Hamba uye ezweni elingelona iqiniso lezibalo
Ngibhale lesi sihloko ngoLwesithathu, ngemuva kwenkulumo nokuzilolonga ekolishi lesayensi yekhompyutha. Ngiyazivikela ekugxekweni kwabafundi balesi sikole, ulwazi lwabo, isimo sengqondo ngesayensi, futhi okubaluleke kakhulu: amakhono okufundisa. Lokhu...akukho muntu obafundisayo.
Kungani ngizivikela kangaka? Ngesizathu esilula - ngineminyaka lapho, mhlawumbe, umhlaba osizungezile ungakaqondwa. Mhlawumbe ngibafundisa ukubopha nokukhulula amahhashi, hhayi ukushayela imoto? Mhlawumbe ngibafundise ukubhala nge-quill pen? Nakuba nginombono ongcono ngomuntu, ngizibheka “njengomlandelayo”, kodwa...
Kuze kube muva nje, esikoleni esiphakeme, bakhuluma ngezinombolo eziyinkimbinkimbi. Futhi kwakungoLwesithathu lokhu lapho ngifika ekhaya, ngayeka - cishe akekho umfundi osefundile ukuthi iyini nokuthi lezi zinombolo zisetshenziswa kanjani. Abanye babheka zonke izibalo njengehansi emnyango opendiwe. Kodwa futhi ngamangala ngempela lapho bengitshela indlela yokufunda. Kalula nje, ihora ngalinye lenkulumo amahora amabili omsebenzi wesikole: ukufunda incwadi, ukufunda ukuxazulula izinkinga esihlokweni esinikeziwe, njll. Ngemva kokulungiselela ngale ndlela, sifika ekuzivocavoca, lapho sithuthukisa khona konke ... Ngokujabulisayo, abafundi, ngokusobala, bacabanga ukuthi ukuhlala enkulumweni - ngokuvamile ebuka efasiteleni - kakade kuqinisekisa ukungena kolwazi ekhanda.
Ima! Sekwanele lokhu. Ngizochaza impendulo yami embuzweni engawuthola ngesikhathi ngisekilasini nozakwethu beSikhwama Sezingane Sikazwelonke, isikhungo eseseka izingane ezinekhono ezivela ezweni lonke. Umbuzo (noma kunalokho isiphakamiso) wawuthi:
— Ungakwazi yini ukusitshela okuthile ngezinombolo ezingezona zangempela?
“Yebo,” ngiphendula.
Iqiniso lezinombolo
"Umngane ungomunye mina, ubungane isilinganiso sezinombolo 220 no-284," kusho uPythagoras. Iphuzu lapha ukuthi isamba sabahlukanisi benombolo engu-220 singu-284, futhi isamba sabahlukanisi benombolo engu-284 singama-220:
1 + 2 + 4 + 71 + 142 = 220
1 + 2 + 4 + 5 + 10 = 11 + 20 + 22 + 44 + 55 + 110 = 284
Okunye ukuqondana okuthakazelisayo phakathi kwezinombolo 220 kanye no-284 yilokhu: izinombolo eziyinhloko eziyishumi nesikhombisa zingu-2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 , futhi 59.
Isamba sabo singu-2x220, kanti isamba sezikwele singu-59x284.
Okokuqala. Awukho umqondo "wenombolo yangempela". Kufana nokuthi ngemva kokufunda isihloko mayelana nezindlovu, uyabuza, "Manje sizocela okungezona izindlovu." Kukhona okuphelele nokungaphelele, okunengqondo nokungenangqondo, kodwa akukho okungelona iqiniso. Ngokuqondile: izinombolo okungezona ezangempela azibizwa ngokuthi ezingavumelekile. Kunezinhlobo eziningi ze "izinombolo" zezibalo, futhi ziyahlukahluka komunye nomunye, njengokuthi - ukuthatha ukuqhathanisa kwe-zoological - indlovu kanye ne-earthworm.
