Izinombolo zangempela kanye nezakhiwo zabo

UPythagoras wathi le nombolo isezansi kwezwe ngokuhambisana nezici eziyinhloko. UPlato wayekholelwa ukuthi inombolo ixhuma lesi simo kanye no-noumenon, isiza ukuqonda, ukukala nokudweba iziphetho. I-Arithmetic ivela egameni elithi "arithmos" - inombolo, ekuqaleni kwaqala ngezibalo. I-Nim ingachaza noma yikuphi into - kusuka e-apula esisekelo kuya ezindaweni ezingabonakali.

Izidingo njengento yokuthuthukiswa

Ezimweni zokuqala zokwakhiwa komphakathi, izidingo zabantu zilinganiselwe isidingo sokugcina amaphuzu - isikhwama esisodwa sokusanhlamvu, izikhwama ezimbili zokusanhlamvu, njll. Ukwenza lokhu, kwakwanele ukuba nezinombolo zemvelo, isethi yayo ehambisana nokulandelana okuhle okungenamkhawulo wezinombolo N.

Kamuva, ngokuthuthukiswa kwezibalo njengezesayensi, kwavela isidingo somkhakha ohlukile wezinombolo Z - kufaka phakathi ubuningi obukhulu kanye nendawo. Ukubonakala kwakhe ezingeni lomndeni kwakushukunyiswa yiqiniso lokuthi emnyangweni we-akhawunti oyinhloko kwakudingekile ngandlela-thile ukulungisa izikweleti nokulahleka. Esikhathini sezesayensi, izinombolo ezimbi zenza kube lula ukuxazulula ukulinganisa okulula okulula . Phakathi kwezinye izinto, manje sekuvele kunokwenzeka ukubonisa uhlelo oluyingcosana lokuxhumanisa, ngoba iphuzu lokubhekisela livele.

Isinyathelo esilandelayo kwakuyisidingo sokufaka izinombolo ezincane, njengoba isayensi engazange imise, ukutholakala okusha okusha nokuningi kwakudinga isisekelo semfundiso yokukhula okusha. Ngakho kwavela insimu yezinombolo ezinengqondo Q.

Okokugcina, ukuhleleka komqondo kwehlile ukwanelisa izicelo, ngoba zonke iziphetho ezintsha zidinga ukulungiswa. Insimu yezinamba zangempela R yabonakala, imisebenzi ka-Euclid ekukhulekeni kwamanani athile ngenxa yokungahleleki kwayo. Ukuthi, izazi zezibalo zakudala zesiGreki zazibeka inani lawo njengento ehlala njalo, kodwa futhi njengenani elibalulekile, elibhekene nenani lamanani angenakulinganiswa. Ngenxa yokuthi izinombolo zangempela zivele, "izindinganiso" ezinjengokuthi "pi" ne "e" zibone "ukukhanya", ngaphandle kokuthi izibalo zanamuhla zingenakwenzeka.

Ukuqanjwa kokugcina kwakuyizinombolo eziyinkimbinkimbi C. Yaphendula imibuzo eminingi futhi ayihambisani nama-postulates asethwe ngaphambilini. Ngenxa yokuthuthukiswa okusheshayo kwe-algebra, umphumela wawubikezelwa - ube nezinamba zangempela, isixazululo sezinkinga eziningi sasingenakwenzeka. Isibonelo, ngenxa yezinombolo eziyinkimbinkimbi, izintambo zochungechunge nezinkomba ziye zahlulwa, ukulingana kwama-hydrodynamics kuye kwanda.

Umbono wezinethi. Cantor

Umqondo wobuningi obungapheli ngaso sonke isikhathi ubelokhu uphikisana, ngoba awukwazanga ukufakazelwa noma ukuphikiswa. Esimweni semathematics, esetshenziswa ngokuthunyelwa okuqinisekisiwe okuqinisekisiwe, lokhu kwabonakala ngokucacile, ikakhulukazi njengoba isici senkolo sasisenesisindo kwisayensi.

Kodwa-ke, ngenxa yomsebenzi wesazi sezibalo uGeorg Cantor, konke kwenzeka ngesikhathi sokuhamba kwesikhathi. Wabonisa ukuthi amasethingi angenamkhawulo anesethi esingapheli, futhi ukuthi insimu R inkulu kunensimu N, vumela kokubili kungapheli. Phakathi nekhulu le-XIX, imibono yakhe yayibizwa ngokuzwakalayo ngokuthi i-delirium nobugebengu ngokumelene nemikhakha yama-classical, engaxhunyiwe, kodwa isikhathi sabeka konke endaweni yaso.

