Imibare
Imibare
nigice cyubumenyi gikubiyemo ingingo zimibare, formula nuburyo bujyanye, imiterere numwanya urimo, numubare nimpinduka zabo. Izi nsanganyamatsiko zigaragara mu mibare igezweho hamwe n’ibice byingenzi by’imibare y’imibare, [1] algebra, [2] geometrie, [1] nisesengura, [3] [4]. Nta bwumvikane rusange mubibare bujyanye nibisobanuro rusange kubijyanye na disipuline yabo. Ibikorwa byinshi byimibare birimo kuvumbura imitungo yibintu bidafatika no gukoresha impamvu zifatika zo kubigaragaza. Ibi bintu bigizwe no gukuramo ibintu biva muri kamere cyangwa - mu mibare igezweho - ibice biteganijwe kugira imitungo runaka, bita axioms. Icyemezo kigizwe nuruhererekane rwo gukurikiza amategeko agabanya ibisubizo bimaze gushingwa. Ibisubizo birimo ibitekerezo byerekanwe mbere, axiom, kandi-mugihe habaye gukuramo ibidukikije - ibintu bimwe na bimwe by'ibanze bifatwa nk'intangiriro nyayo y'igitekerezo kirimo gusuzumwa. Imibare ni ngombwa muri siyansi karemano, ubwubatsi, ubuvuzi, imari, ubumenyi bwa mudasobwa n'ubumenyi mbonezamubano. Nubwo imibare ikoreshwa cyane mugushushanya ibintu, ukuri shingiro kwimibare ntigutandukanye nubushakashatsi ubwo aribwo bwose. Ibice bimwe byimibare, nkibarurishamibare hamwe nimikino yimikino, byatejwe imbere bifitanye isano rya hafi nibisabwa kandi akenshi bishyirwa hamwe mubibare byakoreshejwe. Ibindi bice byatejwe imbere byigenga bivuye mubikorwa ibyo aribyo byose (kandi byitwa imibare itanduye), ariko akenshi bigasanga mubikorwa bifatika. [6] Ikibazo cyo kwishyiriraho integer, kurugero, rusubira muri Euclid muri 300 mbere ya Yesu, ntabwo cyari gifite gahunda ifatika mbere yo gukoreshwa muri cryptosystem ya RSA, ubu ikoreshwa cyane mumutekano wurubuga rwa mudasobwa. Mu mateka, igitekerezo cyo guhamya hamwe n’imibare ifitanye isano n’imibare yagaragaye bwa mbere mu mibare y’ikigereki, cyane cyane mu bice bya Euclid. Kuva yatangira, imibare yagabanijwemo geometrie na arithmetic (manipulation yimibare isanzwe nuduce), kugeza mu kinyejana cya 16 na 17, igihe algebra [a] na calculus itagira ingano yatangijwe nkibice bishya. Kuva icyo gihe, imikoranire hagati yimibare mibare nubuvumbuzi bwa siyansi yatumye ubwiyongere bwihuse bwiterambere ryiterambere ryombi. Mu mpera z'ikinyejana cya 19, ikibazo cy'ifatizo cy'imibare cyatumye habaho uburyo bwa axiomatic, [10] bwatangaje ko ubwiyongere bukabije bw’imibare y’imibare n’imirima yabo ikoreshwa. Imibare Yibihe Byibihe Byiciro Urutonde rurenga 60 murwego rwa mbere rwimibare. Etymology Ijambo imibare rikomoka mu kigereki cya kera máthēma (μάθημα), risobanura "icyigishijwe", [11] "icyo umuntu amenya", bityo rero "kwiga" na "siyanse". Ijambo ryaje kugira ibisobanuro bigufi kandi byinshi bya tekinike y "kwiga imibare" no mubihe bya kera. Inshinga yacyo ni mathēmatikós (μαθηματικός), bisobanura "bijyanye no kwiga" cyangwa "umunyeshuri", ari nako yaje gusobanura "imibare". By'umwihariko, mathēmatikḗ tékhnē (μαθηματικὴ τέχνη; Ikilatini: ars mathica) bisobanura "ubuhanzi bw'imibare". [11] Mu buryo nk'ubwo, rimwe mu mashuri abiri y'ingenzi yatekerezaga muri Pythagoreanism yari azwi ku mibareēmatikoi (μαθηματικοί) - icyo gihe icyo gihe cyasobanuraga "abiga" aho kuba "imibare" mu buryo bwa none. Pythagoreans birashoboka ko abambere babujije gukoresha ijambo kugirango bige gusa imibare na geometrie. Mugihe cya Aristote (384–322 mbere ya Yesu) ubu busobanuro bwarashizweho neza. Mu kilatini, no mu Cyongereza kugeza ahagana mu 1700, ijambo imibare ryakunze gusobanura "inyenyeri" (cyangwa rimwe na rimwe "inyenyeri") aho kuvuga "imibare"; ibisobanuro byahindutse buhoro buhoro kugeza ubu kuva nko mu 1500 kugeza 1800. Iri hinduka ryatumye abantu benshi batavuga rumwe: Urugero, umuburo wa Mutagatifu Agusitini uvuga ko abakristo bagomba kwirinda imibare, bisobanura "abaragurisha inyenyeri", rimwe na rimwe ntibisobanurwa nabi nko kwamagana imibare. 