kg lg enyle` kukh` kmel smtmhultgr km el smsermxt` } pued` dextmk`dg evhluxms`nekte y`r` xu utm.lmq`hmk ygr lgx mksextmo`dgrex, hgk smxt`x ` l` gbtekhmk de kuesgx mkdm.smdugx hgk uk`x h`r`hterxtmh`x deternmk`d`x+ Y`r` ellg xe re`lmq`k fmbrm.d`hmgkex dmrmomd`x, f`hmekdg uk` elehhmk de lgx y`rekt`lex, xeok l`xhu`lmd`dex pue xe buxpuek ek l` dexhekdekhm` } xeok el neh`kmxng de

tr`kxnmxmk de lgx h`r`hterex feredmt`rmgx+De ext` cgrn` ex hgng xe f`k gbtekmdg yrhtmh`nekte tgdgx lgx

ygrt`mkaertgx enyle`dgx ek l` `htu`lmd`d, tgdgx lgx FYD } uk or`k kne.rg de s`rmed`dex de us` de nex`+

L` thkmh` nx hgrreht` ex yglmkmq`r lgx extmon`x de l`x clgrex ce.nekmk`x, yresm`nekte en`xhul`d`x } yrgteomd`x hgk uk` bglx` de yglmkm.q`hmgkex, hgk yglek rehgomdg de l` s`rmed`d pue xe dexe` utmlmq`r hgngy`rekt`l n`xhulmkg+


5/24

5

8+ NULTMYLMH@HMGK @XEVU@L

H`r`hterxtmh`x

L` yrgyò`hmk `xevu`l g seoet`tms` hgkxmxte ek l` yrgduhhmk dekuesgx mkdmsmdugx ` y`rtmr de ygrhmgkex seoet`tms`x de l`x yl`kt`x+

Kg tgd`x l`x exyehmex xgk h`y`hex de yrgyò`rxe ygr exte nedmg, kmtgdgx lgx ro`kgx seoet`tmsgx yuedek xersmr y`r` t`l cmk`lmd`d+

Evmxtek dmxtmktgx ntgdgx de nultmylmh`hmk `xevu`l } lgx yrghexgxxeoumdgx ek h`d` h`xg xgk dmxtmktgx+ @x, ghurre hgk crehuekhm` pue ygr.hmgkex de t`llg tmekek l` h`y`hmd`d de cgrn`r kues`x r`hex } y`rtex der`q yuedek reoeker`r uk kuesg t`llg: l`x fga`x bàg hmert`x hgkdmhmgkex

yuedek cgrn`r kuesgx t`llgx } r`hex+

Uk t`llg } uk` r`q &g dgx t`llgx-, hu`kdg xe lex hgnbmk` de cgrn``dehu`d`, yuedek cuxmgk`rxe yrgduhmekdg uk` hgkevmk s`xhul`r hgktm.ku` } yrgxeoumr xu dex`rrgllg hgng uk xlg mkdmsmdug+

Hgk exte yrghedmnmektg de yrgyò`hmk, l`x yl`kt`x hgkxers`kkteor`nekte l`x h`r`hterxtmh`x oektmh`x de l` yl`kt` n`dre } ex mn.

yrexhmkdmble xu utmlmq`hmk hu`kdg xe yretekd`1 yeryetu`r g nultmylmh`rhlgkex g s`rmed`dex+

NETGDGX DE NULTMYLMH@HMGK @XEVU@L

Ektre lgx dmserxgx ntgdgx de nultmylmh`hmk seoet`tms` pue evmx.tek lgx nx enyle`dgx ek l` smd xgk1

. Ext`h`

. @hgdg

. Mkaertg


6/24

3

4+ NULTMYLMH@HMGK YGR EXT@H@

H`r`hterxtmh`xUk` y`rte del t`llg, de l` r`q g de l` fga` xe xey`r` de l` yl`kt` n`.

dre, xe le hglgh` bàg hgkdmhmgkex `nbmekt`lex c`sgr`blex } xe le mkduhe `

cgrn`r r`hex } t`llgx yrgduhmkdgxe `x uk` kues` yl`kt` mkdeyekdmektemdktmh` ` l` yl`kt` n`dre de l` hu`l yrghede+

Ygr lg t`ktg xe yuede utmlmq`r uk crònektg de t`llg, de r`q g defga`: ek l`x exyehmex pue xe nultmylmh`k hgk c`hmlmd`d ygr ext`h`x, extentgdg tmeke kunergx`x sektà`x, }` pue xe yuedek yrgduhmr nuhf`xyl`kt`x ek uk tmenyg lmnmt`dg y`rtmekdg de uk`x ygh`x yl`kt`x n`drex ùk hgxtg bàg } ygr ntgdgx xekhmllgx, gbtekmkdgxe yl`kt`x hgk uk` ukm.

cgrnmd`d yrhtmh`nekte tgt`l+Ek l` smd evmxte uk` lmnmt`hmk n`rh`d` ygr el fehfg de pue kg ex

ygxmble l` cgrn`hmk de }en`x `dsektmhm`x } ygr t`ktg ex kehex`rmg xmen.yre utmlmq`r ext`h`x de t`llg hgk }en`x, yudmekdg mr `hgny``d`x g kgde fga`x, yerg kukh` xe ygdr utmlmq`r uk` ext`h` de r`q+

Cgrn`hmk de r`hex `dsektmhm`x

L` ygxmbmlmd`d de gbteker uk kuesg mkdmsmdug ` y`rtmr de uk` ext`.

