Cambridge Primary Mathematics Teacher's Resource 4, Emma Low, Cambridge University Press - Public

 

INMHXK@LC VXKMNX[ 

Mntbcmntkis Pcn cnib ibcr crí ís Xcsdur Xcsduric ic

Cmmn Jdw

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I@-XDM Pcrms ng` idg`ktkdgs de usc

Inmhrk`lc Sgkvcrskty Sgkvcrskty Vrcss ks pnrt de tbc Sgkvcrskty de Inmhrk`lc.

Pbks Cg` Sscr Jkicgsc Nlrccmcgt (‒CSJN‛) ks n jclnj nlrccmcgt hctwccg ‒[du‛ (wbkib mcngs tbc kg`kvk`unj iustdmcr) ng` Inmhrk`lc Sgkvcrskty Vrcss (‒tbc Jkicgsdr‛) edr Inmhrk`lc Vrkmnry Mntbcmntkis Pcnibcr's Xcsduric I@-XDM Ytnlc  (‒tbc Vrd`uit‛). Vjcnsc rcn` tbks CSJN inrceujjy. Hy idgtkgukgl td usc tbc Vrd`uit, [du nlrcc td tbc tcrms de tbks CSJN. Ke [du `d gdt nlrcc td tbks CSJN, pjcnsc `d gdt usc tbks Vrd`uit ng` prdmptjy rcturg kt td tbc pjnic wbcrc ydu dhtnkgc` kt.

Kt eurtbcrs tbc Sgkvcrskty‛s mksskdg mksskdg agdwjc`lc kg tbc pursukt de c`uintkdg, jcnrgkgl ng` rcscnrib nthy tbc`ksscmkgntkgl bklbcst kgtcrgntkdg kgtcrgntkdgnj nj jcvcjs de cxicjjcgic.

.

Sgkvcrskty Sgkvcrsk ty Vrkgtkgl Bdusc, Inmhrk`lc     , Sgktc` Akgl`dm

www.inmhrk`lc.drl Kgedrmntkdg dg tbks tktjc? www.inmhrk`lc.drl/ © Inmhrk`lc Sgkvcrskty Vrcss  Pbks puhjkintkdg ks kg idpyrklbt. Yuhocit td stntutdry stntutdry cxicptkdg ng` td tbc prdvkskdgs prdvkskdgs de rcjcvngt idjjcitkvc jkicgskgl nlrccmcgts, gd rcprd`uitkdg de ngy pnrt mny tnac pjnic wktbdut tbc wrkttcg pcrmksskdg de Inmhrk`lc Sgkvcrskty Vrcss. Ekrst puhjksbc`  Vrkgtc` kg Vdjng` hy Dpdjlrne   N intnjdluc rcidr` rcidr` edr tbks puh puhjkintkdg jkintkdg ks nvnkjn nvnkjnhjc hjc erdm tbc Hrktksb J Jkhrnry  khrnry      ----  Vnpcrhnia  Idvcr nrtwdra? Hkjj Hdjtdg Inmhrk`lc Sgkvcrskty Sgkvcrskty Vrcss bns gd rcspdgskhkjkty edr tbc pcrskstcgic dr niiurniy de SXJs edr cxtcrgnj dr tbkr`-pnrty kgtcrgct wchsktcs rcecrrc` td kg tbks puhjkintkdg, ng` `dcs gdt lunrngtcc tbnt ngy idgtcgt dg suib wchsktcs ks, dr wkjj rcmnkg, niiurntc dr npprdprkntc.

Jkicgic Pbc Jkicgsdr lrngts [du [du tbc rklbt td usc tbc Vrd`uit ug`cr tbc tcrms de tbks CSJN ns edjjdws? (n) [ [du du mny dgjy kgstnjj dgc idpy de tbks Vrd`uit (k) dg n skgljc idmputcr dr sciurc gctwdra scrvcr edr usc hy dgc dr mdrc pcdpjc nt `keecrcgt tkmcs, dr (kk) dg dgc dr mdrc idmputcrs edr usc hy n skgljc pcrsdg (prdvk`c` tbc Vrd`uit ks dgjy usc` dg dgc idmputcr nt dgc tkmc ng` ks dgjy usc` hy tbnt skgljc pcrsdg). (h) [du mny dgjy usc tbc Vrd`uit edr gdg-prdekt, c`uintkdgnj c`uintkdgnj purpdscs. (i) [du sbnjj gdt ng` sbnjj gdt pcrmkt ngydgc cjsc td? (k) idpy dr nutbdrksc idpykgl de tbc Vrd`uit, (kk) trngsjntc tbc Vrd`uit, (kkk) rcvcrsc-cglkgccr, `ksnsscmhjc dr `cidmpkjc tbc Vrd`uit, dr (kv) trngsecr, trngsecr, scjj, nssklg dr dtbcrwksc idgvcy ngy pdrtkdg de tbc Vrd`uit. .

Idpyrklbt (n) Njj idgtcgt prdvk`c` ns pnrt de tbc Vrd`uit (kgiju`kgl tcxt, kmnlcs ng` ngikjjnry mntcrknj) ng` njj sdetwnrc, id`c, ng` mctn`ntn rcjntc` td tbc Vrd`uit ks tbc idpyrklbt de tbc J kicgsdr dr bns hccg jkicgsc` td tbc Jkicgsdr, ng` ks prdtcitc` hy idpyrklbt ng` njj dtbcr nppjkinhjc kgtcjjcitunj prdpcrty jnws ng` kgtcrgntkdgnj trcntkcs. (h) [du mny gdt idpy tbc Vrd`uit cxicpt edr mnakgl dgc idpy de tbc Vrd`uit sdjcjy edr hniaup dr nribkvnj purpdscs. [du mny gdt njtcr, rcmdvc dr `cstrdy ngy idpyrklbt gdtkic dr dtbcr mntcrknj pjnic` dg dr wktb tbks Vrd`uit. (i) [du mny c`kt ng` mnac ibnglcs td ngy mntcrknj prdvk`c` kg tbc Vrd`uit kg c`ktnhjc edrmnt (‒C`ktnhjc Mntcrknj‛) ng` stdrc idpkcs de tbc rcsujtkgl ekjcs (‒C`ktc` Ekjcs‛) edr ydur dwg gdg-idmmcri gdg-idmmcriknj, knj, c`uintkdgnj usc, hut [du mny gdt `kstrkhutc C`ktnhjc Mntcrknjs dr C`ktc` Ekjcs td ngy tbkr`-pnrty, dr rcmdvc, njtcr, dr `cstrdy ngy idpyrklbt gdtkics dg C`ktnhjc Mntcrknjs dr C`ktc` Ekjcs, dr idpy ngy pnrt de ngy C`ktnhjc Mntcrknj dr C`ktc` Ekjcs kgtd ngy dtbcr ekjc edr ngy purpdsc wbntsdcvcr. . Jknhkjkty ng` Kg`cmgkekintkdg

Kt ks kjjclnj td rcprd`uic ngy pnrt de bks wdra kg mntcrknj edrm (kgiju`kgl pbdtdidpykgll ng` cjcitrdgki stdrnlc) cxicpt ug`cr tbc edjjdwkgl ikriumstngics? pbdtdidpykg ikriumstngics?

(n) Pbc Vrd`uit ks suppjkc` ‒ns-ks‛ wktb gd cxprcss lunrngtcc ns td kts suktnhkjkty. Pd tbc cxtcgt pcrmkttc` hy nppjkinhjc jnw, tbc Jkicgsdr ks gdt jknhjc edr idsts de prdiurcmcgt de suhstktut suhstktutcc prd`uits, `nmnlcs dr jdsscs de ngy akg` wbntsdcvcr rcsujtkgl erdm tbc usc de tbks Vrd`uit, dr crrdrs dr enujts tbcrckg, ng` kg cvcry insc tbc J kicgsdr‛s jknhkjkty sbnjj hc jkmktc` td tbc sullcstc` jkst prkic dr tbc nmdugt nitunjjy pnk` hy [du edr tbc Vrd`uit, wbkibcvcr ks jdwcr. (h) [du niicpt tbnt tbc Jkicgsdr ks gdt rcspdgskh rcspdgskhjc jc edr tbc pcrskstcgiy pcrskstcgiy,, niiurniy dr nvnkjnhkjkty de ngy SXJs de cxtcrgnj

(k) wbcrc ydu nrc nh nhk`kgl k`kgl hy n jkicgic lrng lrngtc` tc` td ydur sibddj dr kg kgstktutkdg stktutkdg hy hy tbc Idpyrklbt Jkicgskgl Nlcgiy1 (kk) wbcrc gd suib jkicgic cxkst cxksts, s, dr wbcrc ydu wksb ttd d cxicc` tbc tcrms de n jkicgic, ng` ydu bnvc lnkgc` tbc wrkttcg pcrmksskdg de Inmhrk`lc Sgkvcrskty Vrcss1 (kkk) wbcrc ydu nrc njjdwc` td rcprd`uic wktbdut pcrmksskdg ug`cr tbc prdvkskdgs prdvkskdgs de Ibnptcr  de tbc I dpyrklbt, @csklgs ng` Vntcg Vntcgts ts Nit , wbkib idvcrs, edr cxnmpjc, tbc rcprd`uitkdg edr tbc purpdscs de scttkgl cxnmkgntkdg qucstkdgs.

dr tbkr`-pnrty kgtcrgct wchsktcs rcecrrc` td dg tbc Vrd`uit ng` `dcs gdt lunrngtcc tbnt ngy idgtcgt dg suib wchsktcs ks, dr wkjj rcmnkg, niiurntc, npprdprkntc npprdprkntc dr nvnkjnhjc. Pbc J kicgsdr sbnjj gdt hc jknhjc edr ngy idgtcgt mn`c nvnkjnhjc erdm ngy wchsktcs ng` SXJs dutsk`c tbc Vrd`uit dr edr tbc `ntn idjjcitkdg dr huskgcss prnitkics de ngy tbkr`-pnrty kgtcrgct wchsktc dr SXJ rcecrcgic` hy tbc Vrd`uit. (i) [du [du nlrcc td kg`cmgkey tbc Jkicgsdr ng` td accp kg`cmgkekc` tbc Jkicgsdr erdm ng` nlnkgst ngy jdss, idst, `nmnlc dr cxpcgsc (kgiju`kgl wktbdut jkmktntkdg `nmnlcs pnk` td n tbkr` pnrty ng` ngy rcnsdgnhjc jclnj idsts) kgiurrc` hy tbc Jkicgsdr ns n rcsujt de ydur hrcnib de ngy de tbc tcrms de tbks CSJN.

      

  

Pbc pbdtdidpy mnstcrs kg tbks puhjkintkdg mny hc pbdtdidpkc` dr `kstrkhutc` Qcjcitrdgkinjjy] ercc de ibnrlc edr ijnssrddm usc wktbkg tbc sibddj dr kgstktutkdg kgstktutkdg tbnt puribnsc` tbc puhjkintkdg. \drasbccts \drasbccts ng` idpkcs de tbcm rcmnkg kg tbc idpyrklbt de Inmhrk`lc Sgkvcrskty Vrcss, ng` suib idpkcs mny gdt hc `kstrkhutc` dr usc` kg ngy wny dutsk`c tbc puribnskgl kgstktutkdg.

.

Pcrmkgntkdg \ktbdut prcou`kic prcou`kic td ngy dtbcr rklbts, tbc Jkicgsdr mny tcrmkgntc tbks CSJN ke [du enkj td idmpjy wktb ngy de kts tcrms ng` idg`ktkdgs. Kg suib cvcgt, [du must `cstrdy njj idpkcs de tbc Vrd`uit kg ydur pdsscsskdg. . Ldvcrgkgl jnw

Pbks nlrccmcgt ks ldvcrgc` hy tbc jnws de Cgljng` ng` \njcs, wktbdut rclnr` td kts idgejkit de jnws prdvkskdg, ng` cnib pnrty krrcvdinhjy suhmkts td tbc cxijuskvc ourks`kitkdg de tbc Cgljksb idurts. Pbc pnrtkcs `ksijnkm tbc nppjkintkd nppjkintkdg g de tbc Sgktc` Gntkdgs Idgvcgtkdg dg tbc Kgtcrgntkdgnj Ynjc de Ldd`s.

 

Idgtcgts Pbc ctbds de tbc Inmhrk`lc Mntbs prdocit Mntbs prdocit

>I? Bng`jkgl `ntn ng` prdhjcm sdjvkgl

Kgtrd`uitkdg Pcnibkgl npprdnibcs Pnjakgl mntbcmntkis Xcsdurics,, kgiju`kgl lnmcs Xcsdurics

         

Pcrm > >N? Gumhcr ng` prdhjcm sdjvkgl > Gumhcrs ng` tbc gumhcr gumhcr systcm 

                   

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>.> Xcn`kgl, wrktkgl ng` pnrtktkdgkgl gumhcrs >.8 Dr`crkgl, idmpnrkgl ng` rdug`kgl edur-`klkt gumhcrs >.9 Mujtkpjykgl ng` `kvk`kgl hy >6 ng` >66 N``ktkdg ng` suhtrnitkdg (>)  8.> N``ktkdg (>) 8.8 Yuhtrnitkdg (>) 8.9 Vnrtktkdgkgl td n`` ng` suhtrnit Mujtkpjkintkdg ng` `kvkskdg (>)  9.> Jcnrgkgl ng` uskgl mujtkpjkintkdg enits 9.8 Sskgl `duhjcs

  9.9 Mujtkpjykgl n twd-`klkt gumhcr hy n skgljc `klkt >H? Mcnsurc ng` p prdhjcm rdhjcm sdjvkgl   0 \cklbt    0.> Mcnsurkgl wcklbt   3 Pkmc (>)    3.> Pcjjkgl tbc tkmc (>)   3.8 Sskgl tkmctnhjcs   5 Nrcn ng` pcrkmctcr (>)    5.> Nrcn (>)   5.8 Vcrkmctcr (>)

< Lrnpbs, tnhjcs ng` ibnrts (>)  Pnjjy Pnjjy ibnrts ng` hnr ibnrts Inrrdjj `knlrnms 2.8 Ucgg `knlrnms

5; 5 86 80 9> 98 95

8N? Gumhcr ng` prdhjcm sdjvkgl   ; Pbc gumhcr systcm ng` prdpcrtkcs de gumhcr    ;.> @cikmnj gumhcrs kg idgtcxt   ;.8 Vdsktkvc ng` gclntkvc gumhcrs   ;.9 D`` ng` cvcg gumhcrs   >6 N``ktkdg ng` suhtrnitkdg (8)    >6.> N``kgl ng` suhtrnitkgl gcnr mujtkpjcs de >6   >6.8 Ibddskgl tbc mdst ce ﬎ikcgt suhtrnitkdg strntcly   >> Mujtkpjkintkdg ng` `kvkskdg (8)    >>.> Mdrc mujtkpjkintkdg

92 0> 08 0< 02 38 3< 32 58

8H?                

>>.8 @kvk`kgl twd-`klkt gumhcrs hy skgljc-`klkt gumhcrs Lcdmctry ng` p prdhjcm rdhjcm sdjvkgl >8 Ngljcs, pdsktkdg ng` `krcitkdg  >8.> Ngljcs ng` turgkgl >8.8 Vdsktkdg ng` `krcitkdg >9 Yymmctry  >9.> Ybnpcs ng` symmctry >0 8@ ng` 9@ sbnpcs  >0.> 8@ sbnpcs >0.8 9@ sbnpcs

2< 22 ;8 ;0 ;; >66 >60 >>> >>8 >>0 >>< >>2 >86 >89 >80 >8< >82 >96

kkk kkk  

 

8I? Mcnsurc ng` prdhjcm sdjvkgl   >3 Jcgltb    >3.> Mcnurkgl jcgltb   >5 Pkmc (8)    >5.> Pcjjkgl Pcjjkgl tbc tkmc (8)

>93 >95 >09 >00

       

>02 >33 >35 >56

>5.8 Sskgl injcg`nrs >< Nrcn ng` pcrkmctcr (8)  > Nrcn (8) >2 Ypciknj gumhcrs 

kv  kv 

>59

              9H?    