15] Imiterere igaragara mu Cyongereza isubira mu kilatini neuter plural mathica (Cicero), ishingiye ku kigereki cyinshi ta mathēmatiká (τὰ μαθηματικά) kandi isobanura "ibintu byose imibare", nubwo bishoboka ko icyongereza cyatijwe gusa inyito y'imibare (( al) maze akora izina imibare mishya, nyuma yuburyo bwa fiziki na metafizika, twarazwe mu kigereki. Mu cyongereza, izina imibare rifata inshinga imwe. Bikunze kugabanywa kubiharuro cyangwa, muri Amerika ya ruguru, imibare. Ibice by'imibare Mbere yubuzima bushya, imibare yagabanyijwemo ibice bibiri byingenzi: imibare - yerekeranye no gukoresha imibare, na geometrie, bijyanye no kwiga imiterere. Ubwoko bumwebumwe bwa pseudoscience, nka numerologiya na astrologiya, ntabwo noneho byari bitandukanye cyane n'imibare. Mugihe cya Renaissance, hagaragaye utundi turere tubiri . Imibare yerekana imibare iganisha kuri algebra, tuvuze hafi, igizwe nubushakashatsi hamwe no gukoresha formulaire. Ibara, rigizwe nubutaka bubiri butandukanye butandukanye hamwe na calculus yuzuye, ni ubushakashatsi bwibikorwa bikomeza, byerekana ubusanzwe umubano utari umurongo hagati yubwinshi butandukanye, nkuko bigaragazwa nimpinduka. Iri gabana mu bice bine by'ingenzi - imibare, geometrie, algebra, calculus [20] - byakomeje kugeza mu mpera z'ikinyejana cya 19. Ibice nkubukanishi bwo mwijuru hamwe nubukanishi bukomeye noneho bigishijwe nabashinzwe imibare, ariko ubu bifatwa nkibya fiziki. Isomo rya combinatorics ryizwe mu mateka menshi yanditswe, nyamara ntiryabaye ishami ryihariye ry’imibare kugeza mu kinyejana cya cumi na karindwi.
Mu mpera z'ikinyejana cya 19, ikibazo cy'ifatizo mu mibare no gutondekanya uburyo bwa axiomatic bwatumye habaho guturika mu bice bishya by'imibare. [23] Ibyiciro by'Imibare ya 2020 birimo ibyiciro bitarenze mirongo itandatu na bitatu byo mu rwego rwa mbere. Bimwe muribi bice bihuye nigice cyakera, nkuko arukuri kubijyanye numubare wimibare (izina rya kijyambere ryimibare ihanitse) na geometrie. Ibindi bice byinshi byo murwego rwa mbere bifite "geometrie" mumazina yabo cyangwa ubundi bifatwa nkigice cya geometrie. Algebra na calculus ntabwo bigaragara nkibice byo murwego rwa mbere ariko bigabanijwemo ibice byinshi byo murwego rwa mbere. Ibindi bice byo mu rwego rwa mbere byagaragaye mu kinyejana cya 20 cyangwa ntibyari byigeze bifatwa nk'imibare, nka logique y'imibare n'imfatiro.
Umubare wimibare
Ingingo nyamukuru: Igitekerezo cyimibare
Nibwo Ulam spiral, yerekana ikwirakwizwa ryimibare yibanze. Imirongo yijimye ya diagonal yerekana igitekerezo cyubwigenge bugereranijwe hagati yo kuba uwambere no kuba agaciro ka polinomial quadratic, igitekerezo ubu kizwi nka Hardy na Littlewood's Conjecture F.
Umubare wimibare watangiranye no gukoresha imibare, ni ukuvuga imibare isanzwe
(
N.
)
,
{\ kwerekana imiterere (\ mathbb {N}),} hanyuma nyuma yaguka kuri integer
(
Z.
)
{\ kwerekana imiterere (\ mathbb {Z})} nimibare yumvikana
(
Ikibazo
)
.
