Kuyini izibalo? ifayela echaza ifomu temkhakha izibalo. izibalo kanambambili

Kuyini izibalo? Lapho isintu baqala ukusebenzisa izinombolo futhi sisebenze nabo? Kuphi izimpande zawo imiqondo kwansuku like izinombolo, amafraktjhini, ukususa, kwalokho kanye ukubuyabuyelela, lowo muntu uye wenza yingxenye ebalulekile yokuphila kwakhe nesimo sengqondo? izingqondo ngesiGreki bayahlonishwa wesayensi ezifana izibalo, izibalo kanye geometry, njengoba symphony enhle logic womuntu.

Mhlawumbe izibalo akuyona njengoba ejulile njengoba neminye imikhakha yesayensi, kodwa lokho okwakuyokwenzeka ku kubo, abantu ukhohlwe aphansi ukubuyabuyelela amatafula? Ajwayelekile kithi ukucabanga okunengqondo, usebenzisa izinombolo, amafraktjhini, kanye namanye amathuluzi ukunikeza abantu nesikhathi esinzima, futhi isikhathi eside akatholakalanga obaba. Eqinisweni, ngaphambi kokuba ukuthuthukiswa kwe-arithmetic akukho ndawo bolwazi lwabantu kwakungelona yesayensi ngempela.

Arithmetic - Tibalo zamagama

Arithmetic - isayensi izinombolo, kanye noma yimuphi umuntu iqala ukwazana nezwe elithakazelisayo wezibalo. In amazwi M. V Lomonosov, izibalo - yisango yokufunda, okwavula amathuba ukuba Miropoznanie. Kepha kulungile, kuyinto ulwazi oluthuthukayo lomhlaba kungenziwa uhlukane ulwazi izinhlamvu nezinombolo, mathematics kanye inkulumo? Mhlawumbe ezinsukwini ubudala, kodwa hhayi ezweni lanamuhla, lapho ukuthuthukiswa ngokushesha isayensi nobuchwepheshe Sibeka imithetho yaso.

Igama elithi "izibalo" (Gk. "Arifmos") odabuka ngesiGreki, kusho "inombolo". Sidingida inombolo bese konke kungenziwa ahlobene nawo. Lona izwe izinombolo: imisebenzi ehlukahlukene on izinombolo, imithetho zezinombolo, imisebenzi okuhlotshaniswa ukubuyabuyelela, ukususa, njalo njalo ..

Ngokwejwayelekile kwamukelekile ukuthi Isinyathelo sokuqala Lezibalo izibalo futhi base eliqinile eziyinkimbinkimbi ezingxenyeni zalo, ezifana-algebra, izingabunjalo, izibalo ephakeme futhi t. D.

Into eyinhloko izibalo

Ngesisekelo izibalo - kuyinto inamba, izakhiwo kanye nemithetho okuyinto zibhekwa izibalo eliphakeme noma inombolo theory. Eqinisweni, indlela efanele esithathwayo ngokucabangela iyunithi elincane enjalo, njengoba inombolo yemvelo emandleni lesi sakhiwo - wezibalo.

Ngakho-ke, lo mbuzo izibalo, impendulo ilula: kuba isayensi izinombolo. Yebo, mayelana evamile eziyisikhombisa, ayisishiyagalolunye, futhi konke lokhu umphakathi ahlukahlukene. Futhi kanye nje, futhi la mavesi Okuqukethwe akuzwakali kakhulu akakwazi ukubhala ngaphandle zamagama eziyisisekelo, ngaphandle izibalo ayikwazi ixazululwe ngisho imisebenzi eyisisekelo. Yingakho zonke sciences isihambile kuphela ngemva ukuthuthukiswa izibalo nezibalo ezingamaGreki, imfundo nokuba ngokuyinhloko iqoqo ezizathwini.