>2.> Ypciknj gumhcrs >; Ernitkdgs ng` `kvkskdgs  >;.> Cxpdrkgl ernitkdgs >;.8 Ernitkdgs, `cikmnjs ng` mkxc` gumhcrs >;.9 Ernitkdgs ng` `kvkskdg 86 Xntkd ng` prdpdrtkdg  86.> Xntkd ng` prdpdrtkdg Mcnsurc ng` prdhjcm sdjvkgl 8> Inpnikty  8>.> Mcnsurkgl inpnikty

>50 >5< >52 >2> >28

         

88 Pkmc (9)  88.> Mcnsurkgl tkmc 88.8 Injiujntkgl tkmc 89 Nrcn ng` pcrkmctcr (9)  89.> Nrcn ng` pcrkmctcr

>;9 >;0 >;2 86; 8>6

>2< >22

9I? Bng`jkgl `ntn ng` prdhjcm sdjvkgl   80 Lrnpbs, tnhjcs ng` ibnrts (8)    80.> Pn Pnhjcs hjcs ng` hnr ibnrts   80.8 Ercqucgiy tnhjcs ng` trcc `knlrnms   83 Ucgg ng` Inrrdjj `knlrnms 

8>3 8>5 8>; 88<

   

882 896

83.> Inrrdjj `knlrnms (8) 83.8 Ucgg `knlrnms (8)

 

Pbc Ctbds de tbc Inmhrk`lc Vrkmnry Mntbs prdocit Inmhrk`lc Vrkmnry Mntbs ks Mntbs  ks ng kggdvntkvc idmhkgntkdg de iurrkiujum ng` rcsdurics `csklgc` td suppdrt tcnibcrs ng` jcnrgcrs td suiicc` kg prkmnry mntbcmntkis tbrdulb hcstprnitkic kgtcrgntkdgnj mntbs tcnibkgl ng` n prdhjcm-sdjvkgl npprdnib.

Pd lct kgvdjvc` vkskt www.ikc.drl.ua/inmhrk`lcprkmnrymntbs

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Inmhrk`lc Vrkmnry Mntbs hrkgls Mntbs hrkgls tdlctbcr tbc wdrj`-ijnss Inmhrk`lc Vrkmnry mntbcmntkis iurrkiujum erdm Inmhrk`lc Kgtcrgntkdgnj Cxnmkgntkdgs, bklb-qunjkty puhjksbkgl erdm Inmhrk`lc Sgkvcrskty Vrcss  ng` cxpcrtksc kg cglnlkgl dgjkgc cEmcgt mntcrknjs edr tbc mntbcmntkis iurrkiujum erdm GXKIB. Inmhrk`lc Vrkmnry Mntbs deecrs Mntbs deecrs tcnibcrs ng dgjkgc tddj tbnt mnps rcsdurics ng` jkgas td mntcrknjs deecrc` tbrdulb tbc prkmnry mntbcmntkis iurrkiujum, GXKIB ng` Inmhrk`lc Vrkmnry Mntbcmntkis tcxthddas ng` c-hddas. Pbcsc rcsdurics kgiju`c cglnlkgl dgjkgc nitkvktkcs, hcst-prnitkic luk`ngic ng` cxnmpjcs de Inmhrk`lc Vrkmnry Mntbs kg Mntbs kg nitkdg. Pbc Inmhrk`lc iurrkiujum ks `c`kintc` td bcjpkgl sibddjs `cvcjdp jcnrgcrs wbd nrc idg﬎`cgt, rcspdgskhjc, rc﬈citkvc, kggdvntkvc kggdvntkvc ng` cglnlc`. Kt ks `csklgc` td lkvc jcnrgcr s tbc sakjjs td prdhjcm sdjvc ceecitkvcjy, nppj nppjy y mntbcmntkinj agdwjc`lc ng` `cvcjdp n bdjkstki ug`crstng`kgl de tbc suhocit. Pbc Inmhrk`lc Sgkvcrskty Vrcss scrkcs de Pcnibcr‛s Xcsduric prkgtc` Xcsduric prkgtc` hddas ng` I@-XDMs  I@-XDMs prdvk`c hcst-kg-ijnss suppdrt edr tbks prdhjcm-sdjvkgl npprdnib, hnsc` dg pc`nldlkinj prnitkic edug` kg suiicsseuj sibddjs nirdss tbc wdrj`. Pbc cglnlkgl GXKIB dgjkgc rcsdurics bcjp `cvcjdp mntbcmntkinj tbkgakgl ng` prdhjcm-sdjvkgl sakjjs. Pbc hcgc﬎ts de hckgl pnrt de Inmhrk`lc Vrkmnry Mntbs nrc? Mntbs  nrc?   tbc dppdrtugkty td cxpjdrc n mntbs iurrkiujum edug`c` dg tbc vnjucs de tbc Sgkvcrskty de Inmhrk`lc ng` hcst prnitkic kg sibddjs   niicss td ng kggdvntkvc pnianlc de dgjkgc ng` prkgt rcsdurics tbnt ing bcjp hrkgl tbc Inmhrk`lc Vrkmnry mntbcmntkis iurrkiujum td jkec kg tbc ijnssrddm.

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> [du ing cxpjdrc tbc nvnkjnhj nvnkjnhjcc rcsdurics dg tbc Inmhrk`lc Vrkmnry Mntbs wchsktc Mntbs  wchsktc hy iurrkiujum ernmcwdra, sibcmc de wdra, dr tcnibcr rcsdurics. Kg tbks cxnmpjc, tbc ‒Pcnibcr rcsdurics‛ rcsdurics‛ tnh bns hccg scjcitc`. 8 Pbc `rdp-`dwg `rdp-`dwg mcgu njjdws scjcitkdg de rcsdurics hy Ytnlc. Ytnlc. 9 Edjjdwkgl Edjjdwkgl scjcitkdg de tbc ‒Pcnibcr rcsduric‛ rcsduric‛ ng` ‒Ytnlc >‛ >‛, tbc ibnptcrs kg tbc Inmhrk`lc Sgkvcrskty Vrcss tcxthdda ‒Pcnibcr‛s rcsduric >‛ nrc >‛ nrc jkstc`. 0 Ijkiakgl dg n ibnptcr (‒8 Vjnykgl wktb >6‛ >6‛ kg tbks cxnmpjc) rcvcnjs tbc jkst de iurrkiujum ernmcwdra dhocitkvcs idvcrc` kg tbnt ibnptcr ibnptcr.. Ijkiakgl dg n lkvcg dhocitkvc (>Gi> kg tbks cxnmpjc) bklbjklbts tbc mdst rcjcvngt GXKIB nitkvkty edr tbnt dhocitkvc. 3 N jkst de rcjcvngt GXKIB nitkvktkcs edr tbc scjcitc` ibnptcr nrc rcvcnjc`. rcvcnjc`. Ijkiakgl dg n lkvcg GXKIB nitkvkty wkjj bklbjklbt tbc dhocitkvcs tbnt kt idvcrs. [du ing jnugib tbc GXKIB nitkvkty erdm bcrc.

v 

 

Pbc Inmhrk`lc Vrkmnry Mntbs prdocit Mntbs prdocit prdvk`cs n idmpjctc suppdrt pnianlc edr tcnibcrs. Pbc Pcnibcr's Xcsduric ks n stng`njdgc tcnibkgl tcxthdda tbnt ing hc usc` kg`cpcg`cgtjy dr tdlctbcr wktb Inmhrk`lc Vrkmnry Mntbs wchsktc. Mntbs wchsktc. Pbc ercc td niicss wchsktc mnps tbc nitkvktkcs ng` lnmcs kg tbc Pcni Pcnibcr's bcr's Xcsduric Xcsduric td  td tbc Inmhrk`lc Vrkmnry iurrkiujum. Kt njsd bklbjklbts rcjcvngt dgjkgc nitkvktkcs `csklgc` hy tbc GXKIB prdocit tcnm hnsc` nt tbc Sgkvcrskty de Inmhrk`lc. Pbc n``ktkdgnj mntcrknj tbnt tbc Inmhrk`lc Vrkmnry Mntbs prdocit Mntbs  prdocit prdvk`cs ing hc niicssc` kg tbc edjjdwkgl wnys? Ns n Inmhrk`lc Icgtrc?  Ke ydu nrc n rclkstcrc` Inmhrk`lc Icgtrc, ydu lct ercc niicss td njj tbc nvnkjnhjc nvnkjnhjc mntcrknj hy jdllkgl kg uskgl ydur cxkstkgl Inmhrk`lc Kgtcrgntkdgnj Cxnmkgntkdgs jdl kg `ctnkjs. Xclkstcr ns n vksktdr? Ke ydu nrc gdt n rclkstcrc` Inmhrk`lc Icgtrc ydu ing rclkstcr td tbc sktc ns n vksktdr, wbcrc ydu ydu wkjj hc ercc td `dwgjdn` n jkmktc` sct de rcsdurics ng` dgjkgc nitkvktkcs tbnt ing hc scnribc` hy tdpki ng` jcnrgkgl dhocitkvc. Ns ng ugrclkstcrc` vksktdr? [du nrc lkvcg ercc niicss ng kgtrd`uitdry vk`cd ng` sdmc snmpjc rcsdurics, ng` nrc nhjc td rcn` njj nhdut tbc sibcmc.

vk  vk 

 

Kgtrd`uitkdg Pbc Inmhrk`lc Vrkmnry Mntbs scrkcs Mntbs scrkcs de rcsdurics idvcrs tbc cgtkrc idgtcgt de tbc Inmhrk`lc Vrkmnry Mntbcmntkis iurrkiujum ernmcwdra erdm Inmhrk`lc Kgtcrgntkdgnj Cxnmkgntkdgs. Cxnmkgntkdgs. Pbc rcsdurics bnvc hccg wrkttcg hnsc` dg n sullcstc` tcnibkgl ycnr de tbrcc, tcg wcca tcrms. Pbks ing hc nmcg`c` td sukt tbc gumhcr de wccas nvnkjnhjc nvnkjnhjc kg ydur sibddj ycnr. Pbc Inmhrk`lc Vrkmnry Mntbcmntkis ernmcwdra prdvk`cs n idmprcbcgskvc sct de jcnrgkgl dhocitkvcs edr mntbcmntkis. mntbcmntkis. Pbcsc dhocitkvcs `cnj wktb wbnt jcnrgcrs sbduj` agdw ng` hc nhjc td `d. Pbc ernmcwdra ks prcscgtc` kg ﬎vc strng`s? tbc edur idgtcgt strng`s de Gumhcr (kgiju`kgl mcgtnj strntclkcs), Lcdmctry, Mcnsurcs ng` Bng`jkgl @ntn nrc njj ug`crpkggc` hy tbc ﬎etb strng`, Vrdhjcm Ydjvkgl. Vrdhjcm sdjvkgl ks kgtclrntc` tbrdulbdut tbc edur idgtcgt strng`s. \bkjst kt ks kmpdrtngt td hc nhjc td k`cgtkey tbc prdlrcsskdg de dhocitkvcs tbrdulb tbc iurrkiujum, kt ks njsd csscgtknj td hrkgl t dlctbcr tbc `keecrcgt strng`s kgtd n jdlkinj wbdjc wbdjc.. Pbks scrkcs de prkgtc` hddas ng` I@-XDMs puhjksbc` hy Inmhrk`lc Sgkvcrskty Vrcss ks nrrnglc` td cgsurc tbnt tbc iurrkiujum ks idvcrc` wbkjst njjdwkgl tcnibcrs ﬈cxkhkjkty kg npprdnib. Pbc Yibcmc de \dra edr Ytnlc 0 bns hccg eujjy idvcrc` ng` edjjdws kg tbc snmc ‒Sgkt‛ dr`cr ns prcscgtc` hy Inmhrk`lc Kgtcrgntkdgnj Cxnmkgntkdgs (>N’I, 8N’I ng` tbcg 9N’I) hut tbc dr`cr de dhocitkvc idvcrnlc mny vn vnry ry `cpcg`kgl dg n jdlkinj pc`nldly ng` tcnibkgl npprdnib. Pbc idmpdgcgts de tbc prkgtc` scrkcs nrc ns edjjdws?   Pcnibcr‛s Xcsduric (prkgtc` Xcsduric (prkgtc` hdda ng` I@-XDM)   Pbks rcsduric idvcrs njj tbc dhocitkvcs de tbc Inmhrk`lc ernmcwdra tbrdulb jcssdgs rcecrrc` td ns ‒Idrc ‒Idrc nitkvktkcs‛. nitkvktkcs‛. Ns n ‒jcssdg‛ ks n suhocitkvc tcrm (tnakgl mdrc dr jcss tkmc `cpcg`kgl dg tbc sibddj ng` tbc jcnrgcrs) wc prcecr td usc tbc tcrms ‒Idrc ‒ Idrc nitkvkty‛ nitkvkty‛ ng` ‒scsskdg‛ td rckgedric tbnt tbcrc ks sdmc ﬈cxkhkjkty. Cnib Idrc nitkvkty idgtnkgs tbc kgstruitkdgs edr ydu td jcn` tbc nitkvkty ng` idvcr tbc dhocitkvcs, ns wcjj ns prdvk`kgl cxpcitc` dutidmcs, sullcstc` `knjdluc edr `ksiusskdg, ng` jkacjy nrcns

de mksidgicptkdg. N scitkdg injjc` ‒Mdrc ‒Mdrc nitkvktkcs‛ nitkvktkcs‛ prdvk`cs ydu wktb sullcstkdgs edr suppjcmcgtnry dr cxtcgskdg nit kvktkcs. Pbc Pcnibcr‛s Xcsduric ing Xcsduric ing hc usc` dg kts dwg td idmpjctcjy idvcr tbc idursc. (Pbc Jcnrgcr‛s Hdda  ng`  ng` Lnmcs Hdda  sbduj`  sbduj` gdt hc usc` wktbdut tbc nssdikntc` tcnibcr rcsduric, ns tbcy nrc gdt sue ﬎ikcgt dg tbckr dwg td idvcr njj tbc dhocitkvcs.)   Pbc niidmpngykgl I@-XDM I@-XDM idgtnkgs?  n\ \dr` dr` vcrskdg de tbc cgtkrc prkgt c` hdda. Pbks bns hccg suppjkc` sd tbnt ydu ing idpy ng` pnstc rcjcvngt ibugas de tbc tcxt kgtd ydur dwg jcssdg pjngs ke ydu `d gdt wngt td usc dur hdda `krcitjy. [du wkjj hc nhjc td c`kt ng` prkgt tbc \dr` ﬎jcs ns rcqukrc` hut `keecrcgt vcrskdgs de \dr` usc` dg `keecrcgt VIs ng` MNIs wkjj rcg`cr tbc idgtcgt sjklbtjy `keecrcgtjy sd ydu mklbt bnvc sdmc edrmnttkgl kssucs. ^ucstkdgkgl  ’  ’ Pbks `diumcgt dutjkgcs sdmc de tbc `keecrcgt typcs de qucstkdg tcibgkqucs edr mntbcmntkis ng` bdw hcst td usc tbcm, prdvk`kgl suppdrt edr tcnibcrs.   Jcttcrs edr pnrcgts ’ pnrcgts ’ n tcmpjntc jcttcr ks suppjkc` njdgl wktb n mnppkgl lrk` td bcjp ydu td wrktc n jcttcr pcr Sgkt de mntcrknj kg dr`cr td kgedrm pnrcgts wbnt wdra tbckr ibkj` ks `dkgl, ng` wbnt tbcy ing `d td suppdrt tbckr ibkj` nt bdmc.   Vbdtdidpy mnstcrs ’ mnstcrs ’ rcsdurics nrc suppjkc` ns V@Es, ng` ns \dr` ﬎jcs sd tbnt ydu ing c`kt tbcm ns rcqukrc`.   Jcnrgcr‛s Hdda  (prkgtc`  (prkgtc` hdda)   Pbks rcsduric ks suppjcmcgtnry td tbc idursc. Ns tbc ctbds de tbc Inmhrk`lc Mntbs Vrdocit ks Vrdocit ks td nvdk` rdtc jcnrgkgl ng` `rkjj prnitkic, tbcrc nrc gd niidmpngykgl wrktc-kg wdrahddas. Pbc Jcnrgcr‛s Hdda   kgstcn` idmhkgcs idgsdjk`ntkdg ng` suppdrt edr tbc jcnrgcr wktb kgvcstklntkdgs tbnt njjdw ercc`dm de tbdulbt, ng` qucstkdgs tbnt cgidurnlc tbc jcnrgcr td nppjy tbckr agdwjc`lc rntbcr tbng oust rcmcmhcrkgl n tcibgkquc. Pbc kgvcstklntkdgs ng` qucstkdgs nrc wrkttcg Kgtrd`uitkdg

vkk  vkk 

 

td nsscss tbc jcnrgcr‛s ug`crstng`kgl ug`crstng`kgl de tbc jcnrgkgl dutidmcs de tbc Idrc nitkvkty. nitkvkty. Jcnrgcrs ing wrktc `dwg tbckr ngswcrs td kgvcstklntkdgs ng` qucstkdgs kg ng cxcriksc hdda kg dr`cr td kgedrm nsscssmcgt. Pbc dvcrnjj npprdnib npprdnib de tbc Pcnibcr‛s Xcsduric niidmpngkc` hy tbc Jcnrgcr‛s Hdda  njjdws  njjdws n skmpjc wny edr ydu td nsscss bdw wcjj n jcnrgcr

scrkcs. Kt ing hc usc` njdglsk`c tbc Pcnibcr‛s Xcsduric ns scrkcs. Xcsduric ns n sduric de n``ktkdgnj nitkvktkcs td suppdrt jcnrgcrs tbnt gcc` cxtrn rckgedricmcgt, dr td lkvc td n`vngic` jcnrgcrs ns cxtcgskdg. Cnib lnmc idmcs wktb n ‒Mntbs Mntbs edius‛  edius‛ td bklbjklbt tbc kgtcg`c` jcnrgkgl/rckgedricmcgt dutidmc de tbc lnmc, sd tbnt tbc hdda ing hc usc` kg`cpcg`cgtjy de ngy dtbcr

ug`crstng`s n tdpki, wbkjst njsd cgidurnlkgl `ksiusskdg, prdhjcmsdjvkgl ng` kgvcstklntkdg sakjjs.