{\ kwerekana imiterere (\ mathbb {Q}). theory Inyigisho yimibare yigeze kwitwa imibare, ariko muri iki gihe iri jambo rikoreshwa cyane mu kubara. Imibare yatanzwe kuva i Babuloni ya kera kandi birashoboka ko ari Ubushinwa. Abashakashatsi babiri bakomeye bo mu ntangiriro ni Euclid wo mu Bugereki bwa kera na Diyofanti wo muri Alegizandiriya. Ubushakashatsi bugezweho bwibisobanuro byimibare muburyo budasobanutse bwitirirwa Pierre de Fermat na Leonhard Euler. Umurima waje gutanga umusaruro wose hamwe nintererano ya Adrien-Marie Legendre na Carl Friedrich Gauss. [28]
Ibibazo byinshi byavuzwe byoroshye bifite ibisubizo bisaba uburyo buhanitse, akenshi biva mumibare. Urugero rugaragara ni Theorem ya nyuma ya Fermat. Iki gitekerezo cyavuzwe mu 1637 na Pierre de Fermat, ariko byagaragaye mu 1994 gusa na Andrew Wiles, wakoresheje ibikoresho birimo igishushanyo mbonera cya geometrike ya algebraic, teorite y'ibyiciro, na algebra ya homologiya. Urundi rugero ni igitekerezo cya Goldbach, cyemeza ko buri numero irenze 2 nigiteranyo cyimibare ibiri yibanze. Yavuzwe mu 1742 na Christian Goldbach, ntigihamye nubwo hashyizweho ingufu.
Umubare wimibare urimo subareas nyinshi, harimo numero yisesengura yumubare, inyigisho ya algebraic numero, geometrike yimibare (uburyo bwerekanwe), ingero ya diophantine, hamwe nigitekerezo cyo kurenga (ikibazo gishingiye).
Geometrie
Ingingo nyamukuru: Geometrie
Ku buso bwumuzingi, geometrike ya Euclidian ikoreshwa gusa nkaho igereranijwe ryaho. Ku munzani minini igiteranyo cy'imfuruka ya mpandeshatu ntabwo kingana na 180 °.
Geometrie ni rimwe mu mashami ya kera yimibare. Yatangiranye nibisobanuro bifatika bijyanye n'imiterere, nk'imirongo, inguni n'inziga, byakozwe cyane cyane kubikenerwa mu bushakashatsi no mu bwubatsi, ariko kuva ubwo bimera mu bindi bice byinshi.
Agashya k'ibanze ni Abagereki ba kera batangije igitekerezo cyibimenyetso, bisaba ko ibivugwa byose bigomba kwemezwa. Kurugero, ntibihagije kugenzura ukoresheje ibipimo ko, vuga, uburebure bubiri buringaniye; uburinganire bwabo bugomba kugaragazwa hakoreshejwe ibitekerezo bivuye mubisubizo byemewe (theorem) hamwe namagambo make yibanze. Amagambo shingiro ntagengwa nubuhamya kuko arigaragaza (postules), cyangwa ni igice cyibisobanuro byicyigisho (axioms). Iri hame, rishingiye ku mibare yose, ryabanje gusobanurwa kuri geometrie, kandi ryatunganijwe na Euclid ahagana mu mwaka wa 300 mbere ya Yesu mu gitabo cye cyitwa Element. [32]
Geometrie ya Euclidean yavuyemo ni ubushakashatsi bwimiterere nuburyo byateguwe byubatswe kuva kumirongo, indege nizunguruka mu ndege ya Euclidean (indege ya geometrie) hamwe n'umwanya wa Euclidea ufite ibice bitatu. [B] [31]
Euclidean geometrie yatejwe imbere idahinduye uburyo cyangwa urugero kugeza mu kinyejana cya 17, ubwo René Descartes yatangizaga icyo bita coor ya Cartesian Kurya. Ibi byagize impinduka nini ya paradigima: Aho gusobanura imibare nyayo nkuburebure bwumurongo (reba umurongo), yemereye guhagararira amanota ukoresheje imirongo yabo, ari imibare. Algebra (na nyuma, kubara) irashobora gukoreshwa mugukemura ibibazo bya geometrike. Geometrie yagabanyijwemo ibice bibiri bishya: geometrike yubukorikori, ikoresha uburyo bwa geometrike gusa, hamwe na geometrie yisesengura, ikoresha imirongo ikora neza.
Isesengura rya geometrie ryemerera kwiga imirongo itajyanye nuruziga n'imirongo. Imirongo nkiyi irashobora gusobanurwa nkigishushanyo cyimikorere, ubushakashatsi bwerekanwe kuri geometrie itandukanye. Bashobora kandi gusobanurwa nkuburinganire butaziguye, akenshi buringaniza polinomial (byabyaye algebraic geometrie). Isesengura rya geometrie rituma kandi bishoboka gusuzuma umwanya wa Euclidea urenze ibipimo bitatu.
Mu kinyejana cya 19, abahanga mu mibare bavumbuye geometrike itari Euclidea, idakurikiza imyifatire ibangikanye. Mu kwibaza ukuri kw'ukuri, ubu buvumbuzi bwafashwe nko gufatanya na Russell mu kwerekana ikibazo cy'imibare. Iyi ngingo yikibazo yakemuwe no gutunganya uburyo bwa axiomatic, no kwemeza ko ukuri kwa axiom guhitamo atari ikibazo cyimibare. [35] Na none, uburyo bwa axiomatic butuma habaho ubushakashatsi bwa geometrike zitandukanye zabonetse haba muguhindura axiom cyangwa ukareba imitungo idahinduka mugihe cyo guhindura imyanya yihariye.