Arithmetic - isayensi-isipoki

Kuyini izibalo - isayensi yemvelo noma phantom? Eqinisweni, njengoba izazi zefilosofi yamaGreki asendulo wabonisana, lingabe nezinombolo, akukho izibalo empeleni ayikho. Kumane phantom, okudaliwe e ukucabanga lapho ebukela imvelo kanye izinqubo zayo. Empeleni, yini inombolo? Akukho ndawo emhlabeni thina awuboni okunje yayingase ibizwe ngokuthi u inombolo, kunalokho, inombolo - kuyindlela ukuhlola izwe yingqondo yomuntu. Mhlawumbe lolu cwaningo esinalo themselves? Bamafilosofi phikisana lokhu ngoba emakhulwini amaningi eminyaka ilandelana, ngakho ukunikeza impendulo oluphelele asikhuthazi. Kunoma yikuphi, izibalo ngingalenza ngokuqinile ukuba zime ezindaweni zazo ezweni lanamuhla akekho angabhekwa emphakathini ingashintshwa kungesemehlweni izisekelo zalo.

Njengoba sasingekho nenombolo evumayo ephelele

Yiqiniso, into eyinhloko nesebenzela izibalo, - inombolo zemvelo ezifana 1, 2, 3, 4, ..., 152 ... njll Arithmetic izinombolo zemvelo kuwumphumela bangakunaki izinto ezejwayelekile, ezifana izinkomo endaweni emnandi. Noma kunjalo, kwencazelo "okuningi" noma "kancane" uma into isidlulile ukubamba abantu, futhi kwadingeka baziqambela ezingcono nokubala imiklamo.

Kodwa ukufohla yangempela lapho umqondo womuntu ifinyelele eqophelweni lapho ingaba yinye futhi inani elifanayo "ezimbili" ukuqoka futhi 2 kg, futhi 2 isitini futhi izingxenye 2. Iqiniso lokuthi kubalulekile abstract kusuka amafomu, izici kanye nencazelo izinto, khona-ke singazuza ukukhiqiza isenzo esithile nalaba izinto ngesimo integers omuhle. Yavela kanjalo-ke izibalo izinombolo, okuyinto athuthukiswe futhi titfutfukiswe e bangazihluphi ngokuthola isikhundla emphakathini.

Enjalo e-ukujula lwati lwenombolo, njengoba kuqanda nezinombolo negative, amafraktjhini, izinombolo ubheke izinombolo ngezinye izindlela, abe umlando onothileko ezithakazelisayo yentuthuko.

Arithmetic futhi ezingokoqobo abaseGibhithe

Amabili umngane womuntu lasendulo ekuhloleni izwe nokuxazulula izinkinga zansuku zonke - lokhu izibalo futhi geometry.

Kukholakala ukuthi umlando izibalo has kwaqala eMpumalanga sidilizwa: India, Egypt, iBhabhiloni kanye China. Ngakho, Rhind zebungu umsuka waseGibhithe (ibizwa kanjalo ngoba igama elifanayo okuqondene umnikazi) eyaqala ngekhulu XX. BC, ngaphezu neminye imininingo ezibalulekile ihlanganisa ukunwetshwa ingxenyenamba e inani zimbili izingxenye denominators ezahlukene kanye numerator lingana kuze fike.

Ngokwesibonelo: = 1/60 + 2/73 1/219 + 1/292 + 1/365 .

Kodwa uyini inshokutsi yetinhlobo noqhekeko eziyinkimbinkimbi kangaka? Iqiniso lokuthi indlela waseGibhithe akakubekezeleli abstracted ecabanga izinombolo, kunalokho, izibalo enziwa kuphela izinjongo ezifanele. Okungukuthi, abaseGibhithe ngeke behileleke nakwezamabhizinisi ezifana izibalo, kuphela ukuze kwakhiwe ethuneni, isibonelo. Kwakudingekile ukubala ubude isakhiwo fin, futhi wenza ukuba umuntu ahlale-papyrus. Njengoba singabona, intuthuko baseGibhithe ngo izibalo lalibizwa kanjalo, kunalokho omkhulu, ukwakha, kunokuba umuntu abe nothando isayensi.

Ngenxa yalesi sizathu, izibalo ezitholakele ku-papyrus, akakwazi ngokuthi ukucabanga ngendaba izingxenyana zaso. Ezimweni eziningi, lokhu kuyinto ukulungiselela esisebenzayo, okwasisiza ukuba eminye uxazulule izinkinga enamafrakshini. AbaseGibhithe lasendulo angazanga etafuleni ukubuyabuyelela, bakhiqiza izibalo sasivumela ezinde, asakazekela subtasks eziningi. Mhlawumbe lokhu ungomunye walabo subtasks. Kulula ukuqaphela ukuthi izibalo nalezi izikhala kakhulu isikhathi esiningi futhi hhayi ethembisa kakhulu. Mhlawumbe ngenxa yalesi sizathu thina awuboni umnikelo omkhulu ekuthuthukiseni wezibalo baseGibhithe lasendulo.