h` de t`llg r`dmh` ek el fehfg de pue `louk`x hlul`x de teamdgx xehukd`.rmgx tmekek l` h`y`hmd`d de retgrk`r ` uk` hgkdmhmk nermxtentmh` }yrgduhmr uk kuesg xmxten` r`dmhul`r+ El yrghexg del dex`rrgllg de l`x r`.hex `dsektmhm`x ek l`x ext`h`x yuede dmsmdmrxe ek 8 c`xex1

`- Mkmhm`hmkde oruygx de hlul`x nermxtentmh`x,b- Dmcerekhm`hmk de exgx oruygx de hlul`x ek yrmngrdmgx de r`q+h- Dex`rrgllg } eneroekhm` de l`x kues`x r`hex } cgrn`hmk de

hgkevmgkex s`xhul`rex hgk lgx teamdgx hgkduhtgrex de l` ext`h`+

Ek l` n`}gr` de l`x yl`kt`x, l` cgrn`hmk de l`x r`hex xe lles` `h`bg dexyux de pue xe f` fehfg l` ext`h`: el tmenyg ek el pue xe dex.`rrgll`k l`x mkmhm`lex de l` r`q, dexyux de hglgh`r l`x ext`h`x ek el luo`rde ekr`mq`nmektg ex s`rm`ble &8.29 d`x-+ L`x r`hex `y`rehek oeker`l.nekte xgbre l` y`rte ekterr`d` ek el xuelg de lgx yny`kgx } x`rnmek.tgx: ùkpue evheyhmgk`lnekte yuedek k`her xgbre t`llgx cuer` del xuelg

& r`hex `re`x -+


7/24

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Ek deternmk`d`x hgkdmhmgkex lgx yehglgx, e mkhluxg lgx kersmgxyrmkhmy`lex, yuedek t`nbmk enmtmr r`hex+ Ek l` yrhtmh`, l`x r`hex xedex`rrgll`k yrmkhmy`lnekte ek lgx kudgx, ek dgkde l` lmokmcmh`hmk ex ne.kgx mktekx` pue ek lgx ektrekudgx } f`} nx `hunul`hmk de `ou` } xux.t`khm`x de rexers`+ Kg gbxt`kte, `y`rehek hgk crehuekhm` r`hex ` lg l`rogdel ektrekudg+ L` r`q k`he xmenyre ek uk r`dmg nedul`r ` nedm` dmxt`k.hm` de lgx dgx f`hex lmberg.legxgx pue lg bgrde`k } nx r`r`nekte ek elbgrde de uk f`q+ Y`r` x`lmr `l evtermgr, l` r`q debe `tr`sex`r l` hgrteq`, lgpue kg yuede f`her dmreht`nekte: yerg l` h`y` nx mkterk` de l` qgk`xubergx` xe suelse `htms` } xe cgrn` uk` h`y` de hlul`x de y`redex nu}delo`d`x } `l`ro`d`x r`dm`lnekte pue les`kt` f`hm` `rrmb` l` hgrteq`+

L`x hlul`x pueukek l`ter`lnekte lgxr`dmgx nedul`rex } lgxf`hex lmberg.legxgx xesuelsek t`nbmk oeke.r`trmhex } ekoekdr`kuk` h`y` de hlul`x`klgo` ` l` yrehedek.

te, xey`r`kdg ` lgx l`.dgx lgx f`hex+ Ektgk.hex l` r`q kg tmeke de.l`kte de x nx pueuk` ygrhmk de r`dmgnedul`r `mxl`dg ekMkedm` del teamdg dey`redex delo`d`x: exte

teamdg. bl`kdg ex en.yua`dg ygr l` r`qf`hm` cuer` } lg dmomere` nedmd` pue s` hre.hmekdg+ L` r`q `h`b`ygr `y`reher `l evtermgr ek nedmg de uk rgdete+

@l eneroer el t`llg, l` r`q `dsektmhm` oeker`lnekte tmeke dmcerek.hm`d`x uk` hgcm` } lgx xmxten`x de teamdgx grdmk`rmgx de l` r`q, `x hgng l`hgkevmk s`xhul`r hgnylet` hgk el t`llg del hu`l xe grmomk`+


8/24

6

Ygl`rmd`d1 L` ygl`rmd`d mkferekte ek lgx x`rnmektgx ex ngxtr`d`

de cgrn` kgt`ble ek el ekr`mq`nmektg de ext`h`x+ L`x ext`h`x de t`llg cgr.n`k brgtex ek el evtreng dmxt`l &el nx yrvmng ` l` yukt` de l` r`n`-} r`hex ek el evtreng yrgvmn`l &el nx herh`kg ` l` hgrgk` de l` yl`k.t`-+ Exte `xyehtg tmeke nuhf` mnygrt`khm` dexde el yuktg de smxt` yrhtm.hg, }` pue xm l` ext`h` xe hglgh` hgk l` ygl`rmd`d mksertmd`, oeker`lnekteel mkdmsmdug pue xe gbtmeke kg ex sm`ble+

H`llg1 Kgrn`lnekte, uk` seq pue xe f`k fehfg l`x ext`h`x } xe f`khglgh`dg ek hgkdmhmgkex c`sgr`blex y`r` el ekr`mhe, xe cgrn` uk h`llg ekel evtreng b`x`l+ Exte h`llg ext hgkxtmtumdg ygr uk` n`x` mrreoul`r dehlul`x y`rekpumn`tgx`x ek dmserxgx ext`dgx de lmokmcmh`hmk+ El hrehm.nmektg del h`llg xe grmomk` de l`x hlul`x de l` reomk del h`nbmun s`xhu.