Uyu munsi subareas ya geometrie irimo: [25]
Umushinga wa geometrie, watangijwe mu kinyejana cya 16 na Girard Desargues, wagura geometrie ya Euclidean wongeyeho ingingo zitagira iherezo imirongo ibangikanye. Ibi byoroshya ibintu byinshi bya geometrike ya kera muguhuza uburyo bwo kuvura no guhuza imirongo.
Affine geometrie, ubushakashatsi bwimiterere ugereranije nuburinganire kandi bwigenga kubitekerezo byuburebure.
Itandukaniro rya geometrie, ubushakashatsi bwimirongo, hejuru, hamwe nibisanzwe, bisobanurwa hakoreshejwe imirimo itandukanye.
Manifold théorie, ubushakashatsi bwimiterere ntabwo byanze bikunze byinjijwe mumwanya munini.
Riemannian geometrie, ubushakashatsi bwimiterere yintera ahantu hagoramye.
Algebraic geometrie, ubushakashatsi bwimirongo, hejuru, hamwe nibisanzwe, bisobanurwa hakoreshejwe polinomial.
Topologiya, ubushakashatsi bwibintu bibikwa munsi yimiterere ihoraho.
Algebraic topologiya, ikoreshwa muri topologiya yuburyo bwa algebraic, cyane cyane algebra ya homologique.
Discomet geometrie, ubushakashatsi bwibishushanyo bitagira ingano muri geometrie.
Geometrie ya convex, ubushakashatsi bwibisobanuro bya convex, ifata akamaro kayo mubikorwa byayo mugutezimbere.
Geometrie igoye, geometrie yabonetse mugusimbuza imibare nyayo nimibare igoye.
Algebra
Ingingo nyamukuru: Algebra
Inzira ya quadratic, isobanura mu buryo bweruye ibisubizo byuburinganire bwose
Itsinda rya Rubik's Cube ni uburyo bukoreshwa bw'amatsinda [37]
Algebra nubuhanga bwo gukoresha ibingana na formula. Diophantus (ikinyejana cya 3) na al-Khwarizmi (ikinyejana cya 9) ni byo bibiri byabanjirije algebra. [38] Diophantus yakemuye ibigereranyo bimwe birimo imibare karemano itazwi akuramo umubano mushya kugeza abonye igisubizo. Al-Khwarizmi yashyizeho uburyo butunganijwe bwo guhindura ibingana, nko kwimura ijambo kuva kuruhande rumwe rwikigereranyo kurundi ruhande. Ijambo algebra rikomoka ku ijambo ry'icyarabu al-jabr risobanura 'guhuza ibice byacitse' [40] yakoresheje mu kwita bumwe muri ubwo buryo mu mutwe w'igitabo cye cy'ingenzi.
Algebra yabaye akarere mu buryo bwonyine hamwe na François Viète (1540–1603), watangije ikoreshwa ry'ibihinduka mu kwerekana imibare itazwi cyangwa idasobanutse. Ibihinduka byemerera imibare gusobanura ibikorwa bigomba gukorwa kumibare ihagarariwe hakoreshejwe imibare.
Kugeza mu kinyejana cya 19, algebra yari igizwe ahanini no kwiga ku kugereranya umurongo (muri iki gihe umurongo wa algebra), hamwe no kugereranya abagore benshi mu kintu kimwe kitazwi, cyiswe algebraic (ijambo rikoreshwa, nubwo rishobora kuba ridasobanutse). Mu kinyejana cya 19, abahanga mu mibare batangiye gukoresha impinduka zerekana ibintu bitari imibare (nka matrices, integer modular, hamwe na geometrike ihinduka), aho usanga ibikorwa rusange bibarwa byemewe. Igitekerezo cyimiterere ya algebraic gikemura ibi, bigizwe nurutonde rwibintu bitamenyekanye, byimikorere ikora kubintu byashizweho, namategeko ibyo bikorwa bigomba gukurikiza. Ingano ya algebra rero yarushijeho kwiyongera gushiramo ubushakashatsi bwimiterere ya algebraic. Iki kintu cya algebra cyiswe algebra igezweho cyangwa algebra abstract, nkuko byagaragajwe nimbaraga za Emmy Noether. (Ijambo ryanyuma rigaragara cyane cyane mubyerekeranye n'uburezi, mu kurwanya algebra y'ibanze, ireba uburyo bwa kera bwo gukoresha formulaire.)