rcsduric. Edr tbdsc wbd nrc uskgl kt ns pnrt de tbks scrkcs, rcjcvngt lnmcs nrc rcecrrc` td hy tktjc ng` pnlc gumhcr kg tbc ‒ Mdrc nitkvktkcs‛ nitkvktkcs‛ scitkdg de tbc Pcnibcr‛s Xcsduric. Xcsduric. Pbc niidmpngykgl I@-XDM idgtnkgs gcts td mnac rcqukrc` rcsdurics1 kt njsd idgtnkgs n mnppkgl `diumcgt tbnt mnps tbc lnmcs td tbc dtbcr rcsdurics kg tbc scrkcs edr tbdsc wbd rcqukrc kt. Vjcnsc gdtc tbnt tbc Lnmcs Hdda  `dcs  `dcs gdt idvcr njj de tbc Inmhrk`lc dhocitkvcs dg kts dwg1 kt ks edr suppjcmcgtnry usc dgjy.

  Nt Ytnlc 0, cnib Jcnrgcr‛s Hdda  pnlc  pnlc ks `csklgc` td bcjp jcnrgcrs td idgsdjk`ntc ng` nppjy agdwjc`lc. Cnib scitkdg nssdikntc` wktb n Idrc nitkvkty stnrts nitkvkty stnrts wktb ng kgtrd`uitdry kgvcstklntkdg injjc` ‚Jct ‚ Jct‛‛s kgvcstklntc”, kgvcstklntc ”, wbkib ks ng dpcg-cg`c` qucstkdg td lct tbc jcnrgcrs tbkgakgl ng` kgvcstklntkgl. Pbcsc nrc detcg ‒jdw tbrcsbdj`, bklb ickjkgl‛ sd tbnt jcnrgcrs ing npprdnib tbc qucstkdg nt mngy jcvcjs. Pbks ks edjjdwc` hy n scrkcs de qucstkdgs ng`/dr nitkvktkcs td `cvcjdp prdhjcmsdjvkgl sakjjs ng` suppdrt jcnrgkgl tbrdulb `ksidvcry ng` `ksiusskdg. Gcw vdinhujnry ks cxpjnkgc`, ng` wbcrc pdsskhjc tbks ks `dgc uskgl kjjustrntkdgs ns wcjj ns tcxt kg dr`cr td bcjp vksunj jcnrgcrs ng` tbdsc wktb jdwcr jktcrniy jcvcjs. Bkgts ng` tkps prdvk`c `krcit suppdrt tbrdulbdut. K`cnjjy, tbc scsskdg sbduj` hc tnulbt uskgl tbc npprdprkntc Idrc nitkvkty kg nitkvkty kg tbc Pcnibcr Pcnibcr‛‛s Xcsduric wktb Xcsduric wktb tbc Jcnrgcr Jcnrgcr‛‛s Hdda  hckgl   hckgl usc` nt tbc cg` de tbc scsskdg, dr sct ns bdmcwdra, td idgsdjk`ntc idgsdjk`ntc jcnrgkgl.

  nssdikntc` Pbcrc ks lcgcrnjjy n `duhjc pnlc kg tbc Jcnrgcr‛s  edr cnib  edr nssdikntc` Idrc  Idrc nitkvkty kg nitkvkty  kg tbc Pcnibcr‛s XcsduricHdda  Xcsduric edr  edr Ytnlc 0. Pbc Pcnibcr‛s Xcsduric wkjj Xcsduric wkjj rcecr td tbc Jcnrgcr‛s Hdda  pnlc  pnlc hy tktjc ng` pnlc gumhcr, ng` tbc tktjc de tbc Idrc nitkvkty wkjj hc nt tbc hdttdm de tbc Jcnrgcr‛s Hdda  pnlc.  pnlc. Vjcnsc gdtc tbnt tbc Jcnrgcr‛s Hdda   `dcs gdt idvcr njj de tbc Inmhrk`lc dhocitkvcs dg kts dwg1 kt ks edr suppjcmcgtnry usc dgjy.

  Lnmcs Hdda  (prkgtc` hdda ng` I@-XDM)   Pbks rcsduric rcsduric ks idmpjctc idmpjctc kg kkts ts dwg rrklbt klbt ns n sduric de cglnlkgl, kgedrmntkvc mntbs lnmcs. Kt ks njsd n suppjcmcgtnry rcsduric td tbc

vkkk vkkk  

Kgtrd`uitkdg

Cnib ibnptcr kg tbc Pcnibcr‛s Xcsduric kgiju`cs   N ^ukia rcecrcgic scitkdg rcecrcgic scitkdg td jkst tbc tktjc de cnib de tbc Idrc nitkvktkcs  nitkvktkcs  idgtnkgc` wktbkg tbc ibnptcr. Kt prdvk`cs ng dutjkgc de tbc jcnrgkgl dutidmc(s) de cnib Idrc nitkvkty. nitkvkty. (Ycc pnlc vkk ng` jntcr kg tbks jkst, edr n rcmkg`cr de wbnt ks mcngt hy n Idrc nitkvkty.) nitkvkty.)   N jkst de tbc Dhocitkvcs  erdm tbc Inmhrk`lc Vrkmnry Mntbcmntkis Dhocitkvcs erdm iurrkiujum ernmcwdra tbnt nrc idvcrc` nirdss tbc ibnptcr ns n wbdjc. Vjcnsc gdtc tbnt tbks mcngs tbnt gdt njj de tbc jkstc` dhocitkvcs wkj wkjjj hc idvcrc` kg cnib de tbc ibnptcr‛s Idrc nitkvktkcs1 nitkvktkcs1 tbcy nrc idvcrc` wbcg tbc ibnptcr ks tnacg ns n wbdjc. Pbc dhocitkvcs nrc rcecrcgic` uskgl suhbcn`kgls erdm tbc ernmcwdra, edr cxnmpjc ‒ >N? Injiujntkdg (Mcgtnj  ( Mcgtnj strntclkcs)‛ strntclkcs )‛ ng` tbc id`c erdm tbc Yibcmc de \dra, edr cxnmpjc, ‒8Gi9‛.   Vjcnsc hc n nwnrc wnrc tbnt tbnt tbc idgtcgt idgtcgt de ng dhocitkvc dhocitkvc ks detcg spjkt spjkt nirdss nirdss `keecrcgt Idrc nitkvktkcs ng`/dr nitkvktkcs ng`/dr `keecrcgt ibnptcrs edr n jdlkinj prdlrcsskdg prdlr csskdg de jcnrgkgl ng` `cvcjdpmcgt. Vjcnsc hc nssurc` tbnt prdvk`c` ydu cvcgtunjjy idvcr njj de tbc Idrc nitkvktkcs nirdss nitkvktkcs nirdss tbc wbdjc Pcnibcr‛s Xcsduric, Xcsduric, ydu wkjj bnvc idvcrc` njj de tbc dhocitkvcs kg eujj. Kt sbduj` hc ijcnr erdm tbc gnturc de n Idrc nitkvkty wbcg nitkvkty wbcg pnrts de ng dhocitkvc bnvc gdt hccg eujjy idvcrc`. Edr cxnmpjc, n ibnptcr dg jcgltb

 

  Pcnibcr-jc` wbdjc ijnss nitkvkty Pbc mnkg gnrrntkvc rcprcscgts wdra td hc `dgc ns n wbdjc ijnss.   Pcnibcr-Jcnrgcr `ksiusskdg ‚Pcxt tbnt ks sct kg ktnjkis wktbkg `duhjc-qudtntkdg mnras rcprcscgts sullcstc` tcnibcr `knjdluc td kgstklntc Pcnibcr-Jcnrgcr `ksiiuskdg.”

wkjj jkst ‒Mcnsurc‛ dhocitkvcs tbnt njsd kgiju`c wcklbt, suib ns ‒>MK>‛ (Idmpnrc jcgltbs ng` wcklbts hy `krcit idmpnrksdg…) hut tbc wcklbt nspcit de tbc dhocitkvc wkjj gdt hc idvcrc` kg n ibnptcr dg jcgltb(!)1 tbnt pnrt de tbc dhocitkvc wkjj hc idvcrc` kg n ibnptcr dg wcklbt. Dr n ibnptcr ediusskgl dg ug`crstng`kgl tccg gumhcrs ns ‒tcg ng` sdmc mdrc‛ mklbt idvcr tbc nitkdg ‒rciktc gumhcrs kg dr`cr‛ hut dgjy up td 86 ng` tbcrcedrc dgjy pnrtknjjy idvcr dhocitkvc ‒>Gg>‛ (Xciktc gumhcrs kg dr`cr … erdm > td >66…)). Hut pjcnsc hc rcnssurc` tbnt, hy tbc cg` de tbc Pcnibcr‛s Xcsduric, Xcsduric, njj de dhocitkvcs >MK> ng` >Gg> wkjj bnvc hccg idvcrc` kg eujj1 ns wkjj njj dhocitkvcs. Pbc Yummnry Yummnry hujjctc`  hujjctc` jkst nt tbc cg` de cnib Idrc nitkvkty jksts nitkvkty jksts tbc jcnrgkgl dutidmc de tbc nitkvkty ng` ing n`` sdmc ijnrkty de idvcrnlc, idvcrnlc, ke rcqukrc`.   N jkst de acy Vrkdr jcnrgkgl tdpkis ks prdvk`c` td cgsurc jcnrgcrs nrc rcn`y td mdvc dg td tbc ibnptcr, ng` td rcmkg` tcnibcrs de tbc gcc` td hukj` dg prcvkdus jcnrgkgl. Udinhujnry edr  edr tbc ibnptcr ns n wbdjc ks jkstc`.   Kmpdrtngt ng`/dr gcw Udinhujnry \ktbkg tbc Idrc nitkvkty ktscje, rcjcvngt vdinhujnry wkjj hc rcpcntc` njdgl wktb n bcjpeuj `csirkptkdg td suppdrt tcnibkgl de gcw wdr`s. Pbc Idrc nitkvktkcs (wktbkg nitkvktkcs (wktbkg cnib ibnptcr) idjjcitkvcjy prdvk`c n idmprcbcgskvc tcnibkgl prdlrnmmc edr tbc wbdjc stnlc. Cnib Idrc nitkvkty kgiju`cs?   N jkst de rcqukrc` Xcsdurics Xcsdurics td  td inrry dut tbc nitkvkty. Pbks jkst kgiju`cs rcsdurics prdvk`c` ns pbdtdidpy mnstcrs wktbkg tbc Pcnibcr‛s Xcsduric prkgtc` hdda (kg`kintc` hy ‒(pxx)‛), ng` pbdtdidpy mnstcrs prdvk`c` dg tbc I@-XDM (kg`kintc` hy ‒(I@-XDM)‛), ns wcjj ns rcsdurics edug` kg tbc ijnssrddm dr nt bdmc. ‒(Dptkdgnj)‛ rcsdurics nrc tbdsc tbnt nrc rcqukrc` edr tbc nitkvktkcs jkstc` kg tbc ‒Mdrc ‒ Mdrc nitkvktkcs‛ nitkvktkcs‛ scitkdg ng` tbus nrc dptkdgnj.   N mnkg gnrrntkvc tbnt ks spjkt kgtd twd idjumgs. Pbc jcet-bng` (wk`cr) idjumg prdvk`cs kgstruitkdgs edr bdw td `cjkvcr tbc nitkvkty, sullcstkdgs edr `knjdluc td kgstklntc `ksiusskdgs, pdsskhjc rcspdgscs ng` dutidmcs, ns wcjj ns lcgcrnj suppdrt edr tcnibkgl tbc dhocitkvc. @keecrcgics kg edrmnttkgl edrmntt kgl kg tbks scitkdg k`cgtkey `keecrcgt typcs de kgtcrnitkvkty? kgtcrnitkvkty?

 

Jcnrgcr-Jcnrgcr Jcnrgcr-Jcnrgcr kgtcrnitkdg Lrdup ng` pnkr wdra hctwccg jcnrgcrs ks cgidurnlc` tbrdulbdut ng` ks kg`kintc` uskgl n lrcy pngcj hcbkg` tbc tcxt ng` n ibnglc kg edgt.

  Pbc rklbt-bng` (gnrrdw) idjumg prdvk`cs,   tbc vdin vdinhujnry hujnry pngcj   sk`c-gdtcs ng ng` ` cxnmpjcs   n Jdda dut edr! pngcj tbnt deecrs prnitkinj sullcstkdgs edr k`cgtkeykgl ng` n``rcsskgl idmmdg `ke ﬎iujtkcs ng` mksidgicptkdgs, ns wcjj ns bdw td spdt n`vngic` jcnrgcrs ng` k`cns edr cxtcgskdg tnsas td lkvc tbcm   ng Dppdrtugkty edr `kspjny pngcj `kspjny  pngcj td prdvk`c k`cns edr `kspjnys.   N Yummnry Yummnry nt  nt tbc cg` de cnib Idrc nitkvkty td nitkvkty td jkst tbc jcnrgkgl dutidmcs/cxpcitntkdgs edjjdwkgl tbc nitkvkty. Pbks ks niidmpngkc` hy n Ibcia up! scitkdg up! scitkdg tbnt prdvk`cs qukia- ﬎rc prdhkgl qucstkdgs usceuj edr edrmntkvc nsscssmcgt1 ng` n Gdtcs dg tbc Jcnrgcr‛s Hdda  scitkdg   scitkdg tbnt rcecrcgics tbc tktjc ng` pnlc gumhcr de tbc nssdikntc` Jcnrgcr‛s Hdda pnlc, ns wcjj ns n hrkce summnry de wbnt tbc pnlc kgvdjvcs.   N Mdrc nitkvktkcs scitkdg tbnt prdvk`cs sullcstkdgs edr eurtbcr nitkvktkcs1 tbcsc nrc gdt rcqukrc` td idvcr tbc dhocitkvcs ng` tbcrcedrc nrc dptkdgnj nitkvktkcs tbnt ing hc usc` edr rckgedricmcgt ng` `keecrcgtkntkdg. Pbc n``ktkdgnj nitkvktkcs mklbt kgiju`c n rcecrcgic td n lnmc kg tbc Lnmcs Hdda . [du nrc cgidurnlc` td njsd jdda dg tbc Inmhrk`lc Mntbs Vrdocit wchsktc Vrdocit  wchsktc td ﬎g` GXKIB nitkvktkcs jkgac` td tbc Inmhrk`lc dhocitkvcs. Pdlctbcr, tbcsc nitkvktkcs prdvk`c n wcnjtb de mntcrknj erdm wbkib tcnibcrs ing scjcit tbdsc mdst npprdprkntc td tbckr ikriumstngics hdtb kg ijnss ng` edr usc de bdmcwdra ke tbks ks sct.

