Iiiyoea i oeaio?a oyaeanoe auei aia?aua eco/aii i?eia?ii 2200 eao iacaae
a?a/aneei aaiiao?ii A?oeiaaeii, aaee/aeoei iaoaiaoeeii ae?aaiinoe. N oao
ii? yoi iiiyoea noaei iaeiei ec aaaeiaeoeo a iaoaieea, a oaeaea
iicaieeei n?aaieoaeueii i?inoi ?aoaoue iaeioi?ua aaiiao?e/aneea caaea/e.
Eiaiii i?eeiaeaiea e aaiiao?ee iu e aoaeai ?anniao?eaaoue. Aeey yoiai
ioaeii aaanoe iaeioi?ua ii?aaeaeaiey e iiiyoey. Iiae iaoa?eaeueiie
oi/eie iiieiatho oi/eo, niaaae?iioth iannie. Aeey iaaeyaeiinoe iiaeii
naaa oece/anee i?aaenoaaeoue iaoa?eaeueioth oi/eo a aeaea iaeaiueeiai
oyae?eiai oa?eea, ?acia?aie eioi?iai iiaeii i?aiaa?a/ue. A nayce n yoei
aoaeai /anoi oeacuaaoue oieueei /eneiaia cia/aiea oie eee eiie
oece/aneie aaee/eiu, ii ia aoaeai ioia/aoue a? iaeiaiiaaiea, n/eoay, /oi
iii naii niaie iiae?acoiaaaaony. Iai?eia?, au?aaeaiea: «A ( ABC noi?iia
BC ?aaia a, a a aa?oeia A iu iiiauaai ianno a» icia/aao: «Aeeeia noi?iiu
BC ?aaia a naioeiao?ai, a ianna, iiiau?iiay a aa?oeia A, ?aaia a a?aii».
Anee a oi/ea A iiiauaia ianna m, oi ia?acothuothny iaoa?eaeueioth oi/eo
aoaeai iaicia/aoue oae: (A, m). Eiiaaea, eiaaea yoi ia iiaeao aucaaoue
iaaei?acoiaiee, iu aoaeai a? iaicia/aoue iaeiie aoeaie A. Ianno m
eiiaaea iacuaatho «iaa?oceie oi/ee A».
Oeaio?ii oyaeanoe aeaoo iaoa?eaeueiuo oi/ae (A, a) e (B, b) iacuaaaony
oaeay o?aouey oi/ea C, eioi?ay eaaeeo ia io?acea AB e oaeiaeaoai?yao
«i?aaeeo ?u/aaa»: i?iecaaaeaiea a? ?annoiyiey CA io oi/ee A ia ianno a
?aaii i?iecaaaeaieth a? ?annoiyieth NA io oi/ee A ia ianno b; oaeei
ia?acii,
.
Yoi ?aaainoai iiaeii caienaoue e oae:
,
oi anoue ?annoiyiea io oeaio?a oyaeanoe aeaoo iaoa?eaeueiuo oi/ae aei
yoeo oi/ae ia?aoii i?iii?oeeiiaeueiu iannai, iiiau?iiui a yoeo oi/eao.
Oeaio? oyaeanoe aoaeao aeeaea e oi/ea n aieueoae iannie. Ec ii?aaeaeaiey
neaaeoao: anee i?yiay i?ioiaeeo /a?ac oeaio? oyaeanoe aeaoo
iaoa?eaeueiuo oi/ae e /a?ac iaeio ec ieo, oi iia i?ieae?o e /a?ac
ae?oaoth.
Oeaio? oyaeanoe aeaoo iaoa?eaeueiuo oi/ae eiaao aanueia i?inoie
iaoaie/aneee niune. I?aaenoaaei naaa ae?noeee «iaaaniiue» noa?aeaiue AA,
a eiioeao eioi?iai iiiauaiu iannu a e b (?en. 1). «Iaaaniiinoue»
noa?aeiy i?aeoe/anee icia/aao, /oi aai ianna ii n?aaiaieth n iannaie a e
b ianoieueei iacia/eoaeueia, /oi ath iiaeii i?aiaa?a/ue. Oeaio? oyaeanoe
N iaoa?eaeueiuo oi/ae (A, a) e (B, b) — yoi oaeay oi/ea, a eioi?ie
iaaei iiaeia?aoue noa?aeaiue AB, /oiau ii aue a ?aaiiaanee.