l`r } el clgen` `d}`hekte, `ukpue dmserx`x hlul`x de l` hgrteq` } de l`ndul` t`nbmk yuedek hgktrmbumr xu cgrn`hmk+

Hgk crehuekhm` l`x yrmner`x r`hex `y`rehek ` tr`sx del h`llg, hgk.duhmekdg extg ` l` xuygxmhmk de pue l` cgrn`hmk del h`llg ex exekhm`ly`r` el ekr`mq`nmektg+ Xmk enb`rog, l` cgrn`hmk del h`llg } de r`hex xgkyrghexgx mkdeyekdmektex, yerg el fehfg de pue hgk crehuekhm` ghurr`kde n`ker` xmnultke` xe debe ` xu deyekdekhm` de hgkdmhmgkex mkterk`x} `nbmekt`lex `klgo`x, xmekdg sekt`agx` l` yrexekhm` del hmt`dg h`llg

hgng yrgtehhmk de l` b`xe de l` ext`h`+

TMYGX DE EXT@H@X

Xe yuedek yrey`r`r dmserxgx tmygx de ext`h`x, pue xe hl`xmcmh`k de`huerdg hgk l` y`rte de l` yl`kt` de pue xe gbtmekek, yerg `pu xlg s`.ngx ` hgkxmder`r l`x ext`h`x de t`llg, pue xgk l`x kmh`x pue xe yuedek

utmlmq`r ek l` nultmylmh`hmk de l` smd+De `huerdg hgk l` k`tur`leq` de l` n`der` enyle`d`, l`x ext`h`x

de t`llg xe dmsmdek ek s`rmgx oruygx1

. de n`der` dur`+

. de n`der` xenmdur`,

. de n`der` bl`kd`+


9/24

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Ext`h`x de n`der` dur`1

Oeker`lnekte, ex el ntgdg de nultmylmh`hmk nx chml } nekgxhgxtgxg, l`x ext`h`x xgk hngd`x de yrey`r`r, xe hgkxers`k nuhfg tmen.yg hgk uk nkmng de yrehùhmgkex elenekt`lex, xe yuedek tr`kxygrt`r `dmxt`khm`x l`ro`x } kg repumerek epumyg exyehm`l dur`kte el ekr`mq`dg+L`x ext`h`x xe yrey`r`k ek l` ygh` de reygxg seoet`tmsg &cmk`lex de gtg.g, mksmerkg- hgk n`der` del hrehmnmektg de l` ext`hmk `ktermgr lmokmcm.h`d` &n`der` de 0 `g g x`rnmektgx-, ùkpue t`nbmk ex ygxmble f`herext`h`x hgk n`der` de dgx g nx `gx, yudmekdg `dgyt`r dmserx`x cgr.n`x+

L`x ext`h`x "xmnylex" xgk l`x pue oeker`lnekte xe utmlmq`k ek l`yrhtmh`, }` pue xgk l`x nx chmlex de gbteker } n`kmyul`r+ Lgx gtrgx tm.ygx yrexekt`k uk mkterx nx lmnmt`dg } xlg xe enyle`k ek hgkdmhmgkex

nu} exyehm`lex &reygxmhmk de n`rr`x-, enyle`kdg n`der` de S+ smkmcer`rehgomd` dexyux de ecehtu`r l` ygd` kgrn`l del smedg+

El tmyg "t`lk" mkhlu}e uk` yepue` nuexh` de n`der` de dgx `gx,nmektr`x pue l` de "n`qg" mkhlu}e uk` ygrhmk n`}gr+ L`x de "yme debue}" lles`k n`der` `k nx smea` yrghedekte de l` xuyrexmk de `lokbr`qg+ Hgk tgdg exte tmyg de ext`h`x pue lles`k y`rte de n`der` smea` xegbtmekek ukgx ygrhektàex de ekr`mq`nmektg xuyermgrex pue hgk l`x xmn.ylex, yerg tmekek el mkhgksekmekte de pue xgk nx hgxtgx`x de gbteker }xe yrgduhe uk dextrgqg n`}gr ek l` yl`kt` n`dre+


10/24

09

@tekdmekdg ` l` lgkomtud de l` ext`h` xe yuedek yrey`r`r s`rmgx tmygx+

. De }en`1 hgnyuext` ygr uk trgqg de x`rnmektg hgk uk` xgl` }en`+

Exte ntgdg xe utmlmq` hu`kdg f`} yghg n`term`l dmxygkmble } xepumere nultmylmh`r rymd`nekte, hgng ygr eaenylg dexyux de re`lmq`ruk` xelehhmk hlgk`l+

El elenektg de yrgy`o`hmk ex nu} delmh`dg } dmxygke de ygh`h`ktmd`d de rexers`x ygr lg pue ex kehex`rmg utmlmq`r thkmh`x de nultmylm.h`hmk exyehm`lex hglgh`kdg l` ext`h` ek hgkdmhmgkex ytmn`x de tenye.r`tur` &24 . 23 H-, funed`d } `mre`hmk &ek uk xuxtr`tg de yerlmt`, turb`} `rek`, hgk uk yrghedmnmektg de kebulmq`hmk-+ De ext` cgrn`, ` l`x 4xen`k`x xe gbtmeke uk mkdmsmdug bmek dex`rrgll`dg pue ex yrehmxg tr`x.

yl`kt`r y`r` pue yrgxmo` xu hrehmnmektg: exte tr`xyl`kte xe c`hmlmt` xm l`ext`h` xe yuxg ` ekr`mq`r dektrg de uk rehmymekte, oeker`lnekte uk ygtde turb`, }` pue ek exte h`xg el ygt xe hglgh` ek gtrg de dmnekxmgkex n`.}grex yudmkdgxe reyetmr ext` gyer`hmk 0.2 sehex ek cukhmk del dex`.rrgllg `lh`kq`dg ygr l` yl`kt`+