Kgtrd`uitkdg

kx kx  

 

\c wduj` rcidmmcg` tbnt ydu wdra tbrdulb tbc ibnptcrs kg tbc dr`cr tbcy nppcnr kg tbks hdda ns ydu mklbt ﬎g` tbnt jntcr ibnptcrs hukj` dg agdwjc`lc erdm cnrjkcr kg tbc hdda. Ke pdsskhjc, wdra wktb idjjcnlucs ng` sbnrc k`cns ng` dvcr tkmc ydu wkjj eccj idg ﬎`cgt kg md`keykgl ng` n`nptkgl ydur pjngs.

Pcnibkgl npprdnibcs Jcnrgcrs bnvc `keecrcgt jcnrgkgl styjcs ng` tcnibcrs gcc` td nppcnj td njj tbcsc styjcs. [du wkjj ﬎g` rcecrcgics td lrdup wdra, wdrakgl kg pnkrs ng` wdrakgl kg`kvk`unjjy wktbkg tbcsc mntcrknjs. mntcrknjs.

Cxpjnkg mntbcmntkinj tbkgakgl (K tbkga tbnt . . . hcinusc .. .) @cvcjdp ug`crstng`kgl (K ug`cr stng` tbnt . . .) Ydjvc prdhjcms (K agdw tbnt . . . sd . . .) Cxpjnkg sdjutkdgs (Pbks ks bdw K edug` dut tbnt . . .) Nsa ng` ngswcr qucstkdgs (\bnt, wby, wby, bdw, wbcg, ke . . .) Oustkey ngswcrs (K tbkga tbks hcinusc . . .) Pbcrc ks n`vkic dg tbc I@-XDM I@-XDM nhdut tbc typcs de qucstkdgkgl ydu ing usc td lct ydur stu`cgts tnjakgl mntbs (^ucstkdgkgl ).

Pbc lrdupkgl `cpcg`s dg tbc nitkvkty ng` tbc pdkgt rcnibc` wktbkg n scrkcs de scsskdgs. Kt mny hc npprdprkntc td tcnib tbc wbdjc ijnss, edr cxnmpjc, nt tbc hclkggkgl de n scrkcs de scsskdgs wbcg cxpjnkgkgl, `cmdgstrntkgl dr nsakgl qucstkdgs. Netcr tbks kgktknj stnlc, jcnrgcrs detcg hcgc ﬎t erdm dppdrtugktkcs td `ksiuss ng` cxpjnkg tbckr tbdulbts td n pnrtgcr dr kg n lrdup. Yuib nitkvktkcs wbcrc jcnrgcrs nrc wdrakgl idjjnhdrntkvcjy idjjnhdrntkvcjy nrc bklbjklbtc` kg tbc mnkg gnrrntkvc ns `ctnkjc` kg tbc prcvkdus scitkdg. Bklb qunjkty tcnibkgl ks drnj, kgtcrnitkvc ng` jkvcjy ng` ks n twd-wn twd-wny y prdicss hctwccg tcnibcr ng` jcnrgc rs. Jcnrgcrs pjny ng nitkvc pnrt hy nsakgl ng` ngswcrkgl qucstkdgs, idgtrkhutkgl td `ksiusskdgs ng` cxpjnkgkgl ng` `cmdgstrntkgl tbckr mctbd`s td tbc rcst de tbc ijnss dr lrdup. Pcnibcrs gcc` td jkstcg ng` usc jcnrgcr k`cns td sbdw tbnt tbcsc nrc vnjuc`. Jcnrgcrs

Xcsdurics ing suppdrt, nsskst ng` cxtcg` jcnrgkgl. Pbc usc de rcsdurics suib ns Pcg ernmcs,  ernmcs, >66 squnrcs, squnrcs, gumhcr jkgcs, jkgcs, `klkt inr`s ng` inr`s ng` nrrdw inr`s  inr`s  ks prdmdtc` kg tbc Pcnibcr‛s Xcsduric. Xcsduric. Lnmcs prdvk`c n usceuj wny de rckgedrikgl sakjjs ng` prnitkskgl ng` idgsdjk`ntkgl k`cns. Jcnrgcrs lnkg idg﬎`cgic ng` nrc nhjc td cxpjdrc ng` `ksiuss mntbcmntkinj k`cns wbkjst `cvcjdpkgl tbckr mntbcmntkinj jnglunlc.

wkjj mnac crrdrs ke tbcy tnac rksas hut tbcsc nrc ng kmpdrtngt pnrt de tbc jcnrgkgl prdicss.

tbc injiujntdr ks gdt prdmdtc` ns n injiujntkdg tddj hcedrc Ytnlc 3.

Pnjakgl mntbcmntkis \c gcc` td cgidurnlc jcnrgcrs td spcna `urkgl n mntbs scsskdg kg dr`cr td?   idmmugkintc   cxpjnkg ng` try dut k`cns `cvcjdp idrrcit usc de mntbcmntkinj vdinhujnry   `cvcjdp mntbcmntkinj tbkgakgl.

x 

Kt ks kmpdrtngt tbnt jcnrgcrs `cvcjdp mntbcmntkinj jnglunlc ng` idmmugkintkdg kg dr`cr td (uskgl Hjddm‛s tnxdgdmy)?

Kgtrd`uitkdg

Xcsdurics, kgiju`kgl lnmcs

Injiujntdrs sbduj` hc usc` td bcjp jcnrgcrs ug`crstng` gumhcrs ng` tbc gumhcr systcm kgiju`kgl pjnic vnjuc ng` prdpcrtkcs de gumhcrs. gumhcrs. Bdwcvcr, Bdwcvcr,

GXKIB bnvc bnvc ircntc` ng nhug`ngic de cglnlkgl ng` wcjj-tbdulbt-dut mntbcmntkinj rcsdurics, wbkib bnvc bnvc hccg mnppc` td tbc Inmhrk`lc Vrkmnry sibcmc de wdra, ng` nrc nvnkjnhjc nvnkjnhjc erdm tbc Inmhrk`lc Vrkmnry Mntbs wchsktc. Mntbs  wchsktc. Pbckr kgtcrnitkvc ng` `dwgjdn`nhjc nitkvktkcs ing prdvk`c ng njtcrgntkvc jcnrgkgl styjc dr cgrkibmcgt edr sdmc de tbc idrc idgicpts.

 

>N 

> Gumhcrs ng` tbc gumhcr systcm

^ukia rcecrcgic Gumhcr

Idrc nitkvkty >.>? Xcn`kgl, wrktkgl ng` pnrtktkdgkgl gumhcrs (Jcnrgcr‛s hdda? p8)

Dr`crkgl ng` rdug`kgl

Xcn`kgl,wrktkgl ng` pnrtktkdgkgl gumhcrs Vnhjd bnstbcsc`klkt inr`s. Bcmnacs tbrcc-`klkt gumhcrswktb tbcinr`s. \rktc`dwg njj tbcgumhcrs bciduj` mnac.

Jcnrgcrs ug`crstng` tbc pjnic vnjuc de `klkts up td n edur-`klkt gumhcr ng` usc tbks agdwjc`lc td wrktc gumhcrs kg ﬎lurcs, wdr`s ng` cxpng`c` edrm.

2 

\rktccnib  rc` gumhcrkg ﬎lurcs, wdr`s ng` cxpng`c` edrm.   (n) > 6 6 6 8 6 6 6  6  9666 0 6 6 3 66 6 5 6 6 < 6 6 2 6 6 ; 6 6   > 66

8 66

  >6 >6  

 

Idrc nitkvkty >.8? Dr`crkgl, idmpnrkgl ng` rdug`kgl edur-`klkt gumhcrs (Jcnrgcr‛s hdda? p0) Jcnrgcrs idmpnrc gumhcrs dg n gumhcr jkgc uskgl tbc :, 7 ng` = gdtntkdg. Pbcy rdug` wbdjc gumhcrs td tbc gcnrcst >6 dr >66. Idrc nitkvkty >.9? Mujtkpjykgl ng` `kvk`kgl hy >6 ng` >66 (Jcnrgcr‛s hdda? p5) Jcnrgcrs prnitksc mujtkpjykgl ng` `kvk`kgl hy >6 ng` >66, kgiju`kgl kg tbc idgtcxt de mcnsurcs.

 >

>

86  

8

9 66 9 6 9

0 66 0 6 0

(h) >666 8 6 6 9 6 6 0 6 6   > 66   > 6  

>

8 66

366   5 66 3 6

5 6

3



den`klkt `ctcrmkgc` hyktspdsktkdg.

B

P

S

2

9

6

2666

02 td tbcgcnrcst

> Injiujntc. (n) 5<  >6 (`) 0>3  >6 (l) 0>66  >66

Xdug`kgl gumhcrs mnacs tbcm cnskcrtd usc.

>

B 6

P 6

Sgkt>N? Idrcnitkvkty>.> Xcn`kgl, wrktkglng`pnrtktkdgkglgumhcrs

6 36

> 6 9;

> >6 6

;26

5666

3366

(i) 9566  >66 (e) 93  >66 (k) 9;66  >66  

4

S

S 6

P

S 6

>6

>66

  4   >66

9 \rktctbcmksskgl `klkts. (n) 4   4   4    >6  8986 (i) 90 4  6  >6   4   4  5

(h) 05> 4   >6   4  5> (`) 9> 4   >6   4   4  56

0 \rktctbcmksskgl gumhcrs. (n) 4   >6  30 (i) 4   >66  5666

(h) 96<   4    966 (`) 2366  >6

2;;

Sgkt>N? Idrcnitkvkty>.8 Dr`crkgl, idmpnrkglng`rdug`kglgumhcrs

P 3

6

>66

0 \bkib de tbcscgumhcrs ks ijdscst td>6664

6

B 3

(h) 06  >6 (c) 936  >6 (b) 06

8 \bnt ks tbcmksskgl gumhcr4 3066

(`)

9 Xdug`tbcsc gumhcrs tdtbcgcnrcst >66. (n) > 6 56 (h)  < 88 3 (i)  0 52 6 (`)  > 66 <

>6

8

3366

B

P b

3 Bcrcnrcedur gumhcrinr`s.  

tbdusng`?ksnedur-`klktgumhcrtbntks tbdusng`?ksnedur-`klktgumhcrtbntks >6tkmcsjnrlcrtbngn bug`rc`. P b

;666

(i)

P b

tcg, ks 36.

cxpng`c`edrm?035<  0666  366  56  < cxpng`c`edrm?035<

9 \rktctbcscgumhcrs kg ﬎lurcs. pnrtktkdg?hrcnakglupngumhcrkgtdkts hrcnakglupngumhcrkgtdkts pnrtktkdg? (n) gkgct bdusng` n g` gkgc pnrts. Kg 296, t b c 9b n s n vnjucde9 tcgs(96). (h) edurtbdusng`ng` edrty pjnicvnjuc? pjnicvnjuc?tbcvnjuc tbcvnjuc

P

>6

 

\bnt bnppcgswbcg ydu prcsstbccqunjs( ) sklg4 Pryuskgl `keecrcgt stnrt gumhcrs. rs. @d ydu gdtkictbc snmctbkgl4

8 \bnt ks tbcgumhcrsbdwghy ngnrrdw dgcnib gumhcrjkgc4 (n) (h)

Udinhujnry

B

3

 



3  >6    … >>  >66    … >8 366  >6    …

296 td tbcgcnrcst bug`rc`, ks 266.

> \rktctbcscgumhcrs kgdr`cr de skzc,stnrtkgl wktb tbcsmnjjcst. >56 5 >5 56 >6 56 >5 55 (n) >65 5 2;; 6 ;6 6; ;6 ;6 2; ;; (h) ;62 6

`klkt?6,>,8,9,0,3,5,,8,9,0,3,5, (`) 6 ng` >66

Udinhujnry gcnrcst?tt dr dug` t d r dug`tdtbc gcnrcst? tbcgcnrcst bug`rc` ,jdda nt tbc tcgs `klkt ng`ke kt ks … B P S

Jct‛skgvcstk lntc Nbmc`wrktcs n jkstdeedur-`klkt wbdjc gumhcrs. Pbc`klktskg cnib gumhcrn`` up td 9.

Jct‛skgvcstklntc tklntc

5

(h) 3266  >6 (c) 326  >6

Sgkt>N? Idrcnitkvkty>.9 Mujtkpjykglng``kvk`kgl

(i) 3266  >66 (e) 3626  >6

hy>6 ng`>66

Vrkdr jcnrgkgl

gdtc tbnt jkstc` dhocitkvcs mk lbt dgjy hc pnrtknjjy idvcrc` wktbkg ngy lkvcg ibnptcr hut nrc idvcrc` eujjy nirdss tbc hdda wbcg tnacg ns Dhocitkvcs  ’  pjcnsc n wbdjc

Pbks ibnptcr hukj`s dg wdra `dgc kg Ytnlc 9 wbcrc jcnrgcrs wdrac` wktb gumhcrs up td >666 cxpjdrkgl pjnic vnjuc, dr`crkgl, idmpnrkgl ng` rdug`kgl.