Okwesibili, sizokwenza imisebenzi osuvele wazi ukuthi inqatshelwe: ukukhipha izimpande eziyisikwele zezinombolo ezinegethivu. Nokho, izibalo zizoyinqoba leyo migoqo. Ingabe kunengqondo kodwa? Kumathematika, njengakunoma iyiphi enye isayensi, ukuthi ithiyori ingena phakade endaweni yolwazi kuncike ... ekusetshenzisweni kwayo. Uma kungenamsebenzi, khona-ke kugcina kudoti, bese kudoti othile womlando wolwazi. Ngaphandle kwezinombolo engikhuluma ngazo ekupheleni kwalesi sihloko, akunakwenzeka ukuthuthukisa izibalo. Kodwa ake siqale ngezinto ezincane. Yiziphi izinombolo zangempela, uyazi. Bagcwalisa umugqa wezinombolo ngokuminyene futhi ngaphandle kwezikhala. Uyazi futhi ukuthi ziyini izinombolo zemvelo: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …….. - zonke ngeke zilingane inkumbulo ngisho enkulu. Futhi banegama elihle: lemvelo. Banezinto eziningi ezithakazelisayo. Ukuthanda kanjani lokhu:
1 + 15 + 42 + 98 + 123 + 179 + 206 + 220 = 3 + 11 + 46 + 92 + 129 + 175 + 210 + 218
12 15 +2 42 +2 98 +2 123 +2 179 +2 206 +2 220 +2 = 32 11 +2 46 +2 92 +2 129 +2 175 +2 210 +2 218 +2
13 15 +3 42 +3 98 +3 123 +3 179 +3 206 +3 220 +3 = 33 11 +3 46 +3 92 +3 129 +3 175 +3 210 +3 218 +3
14 15 +4 42 +4 98 +4 123 +4 179 +4 206 +4 220 +4 = 34 11 +4 46 +4 92 +4 129 +4 175 +4 210 +4 218 +4
15 15 +5 42 +5 98 +5 123 +5 179 +5 206 +5 220 +5 = 35 11 +5 46 +5 92 +5 129 +5 175 +5 210 +5 218 +5
16 15 +6 42 +6 98 +3 123 +6 179 +6 206 +6 220 +6 = 36 11 +6 46 +6 92 +6 129 +6 175 +6 210 +6 218 +6
17 15 +7 42 +7 98 +3 123 +7 179 +7 206 +7 220 +7 = 37 11 +7 46 +7 92 +7 129 +7 175 +7 210 +7 218 +7
“Kungokwemvelo ukuba nesithakazelo emananini angokwemvelo,” kusho uKarl Lindenholm, noLeopold Kronecker (1823–1891) bakubeka ngamafuphi: “UNkulunkulu wadala izinombolo ezingokwemvelo—konke okunye kuwumsebenzi womuntu!” Izingxenyana (ezibizwa ngokuthi izinombolo ezinengqondo ngochwepheshe bezibalo) nazo zinezakhiwo ezimangalisayo:
nangokulingana:
ungakwazi, kusukela ohlangothini lwesobunxele, uhlikihle ama-pluses bese uwashintsha ngezimpawu zokuphindaphinda - futhi ukulingana kuzohlala kuyiqiniso:
Futhi njalo.
Njengoba wazi, kumafrakshini a/b, lapho u-a no-b beyizinombolo eziphelele, kanye no-b ≠0, bathi inombolo enengqondo. Kodwa ngesiPolish kuphela bazibiza kanjalo. Bakhuluma isiNgisi, isiFulentshi, isiJalimane nesiRashiya. inombolo enengqondo. NgesiNgisi: izinombolo ezinengqondo. Izinombolo ezingenangqondo akunangqondo, akunangqondo. Siphinde sikhulume isiPolish mayelana nemibono engenangqondo, imibono kanye nezenzo - lokhu kuwubuhlanya, okucatshangwayo, okungaqondakali. Bathi abantu besifazane bayawesaba amagundane - akunangqondo yini lokho?