Hgkauo`kdg l` n`}gr xuyercmhme ygxmble de fermd`x } ygr t`ktg uk``bukd`kte yrgduhhmk del h`llg hgk l`x exh`x`x rexers`x de l` ext`h`, xe

buxh`k dmserx`x xgluhmgkex1

. Ext`h` } ext`pumll`1 ex uk trgqg de x`rnmektg hgk s`rm`x }en`x, de uk`lgkomtud s`rm`ble1 29 . 35 hn+ L` dmcerekhm` cukd`nekt`l ektre ext`h` }ext`pumll` ex xu dextmkg }` pue l` ext`pumll` xe dextmk` ` yrgduhmr b`rb`.dgx &y`trgkex xmk mkaert`r-, nmektr`x pue l` ext`h`, oeker`lnekte ll`n`.d` ext`h` mkaert`ble, xe enyle` y`r` yrgduhmr yl`kt`.mkaert`d`, xeokyuede yrehm`rxe hgktmku`hmk1


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L`x ext`h`x } ext`pumll`x xgk el n`term`l utmlmq`dg yrecerektenekteek l` yrhtmh` kgrn`l de nultmylmh`hmk ek l` smd, enylekdgxe hgng ex.t`h`x mkaert`blex lgx crònektgx nx oruexgx de lgx x`rnmektgx } hgngext`pumll`x lgx nx cmkgx, yerg ek exte h`xg hgk uk` lgkomtud uk yghg n`.}gr+

. X`rnmektg1 xe enyle`k ext`h`x nu} l`ro`x gbtekmekdg yrhtmh`nektede h`d` x`rnmektg uk` xgl` ext`h`, lg pue xuygke uk hgkxung or`kde den`der`+ Exte yrghedmnmektg xe utmlmq` ek tgdgx lgx y`xex dgkde kg f`}cmlgver`, } ext b`x`dg ek l` c`hmlmd`d de ekr`mq`nmektg de S+ smkmcer`, en.ylekdgxe bmek x`rnmektgx xmnylex g h`lq`dgx &hgk t`lk-, nx `dehu`.dgx y`r` reygker n`rr`x ygr xu ekr`mq`nmektg nx chml+ L` nultmylmh`.hmk ygr x`rnmektg gcrehe uk `ltg rekdmnmektg, debmdg ek or`k y`rte ` l`

`bukd`khm` de rexers`x } `l or`k knerg de kudgx pue ygxee, ukmkdgxe` extg l` rgtur` de s`xgx pue xe yrgduhek ek dmserx`x qgk`x, lg hu`l grmom.k` uk` dmcmhult`d ek l` hmrhul`hmk de l` x`sm` } uk` n`}gr c`hmlmd`d y`r`l` mkmhm`hmk de r`hex+

Ext`h`x de n`der` xenmdur`1

Ext`x ext`h`x xe yrey`r`k hgk n`der` y`rhm`lnekte lmokmcmh`d`,tgn`d`x dur`kte lgx nexex de ser`kg de lgx brgtex kuesgx, mknedm`t`.

nekte dexyux de pue f` f`bmdg uk hrehmnmektg } l` n`der` f` n`dur`.dg ek y`rte+ L` lgkomtud ex s`rm`ble: xuelek teker de 09 . 29 hn } xe dea`k`louk`x fga`x ek el evtreng xuyermgr, yerg pumt`kdg l`x mkcermgrex: y`r`lmnmt`r l`x yrdmd`x de òu` xe rehgrt`k lmoer`nekte l`x fga`x, yerg exkehex`rmg pue xe fò`k ekr`mq`r bàg hgkdmhmgkex pue reduqh`k `l nkm.ng l` yrdmd` de òu` ygr l`x fga`x, utmlmq`kdg l`x mkxt`l`hmgkex de kmebl`mkternmtekte+ Ek l` yrhtmh` exte tmyg de ext`h`x `yek`x xe utmlmq` ek l`nultmylmh`hmk de l` smd+

Ext`h`x de n`der` bl`kd`1

Exte yrghedmnmektg, `l mou`l pue el de l`x ext`h`x de n`der` dur`de 0 }en`, xe enyle` kmh`nekte hu`kdg xe dmxygke de yghg n`term`l,pue xe pumere yrgyò`r rymd`nekte+

L`x ext`h`x xe gbtmekek de lgx brgtex `k serdex &yny`kgx- } ek.r`q`k hgk nx c`hmlmd`d } r`ymdeq pue l`x de lgx gtrgx tmygx hmt`dgx `kte.rmgrnekte, yerg repumerek nx `tekhmk } uk epumyg `dehu`dg+


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04

Ex kehex`rmg n`kea`rl`x hgk nuhfg humd`dg y`r` esmt`r xu dexeh`.hmk } ygr ellg xe l`x debe ygker ` ekr`mq`r ek uk`x hgkdmhmgkex t`lex puemnymd`k yrdmd`x evhexms`x de òu` xmekdg mnyrexhmkdmble l` utmlmq`hmkde uk xmxten` de kmebl` `rtmcmhm`l+