0Gg>   0Gg8   0Gg9   0Gg; 0Gg>6 0Gg>> 0Gg>8

*

’ ’ ’ ’ ’ ’ ’

0Gi>3 ’ 0Gi83 ’   0Gg< ’

  0Vs0 ’   0Vs3 ’   0Vs; ’ 0Mj8 ’

>N? Gumhcrs ng` tbc gumhcr systcm Xcn` ng` wrktc gumhcrs up td >6 666. Idugt dg ng` hnia kg dgcs, tcgs, bug`rc`s ng` tbdusng`s erdm edur-`klkt gumhcrs. Sg`crstng` wbnt cnib `klkt rcprcscgts kg n tbrcc- dr edur-`klkt gumhcr ng` pnrtktkdg kgtd tbdusng`s, bug`rc`s, tcgs ng` ugkts. Xdug` tbrcc- ng` edur-`klkt gumhcrs td tbc gcnrcst >6 dr >66. Vdsktkdg niiurntcjy niiurntcjy gumhcrs up td >666 dg ng cmpty gumhcr jkgc dr jkgc mnrac` dee kg mujtkpjcs de >6 dr >66. Cstkmntc wbcrc tbrcc- ng` edur-`klkt gumhcrs jkc dg cmpty 6’>666 dr 6’>6 666 gumhcr jkgcs. Idmpnrc pnkrs de tbrcc-`klkt dr edur-`klkt gumhcrs, uskgl tbc 7 ng` : sklgs, ng` ﬎g` n gumhcr kg hctwccg cnib pnkr. >N? Injiujntkdg ( Mcgtnj Mcgtnj strntclkcs ) Sg`crstng` tbc ceecit de mujtkpjykgl ng` `kvk`kgl tbrcc-`klkt tbrcc-`klkt gumhcrs hy >6. >N? Injiujntkdg ( Mujtkpjkintkdg Mujtkpjkintkdg ng` `kvkskdg ) Sg`crstng` tbnt mujtkpjkintkdg ng` `kvkskdg nrc tbc kgvcrsc eugitkdg de cnib dtbcr. 8N? Gumhcrs ng` tbc gumhcr systcm Mujtkpjy ng` `kvk`c tbrcc-`klkt gumhcrs hy >6 (wbdjc gumhcr ngswcrs) ng` ug`crstng` tbc ceecit1 hclkg td mujtkpjy gumhcrs hy >66 ng` pcredrm rcjntc` `kvkskdgs. >N? Vrdhjcm-sdjvkgl ( Sskgl Sskgl ug`crstng`kgl ng` strntclkcs kg sdjvkgl prdhjcms ) Cxpjdrc ng` sdjvc gumhcr prdhjcms ng` puzzjcs, c.l. jdlki prdhjcms. Ssc dr`crc` jksts ng` tnhjcs td bcjp td sdjvc prdhjcms systcmntkinjjy. Cxpjnkg mctbd`s ng` rcnsdgkgl drnjjy ng` kg wrktkgl1 mnac bypdtbcscs ng` tcst tbcm dut. >H? Mcnsurc (Jcgltb, mnss ng` inpnikty) Agdw ng` usc tbc rcjntkdgsbkps hctwccg enmkjknr ugkts de jcgltb (m, im ng` mm). *edr GXKIB nitkvktkcs mnppc` td tbc Inmhrk`lc Vrkmnry dhocitkvcs,

 Udinhujnry   Udin hujnry  `klkt

cxpng`c` edrm

pjcnsc vkskt www.ikc.drl.u www.ikc.drl.ua/inmhrk`lcprkmnrymntbs a/inmhrk`lcprkmnrymntbs

pnrtktkdg

pjnic vnjuc

tbdusng`

Inmhrk`lc Vrkmnry Mntbcmntkis 0 © Inmhrk`lc Sgkvcrskty Vrcss 86>0

rdug` td tbc gcnrcst Sgkt >N 

> 

 

Idrc nitkvkty >.>? Xcn`kgl, wrktkgl ng` pnrtktkdgkgl gumhcrs

JH? p8

Xcsdurics? Gumhcrs njj nrdug` us pbdtdidpy us pbdtdidpy mnstcr (p>>)1 jnrlc vcrskdg edr ijnss `kspjny. Vjnic vnjuc ibnrt? >’;666 pbdtdidpy >’;666 pbdtdidpy mnstcr (I@-XDM)1 jnrlc vcrskdg edr ijnss `kspjny. `kspjny. (Dptkdgnj? 6’; `kic, 6’; spkggcrs dr 6’; `klkt inr`s (I@-X (I@-XDM).) DM).) Jcnrgcrs sbduj` rcidlgksc tbc pjnic vnjuc de tbc `klkts kg >69 ns dgc ‒bug`rc`s‛, zcrd ‒tcgs‛ ng` tbrcc ‒dgcs/ugkts‛ erdm Ytnlc 9 (Sgkt >N, ibnptcr >). Xcmkg` tbcm bdw dur gumhcr systcm ks hnsc` dg >6? tcg jdts de dgcs/ugkts mnacs >61 tcg jdts de tcg mnacs >66. Nsa, ‚\bnt ‚\bnt `d ydu tbkga tcg jdts de dgc bug`rc` mnacs4 ” Cjkikt tbnt tbks mnacs dgc tbdusng`, ng` tbks ks n eurtbcr idjumg kg tbc pjnic vnjuc tnhjc. Cxpjnkg tbnt tbcrc nrc `keecrcgt wnys wc ing `csirkhc n gumhcr uskgl pjnic vnjuc. Edr cxnmpjc, kg >296 wc ing sny tbcrc ks dgc ‒tbdusng`‛, cklbt ‒bug`rc`s‛, 9 ‒tcgs‛ ng` zcrd ‒dgcs‛. Dr wc iduj` sny tbnt nrc >2 ‒bug`rc`s‛, tbrcc ‒tcgs‛ ng` zcrd ‒dgcs‛. Dr, dgc ‒tbdusng`‛ ng` 29 ‒tcgs‛. Ykmkjnrjy, scvcg bug`rc`s (6>6) ks tbc snmc ns >6> tcgs. Lkvc jcnrgcrs sdmc edur-`klkt gumhcrs ng` nsa tbcm td idugt dg ng` hnia kg dgcs, tcgs, bug`rc`s ng` tbdusng`s uskgl tbc pjnic vnjuc tnhjc td bcjp tbcm. Pbcg nsa qucstkdgs suib ns, ‚ \bnt ks > jcss tbng 06664 ” (Ngswcr? 9;;;) ‚  ‚\bnt \bnt ks tcg mdrc tbng 80354 ” (8055), cgidurnlkgl tbcm td tbkga de tbc pjnic vnjuc de cnib `klkt ns tbcy `d sd.

Xcn`kgl gumhcrs @kspjny tbc Gumhcrs njj nrdug` us pbdtdidpy us  pbdtdidpy mnstcr. Jcnrgcrs `ksiuss kg pnkrs wbnt tbcy gdtkic. Xcmkg` jcnrgcrs nhdut pjnic vnjuc hy pdkgtkgl td tbc pkiturc de tbc inr gumhcr pjntc

8 

 Udinhujnry   Udinhujnry  `klkt? 6, >, 8, 9, 0, 3, 5, 296 tbc 9 ks wdrtb 9

tcgs (96). Pb >

B 2

P 9

S 6

Cnib pjnic vnjuc ing dgjy idgtnkg n skgljc `klkt1 ke tbcrc nrc >> ugkts, tbcg n > ks pjnic` kg ‒S‛ ng` tbc dtbcr > ks inrrkc` dvcr td ‒P‛. Edr wbdjc gumhcrs, ydu must njwnys bnvc n `klkt kg n smnjjcr pjnic vnjuc td tbc rklbt kg dr`cr td sbdw tbc skzc de tbc gumhcr1 zcrd ks tbc pjnicbdj`cr. pnrtktkdg? hrcnakgl up n gumhcr kgtd p nrts. cxpng`c` edrm? wbnt ydu lct wbcg ydu pnrtktkdg n

(N93< GGM), ng` rcn`kgl dut tbcsc idmmcgts mn`c hy tbrcc jcnrgcrs? (>) ‚K ‚ K tbkga tbc 9 mcngs tbrcc bug`rc` .”1 .”1 (8) ‚K ‚K tbkga tbc vnjuc de tbc `klkt 9 dg tbc gumhcr pjntc ks tbrcc .”1 (9) ‚K ‚K `dg‛t tbkga tbc 9 bns n pjnic vnjuc tbnt ks kmpdrtngt. Kt‛s oust n gumhcr kg n jkst de jcttcrs ng` gumhcrs.” gumhcrs .” @ksiuss tbcsc idmmcgts wktb tbc ijnss. Cgsurc tbnt jcnrgcrs ug`crstng` tbnt lrdups de `klkts ing bnvc `keecrcgt purpdscs. Edr Edr cxnmpjc, pjnic vnjuc ks kmpdrtngt wbcg rcidr`kgl n sidrc de 9 dr 96, hut ks krrcjcvngt kg n tc jcpbdgc gumhcr.

gumhcr niidr`kgl td pjnic vnjucs, c.l. 035< = 0666 + 366 + 56 + < tbdusng`? n edur-`klkt gumhcr tbnt ks >6 tkmcs jdglcr tbng n bug`rc`.

Xcn`kgl,, wrktkgl ng` pnrtktkdgkgl gumhcrs Xcn`kgl

Dppdrtugkty edr `kspjny

Cxpjnkg tbnt n pnjkg`rdmki gumhcr rcn`s tbc snmc wbcg wrkttcg edrwnr`s dr hniawnr`s. Ybdw bdw 909 ks n pnjkg`rdmki gumhcr ng` nsa jcnrgcrs td lkvc dtbcr cxnmpjcs.

@kspjny cxnmpjcs de gumhcrs erdm gcwspnpcr nrtkijcs, mnlnzkgcs ng` pbdtdlrnpbs. pbdtdlrnpbs.

Sgkt >N 

> Gumhcrs ng` tbc gumhcr systcm

 

Pnac cxnmpjcs, suib ns ;>0, 3663). 3663).

Yummnry    Jcnrgcrs nrc nhjc td rcn` ng` wrktc gumhcrs erdm > td >6 666. Pbcy ug`crstng` tbc vnjuc de cnib `klkt ng` usc tbks agdwjc`lc td wrktc n gumhcr kg cxpng`c` gdtntkdg. Edr cxnmpjc? ;2.”, ‚Pbc `klkt.”, ‚Pbc >8.” Dtbcr jcnrgcrs nttcmpt td lucss tbc gumhcr. Pbc jcnrgcr wbd lkvcs tbc idrrcit ngswcr lnkgs dgc pdkgt ng` `c﬎gcs tbc gcxt gumhcr gumhcr.. Njtcrgntkvcjy, dgc jcnrgcr ibddscs n gumhcr ng` kt ks k`cgtk﬎c` hy dtbcr jcnrgcrs wbd nsa qucstkdgs wktb n ycs/gd ngswcr ngswcr.. @cmdgstrntc bdw jcnrgcrs mklbt usc qucstkdgs td lucss tbc gumhcr gumhcr.. Edr cxnmpjc? ‚Ks ‚Ks tbc tcgs `klkt jcss tbng skx4 ”

Idrc nitkvkty >.>? Xcn`kgl, wrktkgl ng` pnrtktkdgkgl gumhcrs

 

Idrc nitkvkty >.8? Dr`crkgl, idmpnrkgl ng` rdug`kgl edur-`klkt gumhcrs

JH? p0

Xcsdurics? Gumhcr jkgc 6’>666 pbdtdidpy mnstcr (I@-XDM). (I@-XDM). (Dptkdgnj? 6’; `kic1 6’; spkggcrs dr 6’; `klkt i nr`s (I@-XDM)1 (I@-XDM)1 Udjingdcs inr`s pbdtdidpy mnstcr (I@-XDM).) (I@-XDM).)

Lcgcrntc n scqucgic (wbdjc ijnss mcgtnj nitkvkty) Dgc jcnrgcr ks lkvcg n edur-`klkt stnrtkgl gumhcr ng` tbc rcst de tbc ijnss idgtkgucs tbc scqucgic edjjdwkgl n lkvcg rujc. Edr cxnmpjc, n`` dr suhtrnit >, >6, >66 dr >666.

Uksunjksntkdg (wbdjc ijnss nitkvkty) Nsa jcnrgcrs td ijdsc tbckr cycs ng` vksunjksc n hjnga gumhcr jkgc. ‚Sskgl tbc `klkts >, 8, 9 ng` 0, wbnt ks tbc jnrlcst gumhcr ydu ing mnac4 Uksunjksc tbks dg  ydur gumhcr jkgc.” jkgc.” (Ngswcr? 098>)   ‚\bnt ks tbc smnjjcst t wd-`klkt gumhcr ydu ing mnac4 Uksunjksc tbks dg ydur gumhcr jkgc. jkgc.” ” (Ngswcr? >890)   ‚Ssc tbc edur `klkts td mnac n gumhcr kg hctwccg ydur twd gumhcrs. Ks tbks gumhcr ijdscr td tbc jnrlcst gumhcr dr tbc smnjjcst gumhcr4 ”

 Udin  Udinhujnry  hujnry  rdug` td tbc gcnrcst?  td rdug` n gumhcr td tbc gcnrcst bug`rc`  gcnrcst  bug`rc`  jdda  jdda nt tbc tcgs `klkt?

  ke kt ks jcss tbng 3 rdug` `dwg   ke kt ks 3 dr mdrc rdug` up. (Jdda nt tbc pjnic vnjuc td tbc rklbt wbcg `ctcrmkgkgl td rdug` up dr `dwg1 edr cxnmpjc, ke rdug`kgl td tbc gcnrcst >6, ydu wduj` jdda nt tbc ugkt pjnic vnjuc.)

Cxnmpjc?

Jcnrgcrs tbcg `rnw n gumhcr jkgc, mnra tbckr gumhcrs dg kt ng` idmpnrc tbckr rcsujts wktb n pnrtgcr‛s pnrtgcr‛ s hy stntkgl ke tbckr gumhcr ks lrcntcr tbng, jcss tbng dr cqunj td tbckr pnrtgcr‛s. Xcpcnt wktb dtbcr `klkts.

>890

980> gcnrcr td jnrlcr gumhcr

098>

Xdug`kgl Ibnjjcglc pnkrs de jcnrgcrs td `ksiuss wbkib de tbcsc gumhcrs lkvcs 36 wbcg rdug`c` td tbc gcnrcst >6? 06, 30, 3 dg tbks gumhcr jkgc4 ”  ‚Ks tbc gumhcr gcnrcr td 9666 dr 06664 ”  ‚Bdw `d ydu agdw4 ” Cxpjnkg tbnt ke 980> wns rdug`c` td tbc gcnrcst tbdusng` tbc ngswcr wduj` hc 9666 hcinusc 980> ks gcnrcr td 9666 tbng td 0666. Cxnmpjc? Ybdw ngdtbcr gumhcr jkgc mnrac` dgjy erdm 9866 td 9966. ‚\bnt gumhcr ks bnjewny hctwccg 9866 ng` 99664 ” (Ngswcr? 9836)

9666

980>

0666

9 

0 

Sgkt >N 

> Gumhcrs ng` tbc gumhcr systcm

 

‚\bcrc wduj` ydu pjnic 980> dg tbc gumhcr jkgc4 \by4 ” (Ngswcr? jcss tbng tbc bnjewny mnra hcinusc 980> : 9836) ‚Ks kt gcnrcr td 9866 dr 99664 ” (Ngswcr? 9866)

Cxnmpjc? 9866

 

9966

Cxpjnkg tbnt ke 980> wns rdug`c` td tbc gcnrcst bug`rc` tbc ngswcr wduj` hc 9866 hcinusc 980> ks gcnrcr td 9866 tbng td 9966. Nsa?  Jdda nt tbc tcgs `klkt.   Ke kt ks jcss tbng 3 rdug` `dwg.   Ke kt ks 3 dr mdrc rdug` up.

Cxnmpjc? td tbc gcnrcst bug`rc` 9866

980>

9966

Xcpcnt, td rdug` 980> td tbc gcnrcst tcg.

  Jdda nt tbc ugkts `klkt.   Ke kt ks jcss tbng 3 rdug` `dwg.   Ke kt ks 3 dr mdrc rdug` up. (Ngswcr? 9866)

Cxnmpjc? td tbc gcnrcst tcg 980>

9806

Vjnikgl gumhcrs dg n gumhcr jkgc @kspjny tbc Gumhcr jkgc? 6’>666 pbdtdidpy mnstcr edr tbc wbdjc ijnss td scc. Pbc jkgc ks mnrac` dee kg mujtkpjcs de >66, (scc rklbt). rklbt). Cxpjnkg tbnt sdmctkmcs wc gcc` td pjnic gumhcrs mdrc niiurntcjy tbng wc `k` cnrjkcr.   ‚\bcrc wduj` K pjnic 936 dg tbc gumhcr jkgc4 Bdw `k` ydu `cik`c4 ” (Ngswcr kgiju`cs ‒cstkmntkgl‛ bnjewny hctwccg 966 ng` 066)   ‚\bcrc wduj` K pjnic ;864 Bdw `k` ydu `cik`c4 ” (Ngswcr kgiju`cs ijdsc td ;66)   ‚Bdw ing K wrktc n stntcmcgt td sbdw tbnt 936 ks jcss tbng ;864 ” (Ngswcr? 936 : ;86)

Cxnmpjc? 6

>666

Yummnry    Jcnrgcrs idg﬎`cgtjy usc mnrac` ng` ugmnrac` gumhcr jkgcs td jdintc tbc rcjntkvc

Ibcia up!

pdsktkdgs de gumhcrs.   Pbcy idmpnrc pnkrs de gumhcrs uskgl tbc gdtntkdg :, 7 dr =.   \bcg gcicssnry, tbcy rdug` ngy wbdjc gumhcr gumhcr td tbc gcnrcst >6 dr tbc gcnrcst >66.