Ubwoko bumwe bwimiterere ya algebraic ifite akamaro kandi akenshi imitungo yose, mubice byinshi byimibare. Ubushakashatsi bwabo bwabaye ibice byigenga bya algebra, kandi birimo: [25]
itsinda ry'itsinda;
inyigisho;
Umwanya wa vector, ubushakashatsi bwabo burasa cyane na algebra y'umurongo;
impeta;
algebra ya commutative, aribwo kwiga impeta zigenda, zirimo kwiga kuri polinomial, kandi ni igice fatizo cya algebraic geometrie;
algebra;
Kubeshya algebra na Kubeshya itsinda;
Bogean algebra, ikoreshwa cyane mukwiga imiterere yumvikana ya mudasobwa.
Ubushakashatsi bwubwoko bwimiterere ya algebraic nkibintu byimibare niyo ntego ya algebra yisi yose hamwe nibyiciro byicyiciro. Iheruka ikoreshwa kuri buri mibare (ntabwo ari algebraic gusa). Ku nkomoko yacyo, yatangijwe, hamwe na algebra ya homologique yo kwemerera algebraic kwiga ibintu bitari algebraic nkibibanza bya topologiya; kariya gace gakoreshwa cyane bita algebraic topology. [45]
Kubara no gusesengura
Ingingo z'ingenzi: Kubara no gusesengura imibare
Urutonde rwa Cauchy rugizwe nibintu bihinduka uko bishakiye uko bikurikirana (uhereye ibumoso ugana iburyo).
Calculus, yahoze yitwa calculus itagira ingano, yatangijwe mu bwigenge kandi icyarimwe n'abashinzwe imibare bo mu kinyejana cya 17 Newton na Leibniz. [46] Nubusanzwe kwiga isano yimpinduka ziterwa nundi. Calculus yaguwe mu kinyejana cya 18 na Euler atangiza igitekerezo cy'imikorere n'ibindi bisubizo byinshi. Kugeza ubu, "calculus" bivuga ahanini igice cyibanze cyiyi nyigisho, kandi "isesengura" rikoreshwa mubice byateye imbere.
Isesengura ryongeye kugabanywa mu isesengura nyaryo, aho impinduka zerekana imibare nyayo, hamwe nisesengura rigoye, aho impinduka zerekana imibare igoye. Isesengura ririmo subareas nyinshi zisangiwe nibindi bice byimibare zirimo: [25]
Ibiharuro byinshi
Isesengura ryimikorere, aho impinduka zerekana imikorere itandukanye;
Kwishyira hamwe, gupima ibitekerezo hamwe nibishobora kubaho, byose bifitanye isano cyane nibitekerezo bishoboka kuri continuum;
Ingano isanzwe itandukanye;
Ingano itandukanye;
Isesengura ryumubare, ryibanze cyane kubara kuri mudasobwa y ibisubizo byibisanzwe nibice bitandukanye bitandukanijwe bivuka mubikorwa byinshi.
Imibare itandukanye
Ingingo nyamukuru: Imibare itandukanye
Igishushanyo cyerekana urwego rwa leta ebyiri Markov. Leta zihagarariwe na 'A' na 'E'. Imibare nibishoboka byo guhinduranya leta.
Imibare yihariye, nukuvuga muri rusange, ni ubushakashatsi bwibintu byihariye, bibarwa. Urugero ni urutonde rwimibare yose. Kubera ko ibintu byo kwiga hano bidasobanutse, uburyo bwo kubara no gusesengura imibare ntibukurikizwa mu buryo butaziguye.
Amabara ane yerekana amabara hamwe nuburyo bwiza bwo gupakira byari ibibazo bibiri byingenzi byimibare yihariye yakemuwe mugice cya kabiri cyikinyejana cya 20. [50] Ikibazo cya P na NP, gisigaye gifunguye kugeza uyu munsi, nacyo ni ingenzi ku mibare yihariye, kubera ko igisubizo cyacyo gishobora kugira ingaruka ku mubare munini w’ibibazo bigoye kubara.
Imibare yihariye ikubiyemo: [25]
Combinatorics, ubuhanga bwo kubara ibintu byimibare byuzuza inzitizi zimwe. Mu ntangiriro, ibyo bintu byari ibintu cyangwa uduce twa seti runaka; ibi byaguwe mubintu bitandukanye, bishyiraho isano ikomeye hagati ya combinatorics nibindi bice byimibare yihariye. Kurugero, geometrie yihariye ikubiyemo kubara ibishushanyo mbonera bya geometrike
Igishushanyo mbonera hamwe na hypergraphs
Inyandikomvugo ya code, harimo ikosora ikosora hamwe nigice cya kriptografiya
Igitekerezo cya Matroid
Geometrie
Ikwirakwizwa rishoboka
Imikino yimikino (nubwo imikino ikomeza nayo yizwe, imikino isanzwe, nka chess na poker birasobanutse)
Gutezimbere neza, harimo guhuza ibitekerezo, guhuza integer, gutangiza imbogamizi
Imibare yimibare no gushyiraho ibitekerezo
Igishushanyo cya Venn nuburyo bukoreshwa muburyo bwo kwerekana isano iri hagati yimikorere.