RE@LMQ@HMGK YR@HTMH@ DEL EXT@PUMLL@DG

L`x ext`pumll`x xgk uk crònektg de n`der` de 0 `g &x`rnmektg-de uk` lgkomtud s`rm`ble, ek kuextrg y`x ukgx 39 . 69 hn } nx hgrt`x eklgx denx y`xex eurgyegx, pue xe ygkek ek el h`nyg, ek uk` ygh` `de.hu`d`, y`r` pue cgrnek uk xmxten` r`dmhul`r `dsektmhmg } dex`rrgllekukg g s`rmgx brgtex ` y`rtmr de l`x }en`x pue lles`k+ Exte yrghexg xe dex.tmk` ek l` yrgduhhmk de b`rb`dgx y`r` utmlmq`rlgx hgng ygrt`mkaertgx+

Y`r` yrgheder ` l` yrey`r`hmk de l`x ext`pumll`x, xe lmnym`k lgxx`rnmektgx, elmnmk`kdg l`x y`rtex kg ògxt`d`x, lgx kmetgx } lgx q`rhmllgx+@ hgktmku`hmk xe hgrt` 9/5 . 0 hn ygr debàg de uk kudg } hgk uk` s`rm.ll` de l` lgkomtud mkdmh`d` ek h`d` h`xg xe n`rh` el evtreng xuyermgr,d`kdg el xeoukdg hgrte, dea`kdg uk kuesg t`lk+


15/24

05

L`x ext`pumll`x xe dmxygkek ek lke`x xey`r`d`x de 39 . ?9 hn ` 0n, xeok lgx nedmgx nehkmhgx dmxygkmblex ek h`d` h`xg y`rtmhul`r+ Xg.bre l` lke`, xe yl`kt`k de 29 ` 89 ext`pumll`x ygr netrg, lg pue xuygke299 ` 899+999 ext`pumll`x ygr F`+ El n`rpueg de l`x lke`x xe re`lmq` ygr

dmxtmktgx yrghedmnmektgx1 ` n`kg hgk uk hgrdel g hgk uk r`}`dgr `rr`x.tr`dg ygr tr`htgr pue xe`le 8 4 cml`x ` l` seq } yrgsmxtg de uk` yepue`rea` pue yeketr` ukgx 25 hn } tr`q` uk yepueg xurhg dgkde xe hglg.h`rk l`x ext`pumll`x+

L` yl`kt`hmk de l`x ext`pumll`x xe f`he oeker`lnekte ek q`ka`x, l`yrgcukdmd`d de l` q`ka` ext ek cukhmk de l` lgkomtud de l` ext`pumll`:hgk ext`pumll`x hgrt`x xe yuede yrey`r`r uk` q`ka` yghg yrgcukd` } `h`.b`r hl`skdgl`, xmenyre pue l` yrey`r`hmk del terrekg f`}` xmdg buek`:

xm l` ext`pumll` ex nu} l`ro`, ex yrehmxg f`her uk` q`ka` nu} yrgcukd`+ L`yl`kt`hmk yuede re`lmq`rxe bmek ygr h`b`llgkex xmnylex g dgblex, `yrg.sehf`kdg neagr el terrekg ek exte h`xg, yerg l`x l`bgrex de yrey`r`hmkxgk, xmk enb`rog, nx ekogrrgx`x+

L` y`rte ekterr`d` de l` ext`pumll` } l` tmerr` debek pued`r ek ktm.ng hgkt`htg } el òu` debe ghuy`r tgdgx lgx exy`hmgx lmbrex ektre l`xy`rthul`x } l` ext`pumll`: extg xe hgkxmoue ymx`kdg bmek l` q`ka` uk` seqt`y`d` } `ygrt`kdg uk rmeog: ek ext`x hgkdmhmgkex l` `bxgrhmk mkmhm`l de

òu` ex nx chml } xe esmt`k lgx yrmnergx d`gx ygr dexeh`hmk+

@l hglgh`r l`x ext`h`x ek el xuelg ex nu} dmchml pue xte xe ekhuek.tre ek l`x hgkdmhmgkex mdke`x de funed`d, tenyer`tur` } `mre`hmk, y`.r` extmnul`r el ekr`mq`nmektg: ygr ellg xe rehurre ` uk` xerme de yrhtmh`xsmkhul`d`x hgk l` yrgcukdmd`d de yl`kt`hmk1 xm xe yl`kt` yrgcukdg l`funed`d" ex `lt`, yerg bà`k l` tenyer`tur` } l` `mre`hmk } xm xe yl`kt`yghg yrgcukdg, yuede f`ber ygh` funed`d } l` ext`pumll` yuede dexe.

h`rxe ygrpue l` y`rte evyuext` `l `mre tr`kxymr` nuhfg nx+ L` yrhtmh`del `ygrh`dg, g de l` mkhlmk`hmk de l` ext`pumll` yernmtek hgkhmlm`r l`xdmcerektex kehexmd`dex } neagr `k `nb`x ` l` seq: xmk enb`rog, l` yl`k.t`hmk de l`x ext`pumll`x hgk uk` mkhlmk`hmk evhexms`, yuede yrgsgh`r l`cgrn`hmk de b`rb`dgx dexepumlmbr`dgx, }` pue lgx brgtex tmekek uk` mk.xerhmk gblmhu + Lg hgrrmekte ex dea`r dgx }en`x ygr ekhmn` del kmsel delxuelg } `ygrh`rl`x bmek hgk `r`dgx y`x`kdg ukg ygr h`d` l`dg } lueog`rreol`kdg lgx lgngx ` n`kg g enyle`r `ygrh`dgr`x nehkmh`x pue lgdea`k hgnylet`nekte yrey`r`dg+


16/24

03

L` ygh` de yl`kt`hmk s`r` hgk el hlmn` } l` xmtu`hmk del terrekg,yudmekdg hgnekq`r ` cmk`lex de n`rqg } hgktmku`r dur`kte tgdg el nexde `brml: f`hmekdg uk` yl`kt`hmk tenyr`k` xe `yrgsehf` el buek ext`dgde funed`d del xuelg } l`x ext`h`x xe ygkek ek el yergdg de reygxg, ygrlg pue s`k ` ektr`r ek `htmsmd`d ` nedmd` pue s`}` eleskdgxe l` tenye.r`tur`, yerg f`} uk rmexog or`se pue xgk l`x fel`d`x yrmn`ser`lex, pue

`ukpue kg dextru}ek l` yl`kt`, yrgduhek uk or`k ygrhekt`ae de l`x dekg.nmk`d`x "hgrt`x de yme"+


17/24

0?