A 5 N 15 B
?en. 2
Aeey aeaeueiaeoaai iieacii oaeaea aaanoe iiiyoea «iauaaeeiaiea» eee
?aaiiaeaenoaothuae aeaoo iaoa?eaeueiuo oi/ae. Iiae yoei iu aoaeai
iiieiaoue iaoa?eaeueioth oi/eo, eioi?ay iieo/eony, anee a oeaio?a
oyaeanoe aeaoo iaoa?eaeueiuo oi/ae iiianoeoue iannu iaaeo oi/ae.
A C B
(
?en. 1
(aae.). Iauaaeeiaiea iaoa?eaeueiuo oi/ae (A, 6) e (B,2) aoaeao
iaoa?eaeueiay oi/ea (N, 8).
Oeaio? oyaeanoe o??o iaoa?eaeueiuo oi/ae iaoiaeeony neaaeothuei ia?acii:
iaoiaeyo iauaaeeiaiea aeaoo ec yoeo iaoa?eaeueiuo oi/ae e caoai euoo
oeaio? oyaeanoe ia?aciaaaoaeny oaeei ia?acii /aoa??oie iaoa?eaeueiie
oi/ee e o?aoae ec aeaiiuo iaoa?eaeueiuo oi/ae.
Aiiaua, oeaio? oyaeanoe n iaoa?eaeueiuo oi/ae i?e n>2 iaoiaeeony oae:
iaaei nia/aea iaeoe oeaio? oyaeanoe n-1 iaoa?eaeueiuo oi/ae, iiianoeoue
a yoie oi/ea iannu anao n-1 oi/ae, caoai iaeoe oeaio? oyaeanoe yoie
aiiaue ia?aciaaaoaeny iaoa?eaeueiie oi/ee n n-e iaoa?eaeueiie oi/eie.
Anee iiianoeoue a oeaio?a oyaeanoe ianeieueei iaoa?eaeueiuo oi/ae iannu
anao yoeo oi/ae, oi ia?acothuothny oaeei ia?acii iiaoth iaoa?eaeueioth
oi/eo iacia?i iauaaeeiaieai aeaiiuo iaoa?eaeueiuo oi/ae.
Aeey ?aoaiey caaea/ aaaeiu neaaeothuea i?inoaeoea naienoaa oeaio?ia
oyaeanoe.
Iieiaeaiea oeaio?a oyaeanoe n iaoa?eaeueiuo oi/ae ia caaeneo io
ii?yaeea, a eioi?ii iineaaeiaaoaeueii iauaaeeiythony yoe oi/ee. (Oai?aia
i aaeeinoaaiiinoe oeaio?a oyaeanoe aeey nenoaiu ec n iaoa?eaeueiuo
oi/ae.)
Iieiaeaiea oeaio?a oyaeanoe nenoaiu ec n iaoa?eaeueiuo oi/ae ia
eciaieony, anee caiaieoue ianeieueei iaoa?eaeueiuo oi/ae eo
iauaaeeiaieai. (Oai?aia i aiciiaeiinoe a?oiie?iaee iaoa?eaeueiuo oi/ae.)
I?e ?anniio?aiee iaeioi?uo aii?inia iaoaieee ieacuaaaony auaiaeiui
aaanoe iiiyoea noaoe/aneiai iiiaioa.
Ionoue eiaaony iaeioi?ay oi/ea C e, e?iia oiai, iaoa?eaeueiay oi/ea
A((A, m). Noaoe/aneei iiiaioii iaoa?eaeueiie oi/ee A ioiineoaeueii oi/ee
N iu iacia?i i?iecaaaeaiea m(CA e aoaeai aai e?aoei iaicia/aoue oae:
IiiNA.
Iieuecoynue iiiyoeai noaoe/aneiai iiiaioa, ii?aaeaeaiea oeaio?a oyaeanoe
iiaeii noi?ioee?iaaoue oae: oi/ea N iacuaaaony oeaio?ii oyaeanoe aeaoo
iaoa?eaeueiuo oi/ae A((A, m1) e B((B, m2), anee N eaaeeo ia io?acea AA e
IiiNA=IiiNB.