Ingingo nyamukuru: Imibare yimibare no gushyiraho ibitekerezo
Amasomo abiri ya logique yimibare hamwe nigitekerezo cyashyizweho ni aya mibare kuva mu mpera z'ikinyejana cya 19. [52] [53] Mbere yiki gihe, amaseti ntiyabonwaga nkibintu byimibare, kandi logique, nubwo yakoreshwaga kubiharuro, yari iya filozofiya kandi ntabwo yigeze yigwa nabahanga mu mibare.
Mbere y’ubushakashatsi bwa Cantor butagira umupaka, abahanga mu mibare ntibashakaga gutekereza ku byegeranyo bitagira ingano, kandi babonaga ko ubuziraherezo ari ibisubizo by’ibarura ridashira. Igikorwa cya Cantor cyababaje abahanga mu mibare atari ukureba gusa ibice bitagira ingano [55] ahubwo byerekana ko ibyo bisobanura ubunini butandukanye, nkuko bivugwa na Cantor. Ibyo byatumye havuka impaka zishingiye ku gitekerezo cya Cantor.
Muri gihe kimwe, ibice bitandukanye byimibare byasoje ibisobanuro byambere byimbitse byibintu byibanze byimibare ntibyari bihagije kugirango habeho imibare. Ingero zibyo bisobanuro byimbitse ni "igiteranyo ni icyegeranyo cyibintu", "umubare karemano nicyo gikoreshwa mukubara", "ingingo ni ishusho ifite uburebure bwa zeru muri buri cyerekezo", "umurongo ni inzira yasizwe na ingingo igenda ", n'ibindi.
Ibi byabaye ikibazo cyibanze cyimibare.Byaje gukemurwa mubiharuro rusange byimibare hifashishijwe uburyo bwa axiomatic imbere mubitekerezo byemewe. Mu magambo make, buri kintu cyibibare gisobanurwa nuruhererekane rwibintu byose bisa nibintu bigomba kuba bifite. Kurugero, mu mibare ya Peano, imibare isanzwe isobanurwa na "zeru numubare", "buri mubare ufite umusimbura wihariye", "buri mubare ariko zeru ifite uwabanjirije", hamwe namategeko amwe yo gutekereza. [58] Iyi mibare yakuwe mu byukuri ikubiye muri filozofiya igezweho yo kwishyira ukizana, nk'uko yashinzwe na David Hilbert ahagana mu 1910. [59]
"Kamere" yibintu byasobanuwe muri ubu buryo ni ikibazo cya filozofiya abahanga mu mibare basigira abahanga mu bya filozofiya, kabone niyo abahanga mu mibare baba bafite ibitekerezo kuri iyi kamere, bagakoresha ibitekerezo byabo - rimwe na rimwe bita "intuition" - kugira ngo bayobore imyigire yabo n'ibimenyetso. Inzira yemerera gusuzuma "logique" (ni ukuvuga, gushiraho amategeko yemerewe gukuramo amategeko), theorem, ibimenyetso, nibindi nkibintu byimibare, no kwerekana theorem kuri bo. Kurugero, inyigisho za Gödel zuzuye zishimangira, tuvuze hafi, ko muri buri sisitemu ihamye ikubiyemo imibare karemano, hariho theorem nukuri (ibyo bikaba bigaragarira muri sisitemu ikomeye), ariko ntibigaragara imbere muri sisitemu. Ubu buryo bwo gushingira ku mibare bwamaganwe mu gice cya mbere cy’ikinyejana cya 20 n’abashakashatsi mu mibare bayobowe na Brouwer, wateje imbere ibitekerezo bya intuitionistic, bidafite amategeko agenga hagati. [61] [62]
Ibi bibazo n'impaka byatumye habaho kwaguka kwinshi mu mibare, hamwe na subareas nk'icyitegererezo cy'icyitegererezo (kwerekana inyigisho zimwe na zimwe zumvikana imbere mu zindi nyigisho), inyigisho zerekana, ubwoko bw'imiterere, inyigisho yo kubara hamwe no kubara kubara. [25] Nubwo iyi ngingo ya logique yimibare yatangijwe mbere yukuzamuka kwa mudasobwa, ikoreshwa ryayo mugushushanya, kwemeza porogaramu, abafasha mu bimenyetso nibindi bice bya siyanse ya mudasobwa, byagize uruhare mu kwagura izo nyigisho zumvikana.