@ukpue kg xe xuelek re`lmq`r tr`t`nmektgx cmtgx`kmt`rmgx: ek el h`xgde s`rmed`dex xekxmblex ` nmldmu } gdmg, hgng l` 40B ex kehex`rm` l` `ylm.h`hmk de `lok `ktmhrmytgonmhg+ Dur`kte el ser`kg xe ev`nmk`k lgx sm.sergx y`r` sermcmh`r l` yureq` s`rmet`l } ser xm xe f`k yrgduhmdg neqhl`xek el ngnektg de l` yl`kt`hmk, g ek `louk` c`xe `ktermgr: t`nbmk xe`yrgsehf` y`r` hgnyrgb`r el ext`dg x`kmt`rmg de l`x yl`kt`x+

El hrehmnmektg xe yrgxmoue dur`kte or`k y`rte del gtgg, yrgdu.hmkdgxe l` h`d` de l` fga` ek el tr`kxhurxg de l` yrmner` pumkhek` dekgsmenbre ek l` n`}gr` de l`x s`rmed`dex+

L`x yl`kt`x xe hl`xmcmh`k ek dmxtmkt`x h`teogr`x, ek cukhmk delknerg } dmxygxmhmk de l`x r`hex, l` lgkomtud del brgte ògxt`dg } xudmxt`khm` `l t`lk: hu`kdg xt` ex yepue`, l`x yl`kt`x xe ll`n`k "hgrt`xde yme", pue xgk mk`dehu`d`x y`r` el mkaertg ek h`nyg yerg xuelek en.yle`rxe y`r` el ext`blehmnmektg de h`nygx de ymex n`drex+

L`x yl`kt`x, dektrg de h`d` h`teogr`, xeòruy`k ek knerg de 25 59, yrey`r`kdg uky`puete pue xe `t` } etmpuet` hgksekmektenekte+Xu hgkxers`hmk xe rmoe ygr l`x nmxn`x kgrn`xmkdmh`d`x y`r` l`x ext`h`x: xm l` utmlmq`hmk s` `

xer mknedm`t` xe hgkxers`k xmnylenekte ek uk`q`ka` fned` t`y`kdg l`x r`hex } el t`llg } de.a`kdg lgx brgtex `l `mre+ Xm f`} dmcmhult`dex hgner.hm`lex, l` neagr hgkxers`hmk xe hgkxmoue ekhn`r`x crmogrcmh`x ` 9,*0H } 69 % de fune.d`d, dexyux de uk tr`t`nmektg hgk `lok `ktm.hrmytgonmhg } netmd`x ek x`hgx de ylxtmhg: ekext`x hgkdmhmgkex xe hgkxmoue n`kteker lgx b`r.

b`dgx ek buek ext`dg nx de uk `g+ L` n`kmyu.l`hmk } tr`kxygrte de lgx b`rb`dgx debe f`herxehgk nuhf` yrehùhmk, }` pue l` yl`kt` ex nu}xekxmble ` l` dexeh`hmk yerg xgbre tgdg l`x r`.hex, pue `denx xe d``k chmlnekte hgk el crg+El enb`làe xuele f`herxe hgk dgxy`puetex de b`rb`dgx ukmdgx ygr l`xr`hex, } yrgteomekdg exyehm`lnekte

ext` qgk` ygr nedmg de yà`, h`rrm.qgx, h``x, ylxtmhg, eth+


18/24

06

5+ NULTMYLMH@HMGK YGR @HGDG

H`r`hterxtmh`x

El cukd`nektg de exte ntgdg ex f`her dex`rrgll`r r`hex ek ukt`llg pue ext ukmdg ` l` yl`kt` n`dre+ Exte t`llg, uk` seq ekr`mq`dg, xexey`r` de l` yl`kt` n`dre } xe hgksmerte ek uk` yl`kt` mkdeyekdmektepue smse xgbre xux yrgym`x r`hex+

L` sekt`a` pue yrexekt` l` nultmylmh`hmk ygr `hgdg ex pue `l yer.n`keher el t`llg ukmdg ` l` yl`kt` n`dre kg xe mkterrunye l` `lmnekt`.hmk de l` y`rte pue ext ekr`mq`kdg, ygr lg pue kg evmxtek lgx yrgblen`x,pue ek gh`xmgkex xe yrexekt`k ek l`x ext`h`x de dexeh`hmk g c`lt` de ku.trmektex+

Xmk enb`rog exte xmxten` ext `b`kdgk`dg y`r` l` reygxmhmk den`rr`x hgng hgkxehuekhm` de l` yrgfmbmhmk ygr y`rte de l` HE de gbte.ker yl`kt`x xgbre yme cr`khg hgng hgkxehuekhm` de lgx yrgblen`x cmlgv.rmhgx+ Ex, `denx, uk xmxten` de yrgy`o`hmk hgxtgxg ek lgx smsergx deygrt`mkaertgx ygr l` exyehm`lmq`hmk de l` n`kg de gbr` pue repumere+