Izakhiwo eziyisisekelo zensimu R

Izinombolo zangempela azizona izakhiwo ezifanayo kuphela njengezingaphansi, ezifakiwe kuzo, kodwa nazo zengezwa abanye ngenxa yesisindo sezinto zabo:

• I-Zero ikhona futhi ingokwensimu R. c + 0 = c nganoma yimuphi c ku-R.
• I-zero ikhona futhi ingokwensimu R. c x 0 = 0 nganoma yikuphi c ku-R.
• Isilinganiso c: d for d ≠ 0 ikhona futhi ingokoqobo kunoma yimuphi c, d ku-R.
• Insimu R iyalwe, okungukuthi, uma c ≤ d, d ≤ c, bese c = d nganoma yikuphi c, d ku-R.
• Ukwengezwa emkhakheni R kuguqula, okungukuthi, c + d = d + c nganoma yikuphi c, d ku-R.
• Ukuphindaphinda emkhakheni R kuyinto eguquguqukayo, okungukuthi, cx d = dx c nganoma yikuphi c, d ku-R.
• Ukwengezwa emkhakheni R kuhlanganiswa, okungukuthi, (c + d) + f = c + (d + f) kunoma yikuphi c, d, f ku-R.
• Ukuphindaphinda emkhakheni R kuhlanganiswa, okungukuthi, (c x d) x f = c x (d x f) nganoma yimuphi c, d, f ku R.
• Inombolo ngayinye evela ensimini R ikhona enye ehlukile, njengokuthi c + (-c) = 0, lapho c, -c kusuka ku R.
• Kuzo zonke izinombolo ezisensimini R zikhona inverse ezifana c x c ^-1 = 1, lapho c, c ^-1 ka R.
• Iyunithi ikhona futhi ingokwa-R, ukuze c x 1 = c, noma yikuphi c ku-R.
• Umthetho wokusabalalisa ubamba, ukuze c x (d + f) = c x d + c x f, noma yimuphi c, d, f ku-R.
• Emkhakheni R, i-zero ayifani nelinye.
• Insimu R iyaguquka: uma c ≤ d, d ≤ f, bese c c ≤ f nganoma yikuphi c, d, f ku R.
• Emkhakheni R, ukuhleleka nokuhlanganisa kufana: uma c ≤ d, ke c + f ≤ d + f nganoma yikuphi c, d, f ku R.
• Emkhakheni R ukuhlelwa nokuphindaphinda kuhambelana: uma u-0 ≤ c, 0 ≤ d, bese u-0 ≤ c x d kunoma yikuphi c, d kusuka ku-R.
• Zombili izinombolo zangempela nezingokoqobo eziqhubekayo ziqhubekayo, okungukuthi, noma yikuphi c, d ku-R kune-R efana nokuthi c ≤ f ≤ d.

I module in field R

Izinombolo zangempela zihlanganisa into enjalo njengemoduli. Ichazwa ngokuthi | f | Noma yikuphi f ku R. | f | = F, uma 0 ≤ f futhi | f | = -f uma 0> f. Uma sicabanga le moduli njengexabiso lejometri, limelela ibanga elihambako - akunandaba ukuthi "wadlula" nge-zero ekugcineni noma phambili phambili ku-plus.

Izinombolo eziyinkimbinkimbi nezingokoqobo. Yini evamile futhi yini umehluko?

Ngokuvamile, izinamba eziyinkimbinkimbi nezingokoqobo ziyingxenye efanayo, ngaphandle kokuthi i-imaginary unit i, isikwele sayo esingu--1, sijoyine kuqala. Izakhi zamasimu R no C zingamelwa njengefomula elandelayo:

• C = d + f x i, lapho d, f ku-R, futhi ngiyunithi yokucabanga.

Ukuthola c kusuka ku-R kulokhu, f kubhekwa njengokulingana no-zero, okungukuthi, ingxenye kuphela yendima ehlalayo. Ngoba insimu yezinombolo eziyinkimbinkimbi inesethi efanayo sempahla njengensimu yezinombolo zangempela, f x i = 0, uma f = 0.

Ngokuqondene nokungafani okusebenzayo, isibonelo, emkhakheni R ukulinganisa kwe-quadratic akuxazululwa uma ngabe ubandlululo engalungile, kanti insimu C ayifuni ukuvimbela umbandela onjalo ngenxa yokungeniswa kweyunithi yokucabanga i.

Imiphumela

"Izitini" ze-axioms kanye nokuhlelwa kwezibalo lapho kusekelwe khona izibalo azishintshi. Abanye babo, mayelana nokukhuliswa kolwazi nokufakwa kwamakhophi amasha, faka lokhu okulandelayo "izitini", okuzayo esikhathini esizayo kungaba yisisekelo sesinyathelo esilandelayo. Isibonelo, izinombolo zemvelo, naphezu kokuba yi-subset yensimu yangempela R, ungalahlekelwa ukuhambisana kwazo. Kuzo zonke izibalo zokuqala ezisekelwe kuwo, lapho ukuqonda komuntu wezwe kuqala khona.

Ngombono osebenzayo, izinombolo zangempela zibukeka njengomugqa oqondile. Kuyo ungakhetha isiqondiso, khombisa umsuka nesinyathelo. Umugqa uqukethe inombolo engapheli yamaphoyinti, ngayinye ehambisana nenombolo eyodwa yangempela, kungakhathaliseki ukuthi isengqondo noma cha. Kusukela incazelo kucacile ukuthi sikhuluma ngomqondo wokwakha kokubili izibalo ngokujwayelekile, nokuhlaziywa ngezibalo ngokukhethekile.