Ibarurishamibare hamwe nubundi bumenyi
Ingingo z'ingenzi: Imibare n'ibitekerezo bishoboka
Uburyo ubwo aribwo buryo bwo gukwirakwiza abantu ku buryo butemewe (μ), icyitegererezo gisobanura (x̄) gikunda kugabanywa kwa Gawusiya kandi itandukaniro ryacyo (σ) ritangwa n’imipaka yo hagati y’ibitekerezo bishoboka. [64]
Umwanya wibarurishamibare ni imibare ikoreshwa mukusanya no gutunganya amakuru yintangarugero, ukoresheje inzira zishingiye kuburyo bw'imibare cyane cyane inyigisho zishoboka. Abashinzwe ibarurishamibare batanga amakuru hamwe nubushakashatsi bwateganijwe. Igishushanyo mbonera cyibigereranyo cyangwa igerageza bigena uburyo bwisesengura buzakoreshwa. Isesengura ryamakuru yavuye mubushakashatsi bwo kwitegereza bikorwa hifashishijwe urugero rwibarurishamibare hamwe nigitekerezo cyo gufata umwanzuro, ukoresheje guhitamo icyitegererezo. Icyitegererezo hamwe nibisubizo byavuzwe bigomba kugeragezwa hifashishijwe amakuru mashya.
Ibarurishamibare ryiga ibibazo byicyemezo nko kugabanya ingaruka (igihombo giteganijwe) cyigikorwa cyibarurishamibare, nko gukoresha uburyo, urugero, kugereranya ibipimo, gupima hypothesis, no guhitamo ibyiza. Muri ibi bice gakondo byimibare yimibare, ikibazo cyibarurishamibare-cyemezo gishyirwaho mugabanya kugabanya ibikorwa bifatika, nkigihombo giteganijwe cyangwa ikiguzi, mubibazo byihariye. Kurugero, gutegura ubushakashatsi akenshi bikubiyemo kugabanya ikiguzi cyo kugereranya abaturage bivuze nurwego runaka rwicyizere. Kubera gukoresha uburyo bwiza, inyigisho yimibare yimibare ihurira hamwe nubundi bumenyi bwicyemezo, nkubushakashatsi bwibikorwa, igenzura, nubukungu bwimibare. [67]
Imibare yo kubara
Ingingo nyamukuru: Imibare yo kubara
Imibare yo kubara ni ubushakashatsi bwibibazo byimibare mubisanzwe binini cyane kubushobozi bwabantu, mubare. [68] [69] Isesengura ryumubare ryiga uburyo bwibibazo mu isesengura ukoresheje isesengura ryimikorere hamwe nigitekerezo cyo kugereranya; Isesengura ry'imibare rigizwe cyane no kwiga kugereranya no gushishoza hibandwa cyane ku makosa azenguruka. Isesengura rishingiye ku mibare kandi, mu buryo bwagutse, kubara siyanse nabyo byiga ingingo zidasesenguye siyanse yubumenyi, cyane cyane algorithmic-matrix-na-graphique. Ibindi bice by'imibare yo kubara harimo mudasobwa algebra no kubara ibimenyetso.
Amateka
Ingingo nyamukuru: Amateka yimibare
Kera
Amateka yimibare nuruhererekane rwiyongera rwuruhererekane. Ubwihindurize, abstraction igomba kuvumburwa, imwe isangiwe ninyamaswa nyinshi, [71] birashoboka ko yari iyo mibare: kumenya ko, urugero, icyegeranyo cya pome ebyiri hamwe nicyegeranyo cyamacunga abiri (vuga) bifite icyo bihuriyeho, aribyo ko hariya ni bibiri muri byo. Nkuko bigaragazwa nuburebure buboneka ku magufa, usibye kumenya uburyo bwo kubara ibintu bifatika, abantu babanjirije amateka bashobora kuba bari bazi kubara ibintu bidafatika, nkigihe - iminsi, ibihe, cyangwa imyaka. [72] [73]
Ikibaho cy'imibare y'i Babiloni Plimpton 322, yo mu 1800 mbere ya Yesu
Ibimenyetso ku mibare igoye ntibigaragara nko mu 3000 mbere ya Yesu, igihe Abanyababiloni n'Abanyamisiri batangiraga gukoresha imibare, algebra, na geometrie mu gusoresha no kubara amafaranga, mu kubaka no kubaka, no mu bumenyi bw'ikirere. Imyandikire ya kera ya mibare yo muri Mezopotamiya na Egiputa ni kuva 2000 kugeza 1800 mbere ya Yesu. Inyandiko nyinshi zo hambere zivuga inshuro eshatu Pythagorean, nukuvuga, theorem ya Pythagorean isa nkigitekerezo cya kera cyane kandi cyagutse cyane nyuma yimibare yibanze na geometrie. Mu mibare ya Babiloni niho imibare y'ibanze (kongeramo, gukuramo, kugwiza, no kugabana) igaragara bwa mbere mu mateka ya kera. Abanyababuloni na bo bari bafite gahunda-y’ahantu kandi bakoresha sisitemu y’imibonano mpuzabitsina iracyakoreshwa muri iki gihe mu gupima inguni n’igihe.