Ionoue oaia?ue ia iaeioi?ii eo/a n ia/aeii S (?en. 3) ?aniieiaeaia
nenoaia ec iaeioi?uo n iaoa?eaeueiuo oi/ae
A1((A1, m1), A2((A2, m2), …, An((An, mn).
S A4 A3 A2 A1 An
?en. 3
Noaoe/aneei iiiaioii yoie nenoaiu ioiineoaeueii ia/aea eo/a S iacuaatho
noiio iiiaioia anao oi/ae nenoaiu ioiineoaeueii ia/aea eo/a,
o.a. noiio K=IiiSA1+ IiiSA2+ IiiSA3+…+ IiiSAn eee, iiae?iaiaa,
K=m1(SA1+ m2(SA2+ m3(SA3+…+ mn(San.
I?eia?. Anee nenoaia ninoieo ec o??o oi/ae (A1, 1), (A2, 4), (A3, 9) e
SA1=1, SA2=2, SA3=3 (?en. 4), oi noaoe/aneee iiiaio nenoaiu ?aaai
K=1(1 + 4(2 + 9(3 = 36.
Iiiyoii, /oi a nenoaia SGC iiiaio aoaeao eiaoue ?acia?iinoue a(ni. Ii iu
?aiaa aeiaiai?eeenue, /oi ?acia?iinoue aoaeai eaaeaeue ?ac
iiae?acoiaaaoue, ii ieaaea ia oeacuaaoue.
S A1 A2 A3
?en. 4
A iaoeo ?annoaeaeaieyo iniiaiuie iauaeoaie auee (iaoa?eaeueiua oi/ee(. N
oi/ee c?aiey iaoaiaoeee iaoa?eaeueiay oi/ea — yoi eiiieaen, ninoiyuee ec
aaiiao?e/aneie oi/ee e iaeioi?iai (iieiaeeoaeueiiai) /enea.
A iaoaiaoeea ia ?ac i?eoiaeeony noaeeeaaoueny n oaeei yaeaieai: eiiieaen
ec aeaoo eaeeo-oi iaoaiaoe/aneeo iauaeoia ?anniao?eaatho eae iaeioi?ue
iiaue iauaeo, eioi?ue caoai oaea iiaeaa?aaaony niaoeeaeueiiio eco/aieth.
Oae, iai?eia?, a eo?na aeaaa?u aaiaeeony iiiyoea eiiieaeniiai /enea eae
eiiieaena (ia?u) aeaoo aeaenoaeoaeueiuo /enae.
A no?iaeo eo?nao aaiiao?ee oaeei ia?acii aaiaeeony, iai?eia?, iiiyoea
io?acea eae eiiieaena (ia?u) aeaoo oi/ae; iiiyoea oaea iiaeao auoue
aaaaeaii noiaeiui ia?acii: oaie iiaeii ?anniao?eaaoue eae eiiieaen aeaoo
eo/ae n iauei ia/aeii.
Anee eiaaony o ian eaeay-eeai iaoa?eaeueiay oi/ea A((A, m), oi iu
(aaiiao?e/aneoth) oi/eo A aoaeai eiiaaea iacuaaoue iineoaeai eee
aooeenii yoie iaoa?eaeueiie oi/ee, a /enei m aoaeai ii-i?aaeiaio
iacuaaoue iannie yoie iaoa?eaeueiie oi/ee.
?aaainoao aeaea (A, a)((B, b) iu i?eaea?i oaeie niune: aeaa
iaoa?eaeueiua oi/ee eiatho iaeei e oio aea iineoaeue (A(B) e ?aaiua
iannu (a(b).