TMYGX DE @HGDG

Evmxtek s`rmgx yrghedmnmektgx de `hgdg1

El `hgdg xmnyle xe eceht` dgbl`kdg uk x`rnmektg f`xt` el xuelg } hu.brmkdglg y`rhm`lnekte hgk tmerr`, dea`kdg dexhubmertg xu evtreng ter.nmk`l, l` yukt` xe hurs` extrehf`nekte } xe ekdereq`k lgx ltmngx 05.89hn+ Hu`kdg xe re`lmq` el `hgdg } dexyux de ekr`mq`dg el brgte kg xe xe.y`r` de l` yl`kt` n`dre, rehmbe l` dekgnmk`hmk de nuork+


19/24

0=

El `hgdg nltmyle g xeryekt`rmg ex exekhm`lnekte el nmxng pue el `hg.dg xmnyle, evheytg pue el x`rnmektg pued` `lterk`d`nekte hubmertg }dexhubmertg ` lg l`rog de xu evtekxmk+

Ek h`d` uk` de l`x xehhmgkex ekterr`d`x xe cgrn`k r`hex, debmek.

dg teker ek h`d` tr`ng de y`rte dexhubmert` `l nekgx uk` }en` y`r`cgrn`r uk kuesg brgte+ Uk` seq pue lgx `hgdgx f`k ekr`mq`dg, el x`r.nmektg xe hgrt` ek xehhmgkex cgrn`d`x ygr el kuesg brgte } ygr l` ygr.hmk pue lles` l`x r`hex } de ext` cgrn` xe yuedek gbteker s`rm`x yl`kt`xkues`x ` y`rtmr de uk xglg x`rnmektg+

El `hgdg ek ngkthulg g de hgrte } reh`lhe evmoe hgrt`r l` yl`kt` ` r`xdel xuelg ek l` ext`hmk de reygxg } xe `ngktgk` tmerr` `lrededgr de l`b`xe de lgx kuesgx brgtex, y`r` extmnul`r l` cgrn`hmk de r`hex ek ellgx:`l hubrmrlgx hgk tmerr`, xe yrgsgh` el `fml`nmektg } xe extmnul` l` cgrn`.hmk de r`hex+ Ek el mksmerkg xmoumekte xe dexhubre el ngkthulg } xe xe.y`r`k lgx `hgdgx pue xmrsek y`r` el ext`blehmnmektg de kues`x hey`x+

El `hgdg ek q`ka` g de r`n` ekterr`d` hgkxmxte ek hglgh`r uk x`rnmek.tg ek ygxmhmk fgrmqgkt`l ek el cgkdg de uk` q`ka` g xurhg ` 05 hn deyrgcukdmd`d } hubrmr hgk tmerr` &ek s`rm`x gh`xmgkex- lgx brgtex kuesgx `nedmd` pue hreqh`k+ Ek el gtgg.mksmerkg xmoumekte xe hgrt` el x`rnmektgek t`ktgx trgqgx &`hgdgx- hgng brgtex ekr`mq`dgx f`}`+


20/24

29

3+ NULTMYLMH@HMGK YGR MKAERTG

H`r`hterxtmh`x

L` nultmylmh`hmk ygr mkaertg ext b`x`d` ek l` ygxmbmlmd`d pue evmx.te de pue, b`ag hmert`x hgkdmhmgkex, `l ygker ek hgkt`htg dgx y`rtex demkdmsmdugx dmcerektex xe uk`k } hgktmkek xu hrehmnmektg cgrn`kdg ukxglg mkdmsmdug+ @ l` y`rte pue hgkxtmtu}e el xmxten` r`dmhul`r xe le dekg.nmk` yme, y`trk, ygrt`mkaertg g n`xtg, nmektr`x pue ` l` pue cgrn` l`y`rte `re` xe le `ylmh`k lgx kgnbrex de mkaertg, }en`, y` g s`rmed`d+

Yrghexg de xgld`dur`El teamdg rehmk hgrt`dg de l` y`, h`y`q de yrexekt`r `htmsmd`d ne.

rmxtentmh`, ex yuextg ek ktmng hgkt`htg cmag hgk el teamdg de y`trk,

t`nbmk hgrt`dg ek hgkdmhmgkex xmnml`rex, de t`l ngdg pue l`x reomgkexh`nbm`lex de `nb`x y`rtex extk ek hgkt`htg extrehfg+ L`x hgkdmhmgkexde funed`d } tenyer`tur` debek xer t`lex pue extmnulek l` `htmsmd`d del`x hlul`x rehmk evyuext`x } de `pull`x pue l`x hmrhukd`k+ Ek l` reomkh`nbm`l, t`ktg del y`trk hgng del mkaertg, l`x h`y`x evtermgrex de hlul`xevyuext`x yrgduhek hlul`x de y`rkpumn` pue yrgktg xe ektreneqhl`k} ekl`q`k, `l rexult`dg de ext` `htmsmd`d xe le ll`n` teamdg de h`llg+ @lou.k`x de l`x hlul`x del h`llg rehmk cgrn`dg pue xe ekhuektr`k ek l` nmx.

n` lke` de l` h`y` mkt`ht` de h`nbmun del y`trk } del mkaertg xe dmce.rekhm`k f`xt` cgrn`r kues`x hlul`x h`nbm`lex+ Ext`x kues`x hlul`x deh`nbmun yrgduhek teamdg s`xhul`r kuesg, vmlen` f`hm` el mktermgr } clge.n` f`hm` el evtermgr, ext`blehmekdg `x hgkevmk s`xhul`r ektre y`trk emkaertg, repumxmtg mkdmxyekx`ble y`r` pue l` ukmk del mkaertg teko` vmtg+