Ezikhathini zasendulo, izinombolo zazinomphefumulo. Ngalinye lalisho okuthile, ngalinye lifanekisela okuthile, ngalinye libonisa inhlayiya yalokho kuvumelana kwendawo yonke, okungukuthi, ngesiGreki, iCosmos. Lona kanye igama elithi "cosmos" lisho ngqo "ukuhleleka, ukuhleleka". Okubaluleke kakhulu kwakuyisithupha (inombolo ephelele) kanye neshumi, isamba sezinombolo ezilandelanayo 1+2+3+4, ezakhiwe ngezinye izinombolo, uphawu lwazo olusekhona kuze kube namuhla. Ngakho uPythagoras wafundisa ukuthi izinombolo ziyisiqalo nomthombo wayo yonke into, nokutholakala kuphela izinombolo ezingenangqondo waphendulela ukunyakaza kwePythagoras ku-geometry. Siyazazi izizathu zasesikoleni ukuthi
√2 inombolo engenangqondo
Ake sithi kukhona: nokuthi le ngxenyana ayikwazi ukuncishiswa. Ikakhulukazi, kokubili u-p no-q kuyinqaba. Asikwele: 2q2=p2. Inombolo p angeke ibe yinqaba, kusukela lapho p2 kungaba futhi, futhi uhlangothi lwesobunxele lokulingana luyimpinda ka-2. Ngakho-ke, u-p ulinganayo, okungukuthi, p = 2r, yingakho p.2= 4r2. Sehlisa i-equation 2q2= 4r2 ngo 2. Sithola q2= 2r2 futhi siyabona ukuthi u-q kufanele futhi alingane, ebesicabanga ukuthi bekungenjalo. Ukuphikisana okuwumphumela kuqeda ubufakazi - le fomula ngokuvamile ingatholakala kuzo zonke izincwadi zezibalo. Lobu bufakazi besimo buyiqhinga eliyintandokazi lama-sophists.
Lobu bukhulu babungenakuqondwa amaPythagoreans. Konke kufanele kukwazi ukuchazwa ngezinombolo, futhi i-diagonal yesikwele, lapho noma ubani angakwazi ukuyidweba ngenduku esihlabathini, ayinakho, okungukuthi, ukulinganiswa, ubude. “Ukholo lwethu lwaluyize,” kubonakala sengathi amaPythagorean esho. Kanjani? Kuwuhlobo ... okungenangqondo. Inyunyana yazama ukuzisindisa ngezindlela zamahlelo. Noma ubani onesibindi sokuveza ubukhona babo izinombolo ezingenangqondo, kwakufanele ajeziswe ngokufa, futhi, ngokusobala, isigwebo sokuqala sakhishwa inkosi ngokwayo.
Kodwa "umcabango wadlula ngaphandle kokulimala." Isikhathi segolide sesifikile. AmaGreki anqoba amaPheresiya (Marathon 490, Block 479). Intando yeningi yaqiniswa, izikhungo ezintsha zomcabango wefilosofi nezikole ezintsha zavela. AbakwaPythagoraans babesazabalaza nezinombolo ezingenangqondo. Abanye bashumayela bathi: “Asiyikuyiqonda le mfihlakalo; singazindla kuphela futhi simangale nge-Uncharted. Lezi zakamuva zazine-pragmatic futhi zingayihloniphi i-Mystery. Ngaleso sikhathi, kwavela ukwakheka kwengqondo okubili okwenza kube nokwenzeka ukuqonda izinombolo ezingenangqondo. Iqiniso lokuthi siziqonda kahle ngokwanele namuhla lingeka-Eudoxus (ikhulu lesi-XNUMX BC), futhi kwaba kuphela ekupheleni kwekhulu le-XNUMX lapho isazi sezibalo saseJalimane uRichard Dedekind sanikeza imfundiso ye-Eudoxus intuthuko efanele ngokuhambisana nezidingo zokuqina. logic yezibalo.