IGrisi yasendulo futhi izibalo zefilosofi

Abaningi ulwazi East sidilizwa abangu yingcweti ngempumelelo amaGreki asendulo, eyaziwa kubalandeli reflection abstract, abstract nefilosofi. Prakthiza kubo nentshisekelo ukubaqeda kodwa theorists engcono ongqondongqondo okunzima ukubathola. Kwaba kuhle isayensi ngoba izibalo akunakwenzeka ukuya ezijulile, hhayi edwengula ngayo ngokoqobo. Yiqiniso, kungenzeka ukuba azale ande izinkomo 10 namalitha 100 ubisi, kodwa ngeke bakwazi ukuhambisa kude.

AmaGreki Ukucabanga kwesokunxele ngokujulile uphawu abalulekile emlandweni, kanye nemisebenzi yabo, sondela us:

• Euclid futhi "Imisuka".
• Pythagoras.
• Archimedes.
• Eratosthenes.
• Zénon.
• Anaxagoras.

Futhi-ke, uphendulela yonke nefilosofi yamaGreki, futhi ikakhulukazi abalandeli Pythagoras izigameko eziningi abanothando kangaka mayelana izinombolo, okuyinto kubhekwa kubo ngokuvumelana imfihlakalo emhlabeni. Izinombolo ziye ngakho wafunda futhi uphenyo, ukuthi ezinye zazo futhi imibhangqwana zabo kuchasiselwe anezakhi ezikhethekile. Ngokwesibonelo:

• izinombolo Perfect - leyo isamba sazo zonke divisors yayo ngaphandle inombolo uqobo (6 = 1 + 2 + 3).
• izinombolo friendly - lezi zinombolo, enye yazo isamba sonke divisors we nalokho yesibili kungokufanayo (Pythagorean ukwazi pair eyodwa kuphela enjalo: 220 kanye 284).

AmaGreki, owayekholelwa ukuthi isayensi kufanele obathandayo, hhayi ukuba naye ngoba ngenxa yenzuzo, igxathu, ukuhlola, ukudlala futhi unezela izinombolo. Kufanele kuqashelwe ukuthi akuwona wonke yocwaningo lwalo ziye kabanzi, abanye babo babephakathi kuphela "ubuhle."

ongqondongqondo Eastern Ephakathi

Ngokufanayo, ngeNkathi Ephakathi izibalo ukweleta ukuthuthukiswa yayo esikhathi empumalanga. AmaNdiya wasinika izibalo ukuthi ngenkuthalo ukusebenzisa into ngokuthi "zero" enjalo, kanye nesikhundla ukushintshashintsha uhlelo ukubala, evamile Umbono yesimanje. Kusukela Al-iphalishi, okuyinto ngekhulu le-15 wasebenza Samarkand, siye sazuza le amadesimali, ngaphandle okuyinto kunzima ukucabanga izibalo yesimanje.

Ngezindlela eziningi, I-Europe bajwayelane ipumelelo East senziwa Sibonga kungenzeka ukuba umsebenzi usosayensi wase-Italy Leonardo Fibonacci, owabhala incwadi "Liber Abaci", zazi ne emisha asempumalanga. Sekuvele legumbi ekusungulweni kwe-algebra futhi izibalo, ucwaningo kanye nemisebenzi yesayensi eYurophu.

izibalo Russian

Ekugcineni, i-arithmetic, ithole endaweni yalo futhi sisekelwe Europe, waqala ukusakaza kusezweni Russian. Russian izibalo iqale ukunyatheliswa ngo 1703 - kwaba incwadi mayelana izibalo Leontiya Magnitskogo. Kwaphela isikhathi eside kwaba okokufundisa kuphela kwi-mathematics. Iqukethe izikhathi kokuqala kwe-algebra futhi geometry. Izibalo, okuyinto zasetshenziswa izibonelo ebhukwinitifundvo waseRussia lokuqala izibalo, Arabhu. Nakuba zesi-Arabhu aye ahlangana ngaphambi, e kwendandatho eyaqala ngekhulu le-17.