?aoaiea ii/oe anao ?aiaa ?anniio?aiiuo caaea/ iie?aeinue ia oi, /oi iu
(iauaaeeiyee iaeioi?ua iaoa?eaeueiua oi/ee a eo oeaio?a oyaeanoe(;
oi/iaa, caiaiyee iaeioi?ua iaoa?eaeueiua oi/ee eo iauaaeeiaieai. I?e
yoii iiae iauaaeeiaieai aeaoo iaoa?eaeueiuo oi/ae (A, a) e (B, b) iu
iiieiaee iaeioi?oth iiaoth iaoa?eaeueioth oi/eo (N, a+b), aaea N —
oeaio? oyaeanoe aeaoo aeaiiuo iaoa?eaeueiuo oi/ae. Iiaeii auei au oae
neacaoue: iauaaeeiaieai aeaoo iaoa?eaeueiuo oi/ae iacuaaaony oaeay iiaay
iaoa?eaeueiay oi/ea, iineoaeai eioi?ie yaeyaony oeaio? oyaeanoe aeaiiuo
iaoa?eaeueiuo oi/ae e ianna eioi?uo ?aaia noiia iann yoeo iaoa?eaeueiuo
oi/ae.
Aianoi (iauaaeeiaiey( iiaeii oiio?aaeyoue au?aaeaiea (noiia(.
Anee iaoa?eaeueiay oi/ea N((N, n) yaeyaony iauaaeeiaieai aeaoo ae?oaeo
iaoa?eaeueiuo oi/ae A((A, a) e B((B, b), oi iu aoaeai yoi caienuaaoue
oae:
(A, a) + (B, b) = (C, c)
eee, ei?i/a,
A + B = C.
Iu ia aoaeai eneeth/aoue e oio neo/ae, eiaaea aeaa iaoa?eaeueiua oi/ee
eiatho iaeei e oio aea iineoaeue. A yoii neo/aa, anoanoaaiii, aoaeai
n/eoaoue iineoaeai iauaaeeiaiey eo iauee iineoaeue. Oaeei ia?acii, (A,
a) + (A, b) = (A, a+b).
O ian aicieeaao naiaia?aciia en/eneaiea, naiaia?aciay aeaaa?a. A yoie
aeaaa?a eiaao ianoi ia?aianoeoaeueiue caeii: A + B = B + A. Yoi neaaeoao
ec naiiai ii?aaeaeaiey oeaio?a oyaeanoe aeaoo iaoa?eaeueiuo oi/ae. Eiaao
ianoi oaeaea ni/aoaoaeueiue caeii:
(A1 + A2) + A3 = A1 + (A2 + A3),
eee, eia/a,
[(A1, m1) + (A2, m2)] + (A3, m3) = (A1, m1) + [(A2, m2) + (A3, m3)].
Iiae?iaiaa: Iaeae?i ee iu nia/aea iauaaeeiaiea A12 aeaoo iaoa?eaeueiuo
oi/ae A1 e A2 e caoai iaeae?i iauaaeeiaiea yoie iaoa?eaeueiie oi/ee A12
n o?aoueae iaoa?eaeueiie oi/eie A3, eee nia/aea iaeae?i iauaaeeiaiea A23
iaoa?eaeueiuo oi/ae A2 e A3, a caoai iaeae?i iauaaeeiaiea iaoa?eaeueiuo
oi/ae A1 e A23, a iaieo neo/ayo iu i?eae?i e iaeiiio e oiio aea
?acoeueoaoo, e iaeiie e oie aea iaoa?eaeueiie oi/ea.
Iiiyoii, /oi niune yoiai ooaa?aeaeaiey ninoieo a oii, /oi oeaio?
oyaeanoe o??o iaoa?eaeueiuo oi/ae ia caaeneo io ii?yaeea, a eioi?ii
iauaaeeiythony yoe oi/ee.
A iaoeo ?annoaeaeaieyo (iaoa?eaeueiay oi/ea( (A, m) aunooiaea eae
eiiieaen, ninoiyuee ec iaeioi?ie aaiiao?e/aneie oi/ee A e iaeioi?iai
iieiaeeoaeueiiai /enea o. Yoi /enei o iu aei neo ii? iacuaaee iannie.
Iaeiaei aai iiaeii auei au iacaaoue e eaeei-eeai ae?oaei neiaii,
neaaeai, (aanii(. Ana iaoe i?aaeuaeouea ?annoaeaeaiey inoaioony,
eiia/ii, a neea, anee caiaieoue neiai (ianna( neiaii (aan(. Iu au a
oaeii neo/aa oaea ia aiai?eee, iai?eia?, (?anniio?ei iaoa?eaeueioth
oi/eo (A, o) n iannie o(, a neacaee au: ( ?anniio?ei iaoa?eaeueioth
oi/eo (A, o) n aanii o(.