  ‚K rdug`c` n gumhcr td tbc gcnrcst >6. Pbc ngswcr ks

Gdtcs dg tbc Jcnrgcr‛s Hdda Dr`crkgl ng` rdug`kgl (p0)? hclkgs wktb ng kgvcstklntkdg kgv kgvdjvkgl djvkgl dr`crkgl edur-`klkt

tbc mntib. \bnt ks tbc smnjjcst gumhcr tbnt iduj` bnvc nttcg`c`4 \bnt ks tbc jnrlcst gumhcr4 ”   ‚Bdw iduj` ydu ibddsc gumhcrs td mnac tbks gumhcr scgtcgic idrrcit4  + ∅ : >6”

gumhcrs wktbng` `klkts n``7up td tbrcc. Kt tbcg kgiju`cs qucstkdgs tbnt ecnturc rdug`kgl, gumhcr jkgcs tbctbnt : ng` symhdjs.

296. \bnt gumhcr iduj` K bnvc stnrtc` wktb4 ”

  ‚Pbc gcws rcpdrt stntc` tbnt >366 pcdpjc nttcg`c`

Idrc nitkvkty >.8? Dr`crkgl, idmpnrkgl ng` rdug`kgl edur-`klkt gumhcrs

 

Mdrc nitkvktkcs Hdxcs nlnkg (ijnss dr lrdups) [du wkjj gcc` n 6’; `kic, 6’; spkggcr dr sct de 6’; `klkt inr`s (I@-XDM)1 tcmpjn tcmpjntcs tcs edr tbc `kic ng` s pkggcr ing hc edug` dg tbc I@-XDM. I@-XDM.

Cnib jcnrgcr `rnws n lrk` kgiju`kgl tbc ‒jcss tbng‛ symhdj, ns sbdwg dg tbc rklbt. Ssc tbc `kic, spkggcr dr `klkt inr`s td lcgcrntc cklbt `klkts. Jcnrgcrs wrktc cnib `klkt kg tbckr lrk`, nkmkgl td mnac n truc stntcmcgt. Dgic tbc `klkt ks pjnic`, kts pdsktkdg inggdt hc ibnglc`. \bcg idmpjctc, nsa?

  ‚Ks ydur stntcmcgt truc4 Bdw `d ydu agdw4 ”   ‚Bdw `k` ydu `cik`c wbcrc td put tbc `klkts4 ”   ‚\bkib spnics wcrc tbc mdst kmpdrtngt4 \by4 ” Udjingd `ntnhnsc (pnkrs) [du wkjj gcc` tbc Udjingd inr`s pbdtdidpy inr`s pbdtdidpy mnstcr (I@-XDM)1 dgc pcr pnkr de jcnrgcrs. Pbc tcnibcr dr jcnrgcrs prcpnrc, tbcg ngswcr, ngswcr, qucstkdgs rcjntc` td dr`crkgl, idmpnrkgl ng` rdug`kgl gumhcrs. gumhcrs. Edr cxnmpjc?   ‚Gnmc tbc vdjingdcs tbnt nrc 0666 mctrcs bklb wbcg rdug`c` td tbc gcnrcst >66 mctrcs. mctrcs.” (Ngswcr? Idjkmn, Eucld, Euok, Mnugn Jdn, Ycmcru, Pnoumujid) Jkst kg dr`cr de bcklbt, jdwcst jdwcst ﬎  ﬎rst, rst, njj tbc vdjingdcs jcss tbng >666 mctrcs bklb. bklb.” ”   ‚(Ngswcr? Yurtscy >5; m, Arnantdn 2>9 m, Ytrdmhdjk ;85 m)

  ‚Vjnic tbc vdjingdcs dg n gumhcr jkgc niidr`kgl td tbckr bcklbt. bcklbt.” ”

Lnmcs Hdda (KYHG ;>6) ks n lnmc edr twd pjnycrs pjnycrs.. Kt ing hc usc` td prnitksc dr`crkgl edur-`klkt gumhcrs gumhcrs..

3 

5 

 

Sgkt >N 

> Gumhcrs ng` tbc gumhcr systcm

Hjnga pnlc

 

Idrc nitkvkty >.9? Mujtkpjykgl ng` `kvk`kgl hy >6 ng` >66

JH? p5

Xcsdurics? Vjnic vnjuc ibnrt? >’;666 pbdtdidpy >’;666 pbdtdidpy mnstc r (I@-XDM)1 jnrlc vcrskdg edr ijnss `kspjny `kspjny.. \bdjc-gumhcr sjk`cr pbdtdidpy sjk`cr pbdtdidpy mnstcr ng` npprdxkmntcjy >6 sjk`cr >6 sjk`cr strkps (p>8 strkps  (p>8 td >0) edr cnib jcnrgcr ng` n jnrlc vcrskdg edr `cmdgstrntkdg. (Dptkdgnj? >6s ng` >66s >66s  `dmkgdcs `dmkgdcs (I@-XDM).)  (I@-XDM).)

Cxnmpjc? 6 ng` >66 Ybdw tbc stntcmcgt ‒66 im = > m (n enit td hc mcmdrksc`).

Pb

@kspjny tbc Vjnic vnjuc ibnrt? >’;666 pbdtdidpy >’;666 pbdtdidpy mnstcr ng` nsa, ‚\bnt ‚\bnt ks < Ü >64 ” Pnac ngswcrs, ng` tbcg `cmdgstrntc bdw td md`cj tbks dg tbc ibnrt hy pdkgtkgl td 6 Ü >6.

6

6

6

0

6

>

6 Y

‚\bnt ks tbc mksskgl gumhcr kg tbks gumhcr scgtcgic? < Ü  = 66) 0

>

6

Md`cj tbcsc injiujntkdgs dg n pjnic vnjuc lrk` sd jcnrgcrs ﬎g` dut tbnt td mujtkpjy hy >66, tbc `klkts mdvc twd pjnics td tbc jcet. @cmdgstrntc bdw tbc sjk`cr ing hc usc` td md`cj 0> Ü >6  >66 6 hy mdvkgl `klkts twd pjnics td tbc jcet. 0

 

>

6

6

0

>

6

6

N 

> Gumhcrs ng` tbc gumhcr systcm

 

Xcturg td tbc qucst kdg pdsc` cnrjkcr? 6 ≦ >5.386.  >5.386.  

@kvk`kgl hy >6 ng` >66 Ybdw n skmkjnr mksskgl gumhcr prdhjcm? 3966 icgtkmctrcs =  mctrcs. Nsa? ‚Bdw ing K wdra dut tbc ngswcr4 ” Cstnhjksb tbnt ydu must `kvk`c 3966 hy >66. Jcnrgcrs usc tbckr sjk`crs td ‒ug`d‛ tbc prdicss de mujtkpjkintkdg td ﬎g` tbc ngswcr ngswcr.. Xcmkg` jcnrgcrs tbnt mujtkpjkintkdg ng` `kvkskdg nrc kgvcrsc dpcrntkdgs, dgc ug`dcs tbc ceecit de tbc dtbcr. Jcnrgcrs wdra dg tbc gcxt nitkvkty uskgl tbckr sjk`crs. Ybdw tbcsc skx gumhcrs? 0

06

066

0666

>6

>66

Nsa jcnrgcrs td ibddsc tbrcc gumhcrs td mnac n mujtkpjkintkdg stntcmcgt? Nsa tbcm td usc tbc  snmc tbrcc gumhcrs td mnac n `kvkskdg stntcmcgt?

 Ü  =   ü  = 

Md`cj ngswcrs. Xcpcnt wktb `keecrcgt gumhcrs.

Yummnry    Jcnrgcrs usc n sjk`cr td md`cj mujtkpjykgl ng` `kvk`kgl wbdjc gumhcrs hy >6. Pbcy hclkg td mujtkpjy ng` `kvk`c hy >66. Jcnrgcrs hclkg td pcredrm injiujntkdgs mcgtnjjy mcgtnjjy.. Pbcy ug`crstng` tbnt mujtkpjkintkdg ng` `kvkskdg nrc kgvcrsc dpcrntkdgs. Gdtcs dg tbc Jcnrgcr‛s Hdda Mujtkpjykgl ng` `kvk`kgl  (p5)?  (p5)? jcnrgcrs prnitksc mujtkpjykgl ng` `kvk`kgl hy >6 ng` >66, kgiju`kgl kg tbc idgtcxt de mcnsurcs. Vjcnsc gdtc tbnt tbc k`cn de tbc ‚ Jct‛s kgvcstklntc” kgvcstklntc” ks tbnt rcpcntc`jy prcsskgl tbc = sklg wkjj mujtkpjy tbc prcvkdus ngswcr hy >6 (dr >66 c ti) cnib tkmc. Yd, jcnrgcrs sbduj` scc tbc edjjdwkgl pnttcrg?

3 Ü >6 = 36, 366, 3666, 36666 >> Ü >66 = >>66, >>666, >>6666 >8366 ü >6 = >836, >83, >8,3 Vjcnsc gdtc tbnt kg dr`cr edr tbks td wdra dg sdmc injiujntdrs, tbc jcnrgcrs wduj` gcc` td cgtcr

Ibcia up!   ‚Ekjj kg tbc mksskgl gumhcrs.” gumhcrs.”      = 5626 = 52 = TTTT icgtkmctrcs   ‚562 Idpy ng` idmpjctc.” ü idmpjctc.”  09>6mctrcs      

;6 mctrcs = TTTT icgtkmctrcs 66 icgtkmctrcs = TTTT mctrcs >366 icgtkmctrcs = TTTT mctrcs

tbc gumhcrs kg tbc dppdsktc dr`cr k.c., >6 Ü 3 = , rntbcr tbng 3 Ü >6 =.

Idrc nitkvkty >.9? Mujtkpjykgl ng` `kvk`kgl hy >6 ng` >66

 

Mdrc nitkvktkcs @dmkgdcs (pnkrs) [du wkjj gcc` >2 `dmkgdcs erdm tbc @dmkgdcs  pbdtdidpy mnstcr (I@-XDM)1 pcr pnkr de jcnrgcrs. ([du mklbt njsd wngt td ircntc n jnrlcr vcrskdg @dmkgdcs pbdtdidpy edr `cmdgstrntkdg.) Nttnib dr `rnw n jnrlc sct de `dmkgdcs dg tbc hdnr` kg n rng`dm wny. Vjnic dgc `dmkgd kg tbc icgtrc de tbc hdnr` ng` nsa, ‚\bkib `dmkgd ﬎ `dmkgd  ﬎ts ts dg tbks cg`4 ” Pbc idrrcit `dmkgd ks pjnic` kg pdsktkdg. 366

6

66

8    3    6    6  

Pbc lnmc idgtkgucs ugtkj tbc `dmkgdcs edrm n idmpjctc rkgl.

Lnmcs Hdda (KYHG ;>6), Bcxnjkgcs  (p>), n lnmc de strntcly edr twd pjnycrs, pjnycrs, kgvdjvcs rcinjj de gumhcr enits ng` mujtkpjkintkdg hy >6 ng` >66.

; 

>6  >6 

Sgkt >N 

> Gumhcrs ng` tbc gumhcr systcm

 

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Drklkgnj mntcrknj © Inmhrk`lc Sgkvcrskty Vrcss, 86>0

 

\bdjc-gumhcr sjk`cr Pd mnac tbc sjk`cr, ydu wkjj gcc`?        

sjk`cr sjccvc (scc pnlc >9) sikssdrs stkiay tnpc sjk`cr strkp (scc pnlc >0).

Kgstruitkdgs

>. Pd mnac tbc sjccvc, iut njdgl tbc `nsbc` jkgcs td ircntc ircntc edur squnrc ‒wkg`dws ‒wkg`dws‛.‛. 8. Edj` tbc sjccvc kgwnr`s njdgl tbc sdjk` jkgcs, sd tbnt ydu ircntc n tdp ﬈np, ng` tbc zcrds sbdw tbrdulb tbc wkg`dws. 9. Ytkia `dwg tbc ﬈np td tbc hnia de tbc sjk`c sjk`crr sjccvc. 0. Ekt tbc sjk`cr strkp kgtd tbc sjccvc, cgsurkgl tbnt kt sjk`cs erccjy. 3. \bcg uskgl tbc sjk`cr, njwnys njwnys pjnic kt kg tbc stnrt pdsktkdg, wktb tbc idrgcr mnrac` mnrac` ‒Y‛ bk``cg hcbkg` tbc  eurtbcst squnrc td tbc rklbt rklbt dg tbc sjccvc. 6

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Drklkgnj mntcrknj © Inmhrk`lc Sgkvcrskty Vrcss, 86>0

 

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8 N``ktkdg ng` suhtrnitkdg (>)

^ukia rcecrcgic Idrc nitkvkty 8.>? N``ktkdg (>) (Jcnrgcr‛s hdda? p2) Jcnrgcrs prnitksc n``kgl tbrcc ng` edur smnjj gumhcrs uskgl gumhcr pnkrs. Pbcy prnitksc n``kgl dg kg tcgs, tbcg ugkts ng` vkic vcrsn. Pbcy n`` twd-`klkt gumhcrs uskgl tbckr ibdscg mctbd` ng` cxpjnkg tbckr mctbd`s. Idrc nitkvkty 8.8? Yuhtrnitkdg (>) (Jcnrgcr‛s hdda? p>6) Jcnrgcrs prnitksc suhtrnitkdg de twd-`klkt gumhcrs hy idugtkgl hnia, ﬎g`kgl tbc `keecrcgic ng` dtbcr mctbd`s. Pbcy nrc cgidurnlc` td ibddsc tbckr dwg mctbd` ng` cxpjnkg kt. Idrc nitkvkty 8.9? Vnrtktkdgkgl td n`` ng` suhtrnit (Jcnrgcr‛s hdda? p>8) Jcnrgcrs prnitksc n``ktkdg ng` suhtrnitkdg de twd-`klkt ng` tbrcc-`klkt gumhcrs uskgl pnrtktkdgkgl. Pbcy sdjvc wdr` prdhjcms uskgl tbc mctbd` de tbckr ibdkic.

N``ktkdg (>)

Yuhtrnitkdg(>)

Udinhujnry

Jct‛skgvcstklntc

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(n) 028  8>9  

Y d j v c 58’32hy ‒﬎ g`kgltbc`keecrcgic‛ hctwccgtbctwd gumhcrs.

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 ;66



;66 ksnmujtkpjcde>6.Jddaedr twdgumhcrstbntn``tdmnacn mujtkpjcde>6.\cing`dtbkshyjddakgledr gumhcr pnkrstd>6 kg tbc ugkts`klkts.Pbcgibddscnmctbd`tdn`` tbctwdgumhcrstdlctbcr.

Bdw muib mdrcks 4  tbng 4 4

> Ibddscnmctbd` tdsdjvctbcsc n``ktkdgprdhjcms.

(i) >  >>  ;  0   (c) >9  8  >  3  

 882



059 988 2>0 053

Bdw mngy mdrcks 4  tbng 4 4 Bdw mngy ecwcrks 4  tbng 4 4

(n)  (h)  (i)   (`) 



Udinhujnry Ydmcwdr`s tbnt wcuscedr suhtrnitkdg?suhtrnit,suhtrnitkdg, suhtrnitkdg? suhtrnit,suhtrnitkdg, tnac,tnacnwny,mkgus,`circnsc, ecwcr,jcnvc,`keecrcgic. Bdw mngy nrcjcet4 Bdw mngy nrcjcet dvcr4

(h)  >8

>8

4

 

4

 

4

(`) 8  50;  

4

(c) 958  20>  

4

(e) 3;2  5>9  

4

 

Sgkt>N? Idrcnitkvkty 8.9 Vnrtktkdgkgl td n`` ng`

suhtrnit

pjcnsc gdtc tbnt jkstc` dhocitkvcs mklbt dgjy hc pnrtknjjy idvcrc` wktbkg ngy lkvcg ibnptcr hut nrc idvcrc` eujjy nirdss tbc hdda wbcg tnacg ns n wbdjc

Vrkdr jcnrgkgl

Dhocitkvcs  ’ 

Pbks ugkt hukj`s dg jcnrgkgl kg Ytnlc 9?    N``kgl ng` suhtrnitkgl suhtrnitkgl twd-`klkt ng` tbrcc-`klkt gumhcrs.   Sskgl tbc = sklg td rcprcscgt cqunjkty.   Ekg`kgl idmpjcmcgts td >66, sdjvkgl gumhcr cquntkdgs.   Xcdr`crkgl ng n``ktkdg td bcjp wktb n injiujntkdg.  N``kgl ng` suhtrnitkgl suhtrnitkgl mujtkpjcs de >6 ng` mujtkpjcs de >66 td ng` erdm twd- ng` tbrcc-`klkt tbrcc-`klkt gumhcrs.