Incwadi ngokwayo lihlotshiswe izithombe Archimedes futhi Pythagoras, futhi ngolokuqala ikhasi - isithombe izibalo njengomuntu wesifazane. Yena uhlezi esihlalweni sobukhosi, ngaphansi kulotshiwe igama lesiHeberu elisho igama likaNkulunkulu, futhi ezitebhisini eziholela altare, liqoshwe izwi "ukuhlukanisa", "ukwanda", "Ngaphezu kwalokho", njalo njalo. D. Omunye nje ngani isiyakhashelwa maqiniso, okuyinto manje sezibhekwa njengezindawo kuyinsakavukela.

Incwadi ka Amakhasi 600 uchaza njengoba ngesisekelo kwalokho like kanye ukubuyabuyelela amatafula, futhi izicelo wesayensi olwandle.

Akumangalisi ukuthi umbhali ekhethe isithombe ongqondongqondo lesiGreki elisho incwadi yakhe, ngoba yena ngokwakhe Yangihlaba umxhwele ubuhle izibalo, ethi, "Aritmethikhi uye chislitelnitsa lapho wena fair, nezavistnoe ...". Le ndlela izibalo is thungela, ngoba ukwamukelwa kwayo abantu abaningi. angabhekwa ekuqaleni ukuthuthukiswa ngokushesha ukucabanga yesayensi eRussia nemfundo jikelele.

primes ngingakhululeki

inombolo Prime - kuba inombolo yemvelo, okuyinto divisors omuhle 2 kuphela: 1 futhi uqobo. Zonke ezinye izinombolo, ngaphandle 1 libizwa oluyinhlanganisela. Izibonelo izinombolo eziyinhloko: 2, 3, 5, 7, 11, futhi bonke abanye ukuthi akuzona ngaphandle 1 divisors futhi nenombolo uqobo.

Njengoba for inombolo 1, ke singekho - kukhona ukuvumelana ukuthi kufanele kubhekwe futhi elula noma kwakwakhe. Simple Uma unganakile, inombolo elula uyazifihla eziningi eziyimfihlakalo ezingaxazululeki ngaphakathi ngokwabo.

ifayela echaza ifomu Euclid lithi i isibalo esingapheliyo primes, futhi Eratosthenes necebo izibalo "ziye ngapha nangapha ngesisefo" ekhethekile, okwenza kungabi izinombolo eziyinkimbinkimbi, kusale kuphela elula.

Ingqikithi yalo iwukuba ukugcizelela inombolo yokuqala uyeke ukususa izinguqulo, futhi okuphawuleka okwalandela ngaphandle leyo ngeembuyabuyelelo ke. Siphindaphinda le nqubo izikhathi eziningana - bese ithebula izinombolo eziyinhloko.

ifayela echaza ifomu temkhakha izibalo

Phakathi kokuma mayelana izinombolo eziyinhloko kudingeka ngokukhethekile ingasaphathwa eziyisisekelo izibalo theorem.

Basic theorem izibalo lithi noma iyiphi inombolo ephelele engaphezu kuka-1, noma elula noma kungenziwa ukubola ibe umkhiqizo izinombolo eziyinhloko kuze kube oda ukuphindaphinda izici, okuwukuphela kwendlela.

ifayela echaza ifomu temkhakha izibalo wazibonakalisa ngempela nzima, futhi ukuqonda awufani izisekelo nje.

Uma uthi nhlá, izinombolo prime - umqondo aphansi, kodwa akuyona. Physics futhi wake wacabanga athomu aphansi, aze abathole ngaphakathi yonke. Primes lanikezela amahle indaba sezibalo Don Zagier "Eyokuqala kwezigidi ezingu amahlanu izinombolo eziyinhloko."

Kusukela "amhabhula amathathu" imithetho deductive

Lokho ngempela kungenziwa ngokuthi isisekelo kwagcizelela yazo zonke isayensi - imithetho izibalo. Ngisho nalapho ngiseyingane bonke ubuso izibalo, ukutadisha inani imilenze nezingalo ngesikhathi onodoli, inani cubes, apula kanye nokunye. D. Ngakho sitadisha izibalo, okuyinto ke iqhubeka ku imithetho eziyinkimbinkimbi.