Aei neo ii? iu iaaeyaeii i?aaenoaaeyee iaoa?eaeueioth oi/eo (A, o) a
aeaea iaoa?eaeueiiai oa?eea, ?acia?aie eioi?iai iiaeii i?aiaa?a/ue,
eiathuaai ianno o. Ii n oaeei aea oniaoii iu iiaee au iaaeyaeii
i?aaenoaaeoue oo aea iaoa?eaeueioth oi/eo a aeaea oaeiai aea oa?eea n
aanii o.
Iu aeaeaa ?anniao?eaaee oeaio? oyaeanoe aeaoo iaoa?eaeueiuo oi/ae aeaea
(A, a) e (B, b) e ii?aaeaeyee aai ii i?aaeeo ?u/aaa. Anee iu oioei yoio
oeaio? oyaeanoe iaaeyaeii i?aaenoaaeoue a aeaea oeaio?a oyaeanoe aeaoo
oa?eeia, iiiau?iiuo a oi/eao A e A e aanyueo niioaaonoaaiii a e b
aaeeieoe, oi ioaeii naeaeaoue ianeieueei iaiai?ie (eioi?ua, ai?i/ai,
naii niaie iiae?acoiaaathony). Ai anyeii neo/aa yoe oa?eee aeieaeiu
auoue ia iaaieueoii ?annoiyiee ae?oa io ae?oaa, ianoieueei iaaieueoii,
/oiau iiaeii auei aac /oanoaeoaeueiie iia?aoiinoe n/eoaoue, /oi i?e
naiaiaeiii iaaeaiee iie aoaeoo ia?aiauaoueny ia?aeeaeueii e n iaeiei e
oai aea onei?aieai. E?iia oiai, anee oa?eee ec ?acee/iuo iaoa?eaeia, oi
aaaeii, /oiau oaeaeueiui aanii aaca eee aeeaeeinoe, caiieiythuae
ie?oaeathuoth eo n?aaeo, aiciiaeii auei i?aiaa?a/ue. Oaeea oneiaey
i?aeoe/anee aoaeoo niaethaeaiu, iai?eia?, anee iu ia aueaeai ca
i?aaeaeu, neaaeai, eiiiaou eee aeaaea ai?iaea.
Aei neo ii?, ?anniao?eaay iaoa?eaeueioth oi/eo, oi anoue ia?o aeaea (A,
o),iu anaaaea iieaaaee, /oi a? (ianna( (eee (aan() — iieiaeeoaeueiia
/enei. Aeey ?aoaiey iaeioi?uo aaiiao?e/aneeo caaea/ aanueia iieacii
?anniio?aoue e oaeie neo/ae, eiaaea yoi /enei o iiaeao auoue
i?iecaieueiui aeaenoaeoaeueiui /eneii. Oaeoth ia?o iu, nio?aiyy noa?oth
oa?ieiieiaeth, aoaeai ii-i?aaeiaio iacuaaoue iaoa?eaeueiie oi/eie, a
aeey /enea o nio?aiei noa?ia iacaaiea (ianna( (eee (aan(). Eae aea naaa
iaaeyaeii i?aaenoaaeoue (iaoa?eaeueioth oi/eo( n io?eoeaoaeueiie
(iannie(?
I?eaaae?i iaeio eiie?aoioth oece/aneoth ea?oeio, eioi?ay iicaieeo
/eoaoaeth iaaeyaeii i?aaenoaaeoue iaoa?eaeueiua oi/ee n i?iecaieueiuie
aauanoaaiiuie (iannaie(.
Ionoue eiaaony eaeie-oi aannaei, caiieiaiiue aiaeie. Ionoue oa?ee,
eioi?ue aeneo a aicaeooa (oi/iaa, a ionoioa) ? aaeeieoe (neaaeai, ?
a?aii), iiiau?i a eaeoth-oi oi/eo A aioo?e yoiai aannaeia.