>N? N``ktkdg ng` suhtrnitkdg 0Gi>< ’  N`` pnkrs de tbrcc-`klkt tbrcc-`klkt gumhc gumhcrs. rs. 0Gi>2 ’ Yuhtrnit n twd-`klkt gumhcr erdm n tbrcc-`klkt gumhcr. 0Gi>; ’ Yuhtrnit pnkrs de tbrcc-`klkt gumhcrs. >N? Injiujntkdg ( Mcgtnj Mcgtnj strntclkcs )   0Gi5 ’  N`` tbrcc dr edur edur smnjj gu gumhcrs, mhcrs, ﬎g`kgl pnkrs tbnt cqunj >6 dr 86.   0Gi; ’  N`` ngy pnkr de twd-`klkt twd-`klkt gumhcrs, ibddskgl ng npprdprkntc strntcly strntcly.. 0Gi>6 ’ Yuhtrnit ngy pnkr de twd-`klkt gumhcrs, ibddskgl ng npprdprkntc strntcly strntcly.. >N? Vrdhjcm sdjvkgl ( Sskgl Sskgl tcibgkqucs ng` sakjjs kg sdjvkgl mntbcmntkinj prdhjcms )   0Vt> ’ Ibddsc npprdprkntc mcgtnj dr wrkttcg strntclkcs td inrry dut injiujntkdgs kgvdjvkgl n``ktkdg ng` suhtrnitkdg.   0Vt9 ’ Ibcia tbc rcsujts de n``kgl gumhcrs hy n``kgl tbcm kg n `keecrcgt dr`cr dr hy suhtrnitkgl dgc gumhcr erdm tbc tdtnj.   0Vt0 ’ Ibcia suhtrnitkdg hy n``kgl tbc ngswcr td tbc smnjjcr gumhcr kg tbc drklkgnj injiujntkdg.   0Vt2 ’ Cstkmntc ng` npprdxkmntc wbcg injiujnt injiujntkgl, kgl, ng` ibcia wdrakgl. >N? Vrdhjcm sdjvkgl ( Sskgl Sskgl ug`crstng`kgl ng` strntclkcs kg sdjvkgl prdhjcms )   0Vs> ’ Mnac up n gumhcr stdry edr n injiujntkdg, kgiju`kgl kg tbc idgtcxt de mcnsurcs.   0Vs9 ’ Ibddsc strntclkcs td ﬎g` ngswcrs td n``ktkdg dr suhtrnitkdg prdhjcms1 cxpjnkg ng` sbdw wdrakgl.   0Vs; ’ Cxpjnkg mctbd`s ng` rcnsdgkgl drnjjy ng` kg wrktkgl1 mnac bypdtbcscs ng` tcst tbcm dut.

*

*edr GXKIB nitkvktkcs mnppc` td

 Udinhujnry   Udin hujnry  n`` • n``ktkdg • pjus • kgircnsc • sum • tdtnj • njtdlctbcr • suhtrnit  • suhtrnitkdg • tnac • tnac nwny • mkgus • `circnsc • ecwcr • jcnvc • `keecrcgic  • pnrtktkdg • \bnt ks tbc tdtnj gumhcr de …4

tbc Inmhrk`lc Vrkmnry dhocitkvcs, pjcnsc vkskt www.ikc.drl.ua www.ikc.drl.ua/  /  inmhrk`lcprkmnrymntbs

Inmhrk`lc Vrkmnry Mntbcmntkis 0 © Inmhrk`lc Sgkvcrskty Vrcss 86>0

Sgkt >N 

 

Idrc nitkvkty 8.>? N``ktkdg (>)

JH? p2

stdrkcs  ’ ’ Xcsdurics? Idmpjcmcgts-td-86 `dmkgdcs pbdtdidpy mnstcr (p 82)1 Pkmcrs1 dgc pcr jcnrgcr jcnrgcr,, ke nv nvnkjnhjc nkjnhjc.. Gumhcr scgtcgic stdrkcs  N``ktkdg pbdtdidpy N``ktkdg  pbdtdidpy mnstcr (p 8;)1 jnrlc vcrskdg edr ijnss `kspjny. `kspjny. (Dptkdgnj? Hjnga `dmkgdcs pbdtdidpy mnstcr (I@-XDM)1 6’; `klkt inr`s, 6’; `kic dr 6’; spkggcr (I@-XDM)). \nrm-ups (>) Cnib jcnrgcr bns n sct de Idmpjcmcgts-td-86 `dmkgdcs erdm tbc pbdtdidpy mnstcr. mnstcr. Cxpjnkg tbnt tbcy gcc` td ircntc n jddp de cg`-td-cg` `dmkgdcs hy mntibkgl pnkrs de gumhcrs tbnt tdtnj 86. Cxpjnkg ‒idmpjcmcgt td 86‛1 kt ks ngdtbcr wny de snykgl gumhcr pnkr td 86.

Edr jcnrgcrs td hc nhjc td n`` ng` suhtrnit jnrlcr gumhcrs, kt ks kmpdrtngt tbnt?   Pbcy nrc idg﬎`cgt kg tbckr agdwjc`lc de pnkrs de gumhcrs tbnt n`` td mnac n mujtkpjc de >6.   Pbcy ing n`` n mujtkpjc de >6 td ngy gumhcr.

Lnmc >. Cnib jcnrgcr sbue﬈cs tbckr `dmkgdcs. Pbcy ﬎g` bdw qukiajy tbcy ing nrrnglc tbc `dmkgdcs kgtd n jddp sd tbnt cnib pnkr de tduibkgl gumhcrs n``s up td 86. Ssc tkmcrs ke nvnkjnhjc. Lnmc 8. Edr >> pjnycrs pjus dgc jcnrgcr wbd tkmcs tbc dtbcrs. Pbc `dmkgdcs nrc sbue﬈c` ng` `cnjt dut, dgc td cnib pjnycr. Pbc pjnycrs nrrnglc tbcmscjvcs kgtd n jddp sd tbnt cnib pnkr de tbckr tduibkgl gumhcrs n``s up td 86. Ssc n tkmcr td rcidr` bdw qukiajy tbcy ing `d tbks.

(8) Ibddsc n twd-`klkt gumhcr. Nsa jcnrgcrs td n`` >6 td tbks gumhcr ng` td sny wbnt gcw gumhcr tbcy lct. Pbcg nsa jcnrgcrs td n`` >6 td tbc gcw gumhcr ng` sny wbnt tbcy lct1 rcpcnt tbks prdicss cklbt eurtbcr tkmcs. Cxpjnkg tbnt tbks wny de n``kgl gumhcrs bcjps us td n`` n mujtkpjc de >6 td n twd-`klkt gumhcr. Kg tbks cxnmpjc, wc bnvc n``c` >66 (>6 Ü >6) td dur drklkgnj gumhcr. @cmdgstrntc uskgl tbc cxnmpjc? 0< + >< hy n``kgl >6 scvcg tkmcs td n stnrtkgl gumhcr de 00 ks 5. 'Idmpjcmcgt td' ks ngdtbcr wny td `csirkhc ‒gumhcr pnkrs‛. n``, n``ktkdg, pjus, kgircnsc, sum, tdtnj, njtdlctbcr?  

wdr`s wc usc edr n``ktkdg.

Cxnmpjc? uskgl tbc Idmpjcmcgts-td-86 `dmkgdcs. >    3  

 9  

N``kgl tbrcc dr edur smnjj gumhcrs

3

>8

 

2

>6

>    6   >   0  

@rnw tbcsc ikrijcs edr tbc wbdjc ijnss td scc?

Jdda dut edr! >8 ;

< 2

5

> 0

>0

2 5

 >9

3

>3

<

Ibnjjcglc jcnrgcrs td idmc up wktb n mctbd` edr ﬎g`kgl tbc tdtnj de tbc gumhcrs kgsk`c

Jcnrgcrs wbd `d gdt ug`crstng` tbnt n``ktkdg ing hc `dgc kg ngy dr`cr. Ssc pbyskinj dhocits kg tbc ijnssrddm td `cmdgstrntc tbnt tbc dr`cr kg wbkib tbc dhocits nrc n``c` tdlctbcr `dcs gdt njtcr tbc tdtnj gumhcr de dhocits. dhocits.

>3   >3

cnib ikrijc. Idjjcit rcspdgscs.

>5  >5 

Sgkt >N 

8 N``ktkdg ng` suhtrnitkdg (>)

 

Ybnrc usceuj wnys de n``kgl gumhcrs wktbdut uskgl n injiujntdr, suib ns?   n``kgl tbrcc gumhcrs wbcrc tbc sum de twd de tbc gumhcrs ks n gcnr mujtkpjc de >6? ‚K gcc` td n`` >9, >3 ng` 2. K agdw tbnt >9 n`` < ks cqunj td 86, sd >9 n`` 2 must cqunj 8>. 8> . K agdw tbnt 86 n`` >3 ks cqunj td 93, sd 8> n`` >3 must cqunj 95 .” .”   uskgl gcnr `duhjcs ng` idmpcgsntkgl? ‚K gcc` td n`` 5, < ng` 0. 0 . K agdw tbnt `duhjc 5 ks cqunj td >8, sd 5 n`` < must cqunj >9. >9. >9 n`` 0 cqunjs >6 dr 86? ‚K gcc` td n`` 8, 2 ng` 8. 8 . K agdw tbnt 8 n`` 2 cqunjs >6. >6 n`` 8 cqunjs >8.” >8.”  

Ke jcnrgc jcnrgcrs rs ﬎g` tbks nitkvkty `ke ﬎iujt, cgsurc tbcy ug`crstng` tbnt tbcy dgjy gcc` td n`` twd gumhcrs nt n tkmc. Edr cxnmpjc, 0 + 5 + < ks sdjvc` hy n``kgl 0 ng` 5, tbcg n``kgl tbc tdtnj de tbnt (>6) td 8 + ; + 2 = 4 Xcmkg` jcnrgcrs tbnt n``ktkdg ing njwnys hc `dgc kg ngy dr`cr ng` tbnt tbcy sbduj` jdda edr ce ﬎ikcgt wnys de wdrakgl rntbcr tbng njwn njwnys ys tniajkgl n prdhjcm kg tbc dr`cr kt ks lkvcg. Edr cxnmpjc, tbcy iduj` rcidlgksc tbnt ke tbcy n`` >8 ng` 2 ﬎rst tbcy lct 86, tbcg tbcy ing n`` ;.

N``ktkdg hy n``kgl dg tcgs tbcg ugkts Xcn` tbc edjjdwkgl prdhjcm td tbc ijnss? ‚Pbc Pbc ﬎  ﬎rst rst pnrt de my odurgcy wns 32 am. Pbc scidg` ‚Bdw ing wc sdjvc tbks  pnrt de my odurgcy wns 83 am. Bdw enr `k` K trnvcj njtdl njtdlctbcr4  ctbcr4 ” Nsa? ‚Bdw  prdhjcm4 ” Idjjcit rcspdgscs. Pbcg `kspjny tbks gumhcr scgtcgic (ng` ngswcr) dg tbc Gumhcr scgtcgic stdrkcs  stdrkcs  pbdtdidpy mnstcr. Cxpjnkg tbnt dgc wny de n``kgl gumhcrs ks td n`` tbc tcgs pnrt de dgc gumhcr td tbc wbdjc de tbc dtbcr gumhcr, tbcg n`` dg tbc ugkts. Ybdw bdw tbks wdras uskgl tbc cxnmpjc ng` gumhcr jkgcs hcjdw. Idugtkgl dg kg tcgs ng` dgcs >6

86

96

9>

>6

98

>6

>6

>

>

 dr

8> + 98 = 39 8>

9>

0>

3>

38

8>

3 39 9

9>

0>

3>

38

39

@cmdgstrntc tbc snmc strntcly edr tbrcc-`klkt gumhcrs. N`` tbc bug`rc`s, tbcg tbc tcgs ng` tbcg tbc ugkts de dgc gumhcr td tbc wbdjc de tbc dtbcr gumhcr. Idugtkgl dg kg bug`rc`s, tcgs ng` dgcs >66

866

8>6

886

896

89 89> 898

>66

>66

>6

>6

>6

>

905

005

305

33 3 35

355

3< 3 iduj` hc put tdlctbcr td mnac dgc oump de 8.

905

005

305

3 35

355

3< 36, hcinusc kt ks cnskcr td idugt dg kg tcgs erdm n stnrtkgl gumhcr tbnt ks n mujtkpjc de >6. Ns n ijnss, idugt dg tdlctbcr td ﬎g` tbc sdjutkdg td 22 + 08. Pbcg, sbdw tbc ijnss dgc de tbcsc cxnmpjcs ng` gumhcr jkgcs? Idugtkgl dg kg dgcs ng` tcgs >

8

>8

88

98

08

>

>

>6

>6

>6

>6

 dr

52 + 08 = >>6 52

5;

66

>>6

52

5;

66

>>6

Cxpjnkg tbnt tbks mctbd` ing njsd hc usc` ke tbc ugkts n`` up td mdrc tbng >6. Ekrst n`` dg cgdulb ugkts td mnac tbc gumhcr up td n mujtkpjc de >6, tbcg n`` dg tbc tcgs, tbcg n`` dg tbc rcst de tbc ugkts. Cxnmpjc?   88  

>

9

26

89

;6

99

>66

09

> >6

0 00 0

> 86

03

>88

86 >86 + 8 = >88

Dgc gumhcr jkgc sbdws tbc oumps wktb n ruggkgl tdtnj de wbnt vnjuc bns hccg n``c` netcr cnib oump1 tbc dtbcr sbdws tbc scpnrntc vnjuc de cnib oump.

>2  >2 

Sgkt >N 

8 N``ktkdg ng` suhtrnitkdg (>)

 

Yummnry   

 

Jcnrgcrs nrc nhjc td n`` tdlctbcr tbrcc dr edur smnjj gumhcrs hy ﬎g`kgl pnkrs tbnt cqunj >6 dr 86. Pbcy ing n`` ngy pnkr de twd-`klkt gumhcrs gumhcrs,, ibddskgl ng npprdprkn npprdprkntc tc strntcly.

Gdtcs dg tbc Jcnrgcr‛s Hdda N``ktkdg (>) (p2)? (>) (p2)? jcnrgcrs prnitksc n``kgl tbrcc ng` edur smnjj gumhcrs tdlctbcr. Pbcy n`` twd-`klkt gumhcrs ng` cxpjnkg tbckr mctbd`s.

Ibcia up!  

 

‚Pbrdw n `kic edur tkmcs ng` wrktc `dwg tbc edur gumhcrs lcgcrntc`. N`` tbc gumhcrs tdlctbcr. \bkib mctbd` `k` ydu usc td n`` tbc gumhcrs4 \by4 ” ‚Ssc ydur edur gumhcrs td mnac twd twd-`klkt gumhcrs gumhcrs.. N`` tbcsc gumhcrs. \bkib mctbd` `k` ydu usc td n`` tbc gumhcrs4 \by4 ”

Mdrc nitkvktkcs @dmkgd pnkrs (pnkrs) [du wkjj gcc` tbc Hjnga `dmkgdcs pbdtdidpy `dmkgdcs pbdtdidpy mnstcr (I@-X (I@-XDM). DM). Vnkrs de jcnrgcrs ing mnac scts de `dmkgdcs, skmkjnr td tbdsc kg tbc Idmpjcmcgt-td-86 `dmkgdcs lnmc, `dmkgdcs lnmc, edr dtbcr twd-`klkt tdtnjs. Pbcy wkjj gcc` td wdra kg ng drlngksc` wny td mnac surc tbnt tbc `dmkgdcs edrm n idgtkgudus jddp.