Manje ukuphila kwethu konke liqala ukusitshela ngo-imithetho izibalo, ezazingezegceke nomuntu ovamile ewusizo kunazo zonke ukuthi isayensi kunikeza. Lolu cwaningo izinombolo - kuba "Aritmethikhi-ingane", eyethula indoda emhlabeni izinombolo njengoba amadijithi ekuqaleni kuka ebuntwaneni.

Arithmetic Ephakeme - isayensi deductive ukuthi ufunda imithetho izibalo. Iningi lazo siyazi, nakuba mhlawumbe asazi okubekwe ngayo zabo ngqo.

Umthetho kwalokho kanye ukubuyabuyelela

Noma yimuphi integers ezimbili kanye b ungaboniswa njengoba isamba b, + okuyindawo futhi inombolo zemvelo. Ngokuphathelene kwalokho, imithetho elandelayo:

• Commutative, ukuthi lithi permutation imigomo ubeka imali ayishintshi, noma + b = b + a.
• Associative ukuthi kusho isamba akuxhomekile indlela behlukanisa imigomo ezindaweni, noma + (b + c) = (a + b) + c.

Imithetho izibalo, ezifana kwalokho, - omunye eyisisekelo, kodwa benza konke ababengakwenza sciences, ingasaphathwa ekuphileni kwansuku zonke.

Noma yimuphi integers ezimbili kanye b ungaboniswa umkhiqizo noma b * b *, okuyilona elisebenzayo futhi inombolo zemvelo. Ukwenza umkhiqizo imithetho efanayo commutative kanye associative ngokuthi ukwengeza:

• a * b = b * a;
• a * (b * c) = (a b *) * c.

Kuyathakazelisa ukuthi kukhona umthetho, elihlanganisa kwalokho kanye ukubuyabuyelela, eyaziwa ngokuthi yi- ukusatshalaliswa noma umthetho wokusakaza:

a (b + c) = ab + ac

Lo mthetho usifundisa ukusebenza kubakaki, ukuvula kubo, ngaleyo ndlela sikwazi kakade sisebenze amafomula eziyinkimbinkimbi. Lezi imithetho eziyoholela nathi izwe kuyisidala-kuyathandeka kodwa eziyinkimbinkimbi algebra.

Umthetho oda izibalo

ngemithetho logic womuntu lisebenzisa usuku ngalunye, ebheka iwashi nokubala izikweletu. Futhi, nokho, futhi kufanele kwenziwe ngolimi oluthile.

Uma sinalo integers ezimbili omuhle futhi b ke kukho okulandelayo:

• a ilingana b, noma = b;
• a esingaphansi b, noma
• a mkhulu kunezinhliziyo b, noma> b.

Kulezi ezintathu eziphawulwe nje ingaba yinye kuphela. I-Basic Law, okuyinto elawula inqubo, wathi: uma

Kukhona nemithetho ezenza izenzo ukuhlela ukuhlanganisa kanye ukubuyabuyelela: uma

Imithetho izibalo wasifundisa ukusebenza ngezinombolo, izimpawu kanye kubakaki, ukuvula konke ibe symphony avumelanayo izinombolo.

Positional futhi nonpositional uhlelo izinombolo

Singasho ukuthi izinombolo - lena ilimi le-IiMbalo, kusuka kube lula lapho kuxhomeke ezintweni eziningi. Kukhona izinhlelo eziningi ukubala, okuyinto, efana zezilimi izilimu kuhluke.

Cabanga ohlelweni inombolo kusukela iphuzu umthelela izikhundla ku ukubaluleka ambalwa of idijithi kulesi sikhundla. Ngokwesibonelo, uhlelo laseRoma nonpositional lapho inombolo ngayinye ifakwe ngu isethi ezithile izinhlamvu ezikhethekile: I / V / X / L / C / D / M. Ababona, ngokulandelana, izinombolo 1/5/10/50/100/500 / 1000. Kulesi simiso, lesi sibalo ayishintshi ukuzimisela yayo ambalwa, kuye ngasiphi isikhundla kumele: .. I kokuthoma, kwesibili, njll Ukuze uthole ezinye izinombolo, kubalulekile ukubeka phansi base. Ngokwesibonelo:

• DCC = 700.
• CCM = 800.