Mu kinyejana cya 6 mbere ya Yesu, imibare y’Abagereki yatangiye kwigaragaza nka disipuline itandukanye kandi Abagereki bamwe na bamwe nka Pythagore basaga nkaho babonaga ko ari ingingo ubwayo. [76] Ahagana mu mwaka wa 300 mbere ya Yesu, Euclid yateguye ubumenyi bw'imibare hakoreshejwe uburyo bwa posita n'amahame ya mbere, byaje guhinduka uburyo bwa axiomatic bukoreshwa mu mibare muri iki gihe, bugizwe n'ibisobanuro, axiom, theorem, n'ibimenyetso. Igitabo cye cyitwa Element, gifatwa nkigitabo cyatsinze kandi gikomeye mu bihe byose. Imibare ikomeye cyane mu bihe bya kera ikunze gufatwa nka Archimedes (nko mu 287-212 mbere ya Yesu) ya Syracuse. Yateguye uburyo bwo kubara ubuso n'ubunini bwa solide ya revolution kandi akoresha uburyo bwo kunanirwa kubara agace kari munsi ya arc ya parabola hamwe nincamake yuruhererekane rutagira umupaka, muburyo budasa cyane na calculus zigezweho. ] Ibindi byagaragaye mu mibare y’Abagereki ni ibice bya conic (Apolloniyo wa Perga, ikinyejana cya 3 mbere ya Yesu), [81] trigonometrie (Hipparchus wa Nicaea, ikinyejana cya 2 mbere ya Yesu), 83]
Imibare yakoreshejwe mu nyandiko yandikishijwe intoki ya Bakhshali, yanditswe hagati y'ikinyejana cya 2 mbere ya Yesu n'ikinyejana cya 2 nyuma ya Yesu
Sisitemu y'imibare y'Abahindu n'Abarabu n'amategeko yo gukoresha ibikorwa byayo, ikoreshwa ku isi yose muri iki gihe, yahindutse mu gihe cy'ikinyejana cya mbere nyuma ya Yesu mu Buhinde kandi yanduzwa mu bihugu by'iburengerazuba hakoreshejwe imibare ya kisilamu. Ibindi bintu byingenzi byagaragaye mu mibare yo mu Buhinde harimo ibisobanuro bigezweho no kugereranya sine na cosine, hamwe nuburyo bwambere bwuruhererekane rutagira umupaka. [85]
Hagati na nyuma
Urupapuro rwo muri Algebra ya Al-Khwārizmī
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Mu gihe cya Zahabu ya Islamu, cyane cyane mu kinyejana cya 9 n'icya 10, imibare yabonye udushya twinshi twubakiye ku mibare y'Abagereki. Ikintu cyagaragaye cyane mu mibare ya kisilamu ni iterambere rya algebra. Ibindi byagezweho mu gihe cya kisilamu harimo gutera imbere muri trigonometrie ya sphericale no kongeramo ingingo icumi kuri sisitemu y’imibare y’abarabu. Abahanga mu mibare benshi bazwi muri iki gihe bari Abaperesi, nka Al-Khwarismi, Omar Khayyam na Sharaf al-Dīn al-Ṭūsī. [88] Imyandikire y’imibare y’ikigereki n’icyarabu nayo yahinduwe mu kilatini mu gihe cyagati cyo hagati kandi iboneka mu Burayi.
Mugihe cyambere cya kijyambere, imibare yatangiye gutera imbere muburyo bwihuse muburayi bwiburengerazuba, hamwe nudushya twahinduye imibare, nko kwinjiza impinduka hamwe nibimenyetso byikigereranyo byanditswe na François Viète (1540–1603), hashyizweho imirongo ya René Descartes ( 1596–1650) yo kugabanya geometrie kuri algebra, no guteza imbere kubara na Isaac Newton (1642–1726 / 27) na Gottfried Leibniz (1646–1716) mu kinyejana cya 17. Leonhard Euler (1707–1783), umunyamibare uzwi cyane mu mibare yo mu kinyejana cya 18, yahujije ibyo bishya mu mubiri umwe hamwe n’amagambo asanzwe, maze abirangiza avumbura kandi ahamya ibimenyetso byinshi. Birashoboka ko imibare ya mbere mu mibare yo mu kinyejana cya 19 yari umuhanga mu mibare w’umudage Carl Gauss, wagize uruhare runini mu bice nka algebra, isesengura, geometrie itandukanye, inyigisho ya matrix, inyigisho z’imibare, n'imibare. Mu ntangiriro z'ikinyejana cya 20, Kurt Gödel yahinduye imibare atangaza ibitekerezo bye bituzuye, byerekana igice ko sisitemu iyo ari yo yose ihamye - niba ifite imbaraga zihagije zo gusobanura imibare - izaba irimo ibitekerezo nyabyo bidashobora kwemezwa.
Imibare kuva yaguwe cyane, kandi habaye inte yera imbuto