Xcvcrsc ng` n`` (pnkrs) [du wkjj gcc` n sct de 6’; `klkt inr`s (I@-XDM). Nsa jcnrgcrs td lcgcrntc pnkrs de twd-`klkt gumhcrs hy scjcitkgl twd inr`s nt n tkmc. Edr Edr cxnmpjc, tbc inr`s 5 ng` < mnac tbc gumhcrs 5< ng` 6? N``ktkdg (>)

 

Idrc nitkvkty 8.8? Yuhtrnitkdg (>)

JH? p>6

Xcsdurics? Gumhcr scgtcgic stdrkcs ’ Yuhtrnitkdg pbdtdidpy Yuhtrnitkdg  pbdtdidpy mnstcr (p96)1 jnrlc vcrskdg edr ijnss `kspjny `kspjny.. (Dptkdgnj? 6’; `klkt inr`s, 6’; `kic dr 6’; `klkt spkggcr (I@-XDM).) (I@-XDM).)

\nrm-up Ibddsc n twd-`klkt gumhcr lrcntcr tbng 56, edr cxnmpjc, 26 erdm tbks gumhcr ﬎vc tkmcs ng` td sny wbnt gumhcr tbcy lct cnib tkmc. Ssc tbks mctbd` de suhtrnitkdg td bcjp jcnrgcrs suhtrnit n mujtkpjc de >6 erdm n twd-`klkt gum gumhcr. hcr. Yct skmkjnr qucstkdgs tbnt tbc wbdjc ijnss ing ngswcr uskgl sjntcs, smnjj wbktchdnr`s wbktchdnr`s dr ng KIP rcsduric. Xcpcnt wktb tbrcc-`klkt gumhcrs.

Cxnmpjc? 2< ∘ 36 = 9< 2< ∘ >6 = 6 ﬎vc tkmcs 5< ∘ >6 = 3< 3< ∘ >6 = 0< 0< ∘ >6 = 9<

Yuhtrnitkdg hy idugtkgl hnia

Xcn` dut tbks prdhjcm? ‚K bnvc 23 mj de dkj kg n hdttjc, tbcg K usc >3 mj kg n rcikpc. Bdw muib dkj ks jcet kg tbc hdttjc4 ” Ibnjjcglc jcnrgcrs td cxpjnkg bdw tbcy mklbt s djvc tbks prdhjcm. Idjjcit ng` `ksiuss rcspdgscs. Pbcg `kspjny tbks stdry erdm tbc Gumhcr scgtcgic stdrkcs pbdtdidpy stdrkcs pbdtdidpy mnstcr. Cxpjnkg tbnt wbcg tbc gumhcr td hc suhtrnitc` ks smnjj (edr cxnmpjc 23  ∘ >3), dr n mujtkpjc de >6, kt ks ce ﬎ikcgt td idugt hnia td ﬎g` tbc ngswcr. Ybdw tbc edjjdwkgl gumhcr jkgcs td `cmdgstrntc tbks mctbd`? Idugtkgl hnia kg2 dgcs <

38 ∘ 2 = 00

5

3

00

0

9

8

38

8>

Idugtkgl hnia kg tcgs >6

>6

>6

>6

28 ∘ 56 = 88 88

98

08

38

`circnsc, ecwcr, jcnvc, `keecrcgic? sdmc wdr`s tbnt wc usc edr suhtrnitkdg.

>6

Cxpjnkg td jcnrgcrs tbnt, kg tbks cxnmpjc, kt ing hc usceuj td hrcna tbc 2 kgtd 8 ng` 5. 8 ks suhtrnitc` sd tbnt jcnrgcrs rcnib n mujtkpjc de >6, tbcg tbc rcmnkgkgl 5 ks suhtrnitc`.

>6

suhtrnit, suhtrnitkdg, tnac, tnac nwny, mkgus,

>

 

>6

 Udin  Udinhujnry  hujnry 

58



96

0>

9>

3>

98

39

>;  >; 

86  86 

Sgkt >N 

8 N``ktkdg ng` suhtrnitkdg (>)

 

@rnw tbcsc twd ikrijcs de gumhcrs edr tbc wbdjc ijnss td scc. Nsa jcnrgcrs td suhtrnit gumhcrs kg tbc scidg` ikrijc erdm gumhcrs kg tbc ﬎rst ikrijc, c.l.

3>

39

@rnw tbcsc twd ikrijcs de gumhcrs edr tbc wbdjc ijnss td scc. 02

33

56

09

92

39 32 50<

  83 95

9> 8<

Lct jcnrgcrs td try tbc ‒ ﬎g`kgl tbc `keecrcgic‛ mctbd` hy suhtrnitkgl ngy gumhcr kg tbc scidg` ikrijc erdm ngy gumhcr kg tbc ﬎rst ikrijc, c.l. 32 ∘ 9>.  9>. Jcnrgcrs ing sdjvc t bc

Jcnrgcrs wbd idgtkguc td usc gumhcr jkgcs edr qucstkdgs tbcy ing cnskjy ngswcr wktbdut dgc, c.l. 5< ∘ 53 = 8. Cgidurnlc jcnrgcrs td `d hkllcr oumps dg tbc gumhcr jkgc, dr td `cvcjdp tbckr dwg wnys de rcidr`kgl tbckr wdrakgl ng` kgtcrmc`kntc stcps kg tbckr injiujntkdg. Pbcsc must hc ce  ﬎  ﬎ikcgt ikcgt ng` rcjknhjc. rcjknhjc.

suhtrnitkdg prdhjcms mcgtnjjy, mcgtnjjy, wktb tbc suppdrt de n gumhcr jkgc ke npprdprkntc npprdprkntc.. Idrc nitkvkty 8.8? Yuhtrnitkdg (>)

 

Yummnry  Jcnrgcrs ing `d n suhtrnitkdg uskgl ngy pnkr de twd-`klkt gumhcrs, gumhcrs, ibddskgl ng npprdprkntc strntcly. Gdtcs dg tbc Jcnrgcr‛s Hdda Yuhtrnitkdg (>) (p>6)? (>) (p>6)? jcnrgcrs prnitksc suhtrnitkdg hy ibddskgl ng` uskgl `keecrcgt mctbd`s, tbcg wrktc n stdry td kjjustrntc n suhtrnitkdg gumhcr scgtcgic.

Ibcia up!  

 

‚Dur ijnss bn` 9; pcgikjs nt tbc stnrt de tbc ycnr. Gdw wc bnvc 98. Bdw mngy bnvc wc jdst4 \bkib mctbd` `k` ydu usc td suhtrnit tbc gumhcrs4 \by4 ” ‚Dur ijnss bn` 82 pcgs nt tbc stnrt de tbc ycnr, hut wc jdst 0. Bdw mngy `d wc bnvc gdw4 \bkib mctbd` `k`  ydu usc td suhtrnit tbc gumhcrs4 \by4 ”

Mdrc nitkvktkcs Yuhtrnitkgl gumhcrs (wbdjc ijnss) [du wkjj gcc` 6’; `klkt inr`s, n 6’; `kic dr n 6’; spkggcr (I@-XDM). Jcnrgcrs usc n `kic/ spkggcr/`klkt inr`s td lcgcrntc twd-`klkt gumhcrs. Nsa jcnrgcrs td suhtrnit dgc gumhcr erdm tbc dtbcr uskgl ng npprdprkntc strntcly strntcly..

>66 `dwg lnmc (edr twd dr mdrc pjnycrs) Pbc pjnycrs cnib stnrt wktb >66. Pbcy tnac turgs td ibddsc wbctbcr td tbrdw dgc, twd dr tbrcc `kic. Pbcy suhtrnit tbc tdtnj dg tbc `kic erdm tbckr tdtnj gumhcr.. Pbc wkggcr ks tbc pjnycr wbd stdps pjnykgl tbc lnmc wbcg tbckr gumhcr ks ijdscst t d 6, wktbdut ldkgl pnst 6. gumhcr

Gumhcr scgtcgic stdrkcs (edr kg`kvk`unjs) Ssc tbc Gumhcr Ycgtcgic Ytdrkcs pbdtdidpy mnstcr edr cxnmpjcs. Nsa jcnrgcrs td cnib wrktc twd gumhcr scgtcgic stdrkcs edr suhtrnitkdg enits. Dgc sbduj` hc kg tbc idgtcxt de mcnsurcs.

Lnmcs Hdda (KYHG ;>6   8>

88  88 

 

Sgkt >N 

8 N``ktkdg ng` suhtrnitkdg (>)

Hjnga pnlc

 

Idrc nitkvkty 8.9? Vnrtktkdgkgl td n`` ng` suhtrnit

JH? p>8

Xcsdurics? Vjnic vnjuc ibnrt? > td ;666 pbdtdidpy ;666  pbdtdidpy mnstcr (I@-XDM)1 jnrlc vcrskdg edr ijnss `kspjny. `kspjny. Gumhcr scgtcgic stdrkcs pbdtdidpy stdrkcs pbdtdidpy mnstcr (p8; ng` 96)1 jnrlc vcrskdg edr ijnss `kspjny. Mntcrknjs td `cmdstrntc pjnic vnjuc, suib ns hjdias dr strnws hug`jc` kg lrdups de bug`rc`s, tcgs ng` dgcs. (Dptkdgnj? 6’; `klkt inr`s, 6’; `kic dr 6’; spkggcr (I@-X (I@-XDM).) DM).)

\nrm-ups (>) Jcnrgcrs pkia twd twd-`klkt mujtkpjcs mujtkpjcs de >6. Pbcy ﬎g` tbc tdtnj de tbc gumhcrs ng` ﬎g` tbc `keecrcgic hctwccg tbc gumhcrs. Pbc tdtnj ng` `keecrcgic nrc tbc twd gcw gumhcrs. Jcnrgcrs rcpcnt tbc prdicss de ﬎g`kgl tbc tdtnj ng` `keecrcgic uskgl tbc twd gcw gumhcrs.. Pbcy idgtkguc ugtkj hdtb gumhcrs nrc lrcntcr tbng >66, (scc cxnmpjc, rklbt). gumhcrs rklbt).

Cxnmpjc? 86 ng` 26 86 + 26 = >66 26 ∘ 86 = 56

(8) Ns pcr \nrm-up >, hut wktb twd tbrcc-`klkt gumhcrs. gumhcrs. Jcnrgcrs idgtkguc ugtkj hdtb gumhcrs nrc lrcntcr tbng >666.

>66 + 56 = >56 >66 ∘ 56 = 06

Xcmkg` jcnrgcrs nhdut tbc wdra tbcy bnvc `dgc dg pjnic vnjuc (ibnptcr >) ng` pnrtktkdgkgl gumhcrs kgtd bug`rc`s, tcgs ng` dgcs. @kspjny tbc Vjnic vnjuc ibnrt? > ∘ ;666  ;666 edr  edr jcnrgcrs td rcecr td `urkgl tbc gcxt nitkvkty. Nsa jcnrgcrs td pnrtktkdg sdmc tbrcc-`klkt gumhcrs suib ns 89;, ;98 ng` 9;8. Pbc wbdjc ijnss ing rcspdg` kg gumhcr scgtcgics suib ns 89; = 866 + 96 + ;, uskgl sjntcs, smnjj wbktchdnr`s dr ng KIP rcsduric.

N``ktkdg hy pnrtktkdgkgl Xcn` dut tbks prdhjcm? K bn` ; mj. Bdw muib Cgidurnlc jcnrgcrs td mnac odttkgls td bcjp sdjvc tbks prdhjcm. (Gdtc  oukic ks kg tbc oul kg tdtnj4   Cgidurnlc tbnt jcnrgcrs nrc gdt cxpcitc` td ibnglc tbc mcnsurc de jkquk` erdm mkjjkjktrcs td jktrcs. Mnac n gdtcs de ngy tbnt `d, td kgedrm pjnggkgl edr Sgkt >H). Nsa tbcm td cxpjnkg tbckr mctbd` edr wdrakgl dut tbc sdjutkdg ng` `ksiuss njj mctbd`s ns n ijnss. Pbcg `kspjny tbks gumhcr scgtcgic stdry erdm tbc Gumhcr scgtcgic stdrkcs pbdtdidpy stdrkcs pbdtdidpy mnstcr mnstcr,, wbkib kgiju`cs tbc ngswcr. Pcjj jcnrgcrs tbnt ns gumhcrs kg n injiujntkdg lct jnrlcr, kt ks bnr`cr td accp trnia de tbc `keecrcgt pnrts de tbc injiujntkdg. N wrkttcg mctbd`, rntbcr tbng mcgtnj strntclkcs, ks detcg gcc`c`. Cxpjnkg tbnt ke twd gumhcrs nrc pnrtktkdgc` kgtd bug`rc`s, tcgs ng` ugkts tbcy ing hc rcidmhkgc` kg `keecrcgt wnys ng` njwnys prd`uic tbc snmc tdtnj. Cxpjnkg tbnt tbks ks usceuj hcinusc kt ks cnskcr td n`` tdlctbcr mujtkpjcs de >66 dr >6, ns jcnrgcrs `k` kg tbc wnrm-

>56 + 06 = 866 >56 ∘ 06 = >86 cg`.

 Udin  Udinhujnry  hujnry  pnrtktkdg? hrcnakgl up n gumhcr kgtd kts pnrts. Edr

cxnmpjc, 562 = 5 bug`rc`s, 6 tcgs, 2 dgcs = 566 + 2

89   89

up. @cmdgstrntc tbks uskgl pjnic vnjuc mntcrknjs ng` tbc sum )

 

Cxnmpjc? (>) wrkttcg pnrtktkdgkgl mctbd` 866   96 + 06 = 6   >866 + 6 = >826, tbcrcedrc 826

  366 + 866 >866 + 26 = >826 tbcrcedrc, 826

  8 + 2 = >6 >6 + 06 + 96 = 26

Lkvc jcnrgcrs tkmc td prnitksc tbks mctbd` de n``ktkdg? `rnw tbcsc ikrijcs edr tbc wbdjc ijnss td scc. Nsa jcnrgcrs td n`` gumhcrs kg tbc ﬎rst ikrijc td gumhcrs kg tbc scidg` ikrijc, c.l. 5>; + 092.

902 >;

>53 369 32<

829   092 ;80 258

>;> 56<

Yuhtrnitkdg hy pnrtktkdgkgl Lkvc jcnrgcrs tbks prdhjcm? K bn` ; mj. Bdw muib oukic ks jcet kg tbc oul4  Cgidurnlc jcnrgcrs td wrktc `dwg kgtcr mc`kntc stnlcs td bcjp sdjvc tbks prdhjcm. Nsa tbcm td cxpjnkg tbckr mctbd` edr wdrakgl dut tbc sdjutkdg. @ksiuss njj mctbd`s ns n ijnss. Pbcg `kspjny tbks prdhjcm erdm tbc Gumhcr scgtcgic stdrkcs pbdtdidpy stdrkcs pbdtdidpy mnstcr, wbkib kgiju`cs tbc ngswcr ngswcr.. Pcjj jcnrgcrs tbnt, jkac kg n``ktkdg, wbcg tbc gumhcrs kg n injiujntkdg nrc jnrlcr kt ks bnr`cr td accp trnia de tbc `keecrcgt pnrts de tbc injiujntkdg. Edr tbcsc typcs de prdhjcm n wrkttcg mctbd`, rntbcr tbng mcgtnj nrktbmctki, ks detcg gcc`c`. Cxpjnkg tbnt sdmc suhtrnitkdg injiujntkdgs ing hc npprdnibc` uskgl skmpjc pnrtktkdgkgl kgtd bug`rc`s, tcgs ng` ugkts kg tbc s nmc wny ns wktb n``ktkdg injiujntkdgs. injiujntkdgs. \rktc tbc edjjdwkgl scqucgic ng` `cmdgstrntc kt wktb pjnic vnjuc mntcrknjs? 0, tbc rcedrc 0 Idrc nitkvkty 8.9? Vnrtktkdgkgl td n`` ng` suhtrnit

 

Pcjj jcnrgcrs tbnt sdmctkmcs wc gcc` td hc mdrc ircntkvc wbcg wc pnrtktkdg tbc gumhcrs. Edr cxnmpjc, cxnmpjc, edr tbc sum

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