21/24

20

TMYGX DE MKAERTGX

Hl`xmcmh`hmk de lgx mkaertgx

Ek l` yrhtmh` xe utmlmq`k dmxtmktgx tmygx de mkaertgx pue xe yuedek

hl`xmcmh`r de s`rm`x cgrn`x1. ygr l` ygh` de re`lmq`hmk1

. yrmn`ser` &de gag sel`kdg-

. gtgg &de gag durnmekdg-

. ygr el n`term`l enyle`dg1. y`

. }en`

l` y` lles` 0, 2, 8 }en`x } yuede xer de yrmn`ser` g de gtgg, nmektr`x

pue l` }en` ex de gtgg+.ygr el xmtmg ek pue xe mkaert`1

. h`beq`

. hgxt`dg

lgx de h`beq` xgk de yrmn`ser` } el de hgxt`dg de gtgg+

. ygr el luo`r ek pue xe re`lmhek1

. `xmektg g h`nyg

. t`ller

el de `xmektg yuede xer de yrmn`ser` u gtgg } el de t`ller de yrmn`ser`+

. ygr el ntgdg de re`lmq`hmk1

. ` n`kg

. ` npumk`


22/24

22

Mkaertgx de yrmn`ser`1

Fekdmdur`1 Hgkxmxte ek

hgrt`r l`x y`x ek cgrn`de hu` y`r` mktrgduhmr.l`x ek el y`trk, fekdmdgdm`netr`lnekte+

@yrgvmn`hmk1 Xe b`xèk ygker ek hgkt`htgxehhmgkex gblmhu`x puedebek xer elmyxex y`rehm.d`x hgk y`trk } y` del

nmxng dmnetrg } uk` mkhlmk`hmk ek el hgrte mdktmh`+ Evmxtek dgx tmygxpue xe utmlmq`k hgk nuhf` crehuekhm`1 el dekgnmk`dg ymhg de clùt` g

`yrgvmn`hmk } el mk.aertg mkolx+

Mkaertg de `xmektg g

h`nyg1 Mktersmekek ekl uk y`trk dex.

`rrgllkdgxe "mk xmtu",pue ext ek hrehmnmektgg ` yuktg de ektr`r ek`htmsmd`d, } uk` y` hgkl`x }en`x ek reygxg: l`

xgld`dur` xe re`lmq` ek uk `ygrh`dg de tmerr`, ektr`kdg l`x }en`x rym.d`nekte ek `htmsmd`d+ L` ygh` nx `dehu`d` ex l` hgnyrekdmd` ektrelgx llgrgx } l` brgt`hmk, yerg yuede `l`ro`rxe &yrmnergx n`rqg nedm`.

dgx n`}g-+ El y`trk ex deh`ymt`dg `l kmsel del xuelg } dexyux de lgxllgrgx ex yrehmxg f`her ext` gyer`hmk 6.05 d`x `ktex del mkaertg y`rèsmt`r el `cluag de x`sm` brut` pue ektgryeher` l` ukmk+ L` y` ex uktrgqg de x`rnmektg del `g hgk dgx }en`x, ` sehex 0.8, ek reygxg: l`x deuk` xgl` }en` tmekek yghg sglunek } extk nu} evyuext`x ` dexeh`hmk:l`x evhexms`nekte l`ro`x evmoek `ygrh`dgx or`kdex, nu} dmchmlex de re`.lmq`r } n`kteker+ Lgx mkaertgx brgt`k `l h`bg de 4.3 xen`k`x, debmkdg.lex yrext`r exyehm`lex humd`dgx1 ref`her el `ygrh`dg, humd`r de `t`puex

de oux`kgx de `l`nbre, hglgh`r uk xgygrte y`r` `t`r lgx brgtex } esmt`rlgx d`gx del smektg, tr`t`nmektgx hgktr` ekcerned`dex } elmnmk`hmk den`l`x fmerb`x+


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Mkaertg de t`ller1 Ex uk` thkmh` ux`d` ek l` yrgduhhmk de yl`kt`x.mkaertg1xe mkaert` uk` y` xgbre uk` ext`h` } exte mkdmsmdug, ext`h` mk.aert`d`, xe lles`r ` l`y`rhel` de smserg dgk.de xe yrgduhmr el ek.r`mq`nmektg del y`trk} el dex`rrgllg de l`x}en`x de l` y`+ Elmkaertg oeker`lnektexe re`lmq` ` npumk`,yerg kg evmxte kmkokmkhgksekmekte ekf`herlg ` n`kg+ L`

ext`h` mkaert`d` kg xelles` dmreht`nekte `lh`nyg, xmkg pue `ktexex xgnetmd` ` l` ll`n`.d` "extr`tmcmh`hmk" ekpue xe hgkxmoue l` xgl.d`dur` del mkaertg+

Mkaertgx de Gtgg1Xgk mkaertgx de `xmektg} de hgxt`dg, yudmk.dgxe utmlmq`r y`x g}en`x, xeok lgx h`xgx+Xe re`lmq` ` cmk`lex deser`kg yerg l`x }en`xkg ektr`k ek hrehm.

nmektg f`xt` l` yrmn`.ser` xmoumekte+


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Kgrn`x de hgnerhm`lmq`hmk de n`der`x } yl`kt`x de smd+

Mkaert`kdg ek L` Rmga`

6. Multiplicación de La Vid

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