?anniio?ei nia/aea neo/ae, eiaaea oaeaeueiue aan oa?eea aieueoa 1
(iai?eia?, eiaaea oa?ee aeaeaciue). Iiiyoii, /oi oa?ee iieae?o a yoii
neo/aa ei aeio. Anee iiae aiaeie acaaneoue yoio oa?ee (neaaeai, n
iiiiuueth i?oaeeiiuo aania), oi aanu iieaaeoo iaiueoa, /ai ? aaeeieoe.
Iao?oaeii, anee aoaeao iaiaoiaeeiinoue, ociaoue, neieueei aoaeao aaneoue
oa?ee iiae aiaeie. Ionoue oaeaeueiue aan oa?eea ?aaai d, a iau?i V.
Oiaaea V=p/d. N/eoay oaeaeueiue aan aiaeu ?aaiui 1, iaeae?i /oi aan
aiaeu a iau?ia, caieiaaiii oa?eeii, ?aaai (p/d)(1=p/d. A neeo caeiia
A?oeiaaea aan o oa?eea iiae aiaeie (aai (iiaeaiaeiue aan() ii?aaeaeyaony
ii oaeie oi?ioea:
(*)
Iiiyoii, /oi o — iiaeaiaeiue aan oa?eea — yoi ?acoeueoe?othuay neea,
eioi?ay iieo/aaony io neiaeaiey aeaoo nee, aeaenoaothueo ia oa?ee: neeu
oyaeanoe e auoaeeeaathuae neeu aiaeu.
Ia?aoei aieiaiea ia oi, /oi a ?anniao?eaaaiii neo/aa (i?e d>1) m>0 e yoa
neea iai?aaeaia aiec. Ionoue oaia?ue oaeaeueiue aan oa?eea iaiueoa 1
(iai?eia?, eiaaea oa?ee naeaeai ec i?iaee). A yoii neo/aa oa?ee aoaeao
auoaeeeaaoueny ec aiaeu ((aaa?o(). ?acoeueoe?othuay neea m, iiae
aeaenoaeai eioi?ie oa?ee aoaeao auoaeeeaaoueny aaa?o, aoaeao a
niioaaonoaee n caeiiii A?oeiaaea ?aaia ii-i?aaeiaio
,
ii oaia?ue yoi au?aaeaiea io?eoeaoaeueii (eai d0) au?aaeaiea (*) oa?aeoa?ecoao aaee/eio
?acoeueoe?othuae neeu, eioi?ay aeaenoaoao ia oa?ee; iia iai?aaeaia
(aiec( i?e o>0 (o.a. i?e d>1) e (aaa?o( i?e o0
iu yoo neeo iacaaee (iiaeaiaeiui aanii oa?eea(. Oi aea iacaaiea iu
nio?aiei e i?e o(0. Oaeei ia?acii, iiaeaiaeiue aan oa?eea iiaeao
au?aaeaoueny eae iieiaeeoaeueiui, oae e io?eoeaoaeueiui /eneii eee
ioe?i.
Ia?aeae?i oaia?ue e iaaeyaeiiio enoieeiaaieth (iaoa?eaeueiuo oi/ae(.
Iaoa?eaeueioth oi/eo (A, o) i?e ethaii o (iieiaeeoaeueiii,
io?eoeaoaeueiii eee ?aaiii ioeth) iu iiaeai iaaeyaeii i?aaenoaaeyoue a
aeaea oa?eea, ?acia?aie eioi?iai iiaeii i?aiaa?a/ue, iiiau?iiiai a oi/ea
A e eiathuaai iiaeaiaeiue aan o.
Cia/eo, oi /enei o, eioi?ia iu oneiaeeenue iacuaaoue (iannie(
iaoa?eaeueiie oi/ee, iu enoieeiauaaai, eae (iiaeaiaeiue aan oa?eea(. I?e
o>0 iu iaoa?eaeueioth oi/eo (A, o) iaaeyaeii i?aaenoaaeyai a aeaea
oa?eea, oiiouaai a aiaea (iai?eia?, aeaeaciiai). I?e o0, e i?ioeaiiieiaeii iai?aaeaiu, anee o
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