Inqwaba yezibalo noma ukuhlukunyezwa
Ungaphila ngaphandle kwezinombolo? Ngisho noma ngabe impilo izoba yini... Kwakuzodingeka siye esitolo siyothenga izicathulo ngenduku, esasilinganisa ubude bonyawo ngaphambili. “Ngingathanda ama-apula, ah, nali!” - sizobonisa abathengisi emakethe. "Kukude kangakanani ukusuka eModlin kuya e-Nowy Dwur Mazowiecki"? “Iseduze!”
Izinombolo zisetshenziselwa ukukala. Ngosizo lwabo, siveza neminye imiqondo eminingi. Isibonelo, isikali semephu sibonisa ukuthi indawo yezwe yehle kangakanani. Isilinganiso sokubili ukuya kwesinye, noma u-2 nje, siveza iqiniso lokuthi into iphindwe kabili ngosayizi. Ake sithi ngokwezibalo: i-homogeneity ngayinye ihambisana nenombolo - isikali sayo.
Umsebenzi. Senza ikhophi ye-xerographic, khulisa isithombe izikhathi eziningana. Khona-ke isiqephu esikhulisiwe saphinde sandiswa izikhathi b. Siyini isikali sokukhulisa esijwayelekile? Impendulo: a × b iphindwe ngo-b. Lezi zilinganiso zidinga ukuphindaphindwa. Inombolo "minus one", -1, ihambisana nokunemba okukodwa okumaphakathi, okungukuthi kuzungeziswe ngamadigri ayi-180. Iyiphi inombolo ehambisana nokujika kwe-90 degree? Ayikho inombolo enjalo. Kunjalo, ku… noma kunalokho, kuzoba maduze. Ingabe usukulungele ukuhlushwa ngokuziphatha? Yiba nesibindi uthathe impande yesikwele yokususa okukodwa. Ngilalele? Yini ongakwazi? Phela ngakutshela ukuthi yiba lesibindi. Yikhiphe! Hhayi-ke, donsa, donsa... ngizosiza... Lapha: -1 Manje njengoba sesinayo, ake sizame ukuyisebenzisa... Kunjalo, manje singakhipha izimpande zazo zonke izinombolo ezinegetive, ngokuba isibonelo.:
√-4 = 2√-1, √-16 = 4√-1
"Ngaphandle kobuhlungu bengqondo okuhlanganisayo." Yilokhu uGirolamo Cardano akubhala ngo-1539, ezama ukunqoba ubunzima obungokwengqondo obuhlotshaniswa nayo - njengoba ngokushesha yabizwa ngokuthi - amanani acatshangelwayo. Wazicabangela lezi...
...Umsebenzi. Hlukanisa i-10 ibe izingxenye ezimbili, umkhiqizo wawo 40. Ngikhumbula ukuthi kusukela esiqeshini esedlule wabhala into enjengale: Ngokuqinisekile akunakwenzeka. Nokho, masenze lokhu: hlukanisa u-10 ube izingxenye ezimbili ezilinganayo, ngayinye ilingane no-5. Ziphindaphinde - kuvele 25. Kusukela ku-25, manje khipha u-40, uma uthanda, futhi uthola -15. Manje bheka: √-15 engezwe futhi esuswa ku-5 ikunikeza umkhiqizo wama-40. Lezi izinombolo 5-√-15 kanye no-5 + √-15. Ukuqinisekiswa komphumela kwenziwa nguCardano kanje:
“Ngaphandle kobuhlungu benhliziyo okuhlanganisayo, phindaphinda 5 + √-15 ngo-5-√-15. Sithola u-25 - (-15), olingana no-25 + 15. Ngakho, umkhiqizo ungama-40 .... Kunzima impela."
Nokho, yimalini: (1 + √-1) (1-√-1)? Masiphindaphindeke. Khumbula ukuthi √-1 × √-1 = -1. Kuhle. Manje