Okuningi ajwayelekile kithi uhlelo inani usebenzisa zesi-Arabhu kuyinto positional. Esimisweni ezifana nenani osenyameni ichaza inombolo kwamadijithi, isibonelo, enamadijithi amathathu-izinombolo: 333, 567, njll Isisindo yimiphi osenyameni incike isikhundla lapho bangamaphesenti omunye, isb isibalo 8 kule sikhundla yesibili inenani 80 Kuyinto ejwayelekile uhlelo idesimali, kunezinye ohlelweni positional ezifana kanambambili.

izibalo kanambambili

Singabantu ajwayelekile idesimali uhlelo, ehlanganisa olulodwa-bit ne-multi-bit izinombolo. Isibalo ngakwesokunxele e inombolo enamadijithi aphindwa izikhathi eziyishumi enkulu e ibalulekile kulowo ngakwesokudla. Ngakho, sasivame ukufunda 2, 17, 467, nokunye. D. Kuyinto yikuphi ukuhlakanipha indlela kwesigaba ezahlukene, okuthiwa "izibalo kanambambili." Lokhu akusona isimanga, ngoba izibalo kanambambili ayidaliwe ngoba logic womuntu, futhi computer. Uma izibalo izinombolo uvela nokubala, elibuye adonswa kusukela impahla kuncike ngokuthi "enqunu" izibalo ke lokhu ngeke isebenze nge computer yakho. Ukuze sikwazi ukudlulisela ulwazi lwabo ne-computer, indoda kwadingeka ukuba usungula ukubala imodeli.

izibalo Binary isebenza nge zamagama kanambambili, esakhiwa kuphela 0 no 1. Futhi ukusetshenziswa kwalesi zamagama ibizwa ngokuthi uhlelo kanambambili.

Ngokungafani kanambambili idesimali izibalo ukuthi ukubaluleka isikhundla kwesokunxele Akasekho 10, nezikhathi 2. izinombolo kanambambili yefomu D. Kufanele siwaqonde kanjani la izinombolo 111, 1001 kanye nokunye.? Ngakho, sicabanga inombolo 1100

1. Idijithi yokuqala ngakwesokunxele - 1 * 8 = 8, akhumbula bona idijithi wesine, okusho ukuthi kumele iphindwe 2, sithola 8 isikhundla.
2. idijithi Okwesibili 1 * 4 = 4 (isikhundla 4).
3. Idijithi lwesithathu 0 * 2 = 0 (isikhundla 2).
4. Idijithi wesine 0 * 1 = 0 (isikhundla 1).
5. Ngakho inombolo yethu 1100 = 8 + 4 + 0 + 0 = 12.

Okungukuthi, kwathatha isigaba esisha kwesokunxele okukushoyo ohlelweni kanambambili iyanda yi 2 kanye idesimali - ukufika e-10 uhlelo okunjalo yayinesithiyo esisodwa:-ke likhulu kakhulu ukukhula ibhithi ukuthi ziyadingeka ukurekhoda izinombolo. Izibonelo izinombolo zamadesimali dvochinyh njengoba kungabonakala ithebula.

izinombolo zamadesimali ivezwa ngezindlela ezihamba ngakubili ngezansi.

Iphinde isetshenziselwe octal, kanye uhlelo i-hexadecimal izinombolo.

Lokhu izibalo ongaqondakali

Kuyini izibalo, "ezimbili plus ezimbili" noma izimfihlakalo nhlobo izinombolo? Njengoba ubona, izibalo, can, futhi kubonakala efika kuqala elula, kodwa boni kalula okukhohlisayo. Kungenzeka ukufunda izingane, futhi kanye Anti Owl kusukela opopayi "Aritmethikhi-ingane", futhi ungakwazi bagxuma ucwaningo lwesayensi olujulile oda cishe zefilosofi. Emlandweni ke imukile nokubala izinto ukukhulekela ubuhle izinombolo. Iyodwa into esiqiniseka ngayo: nge ukusungulwa the postulates eziyisisekelo izibalo, isayensi bangathembela ehlombe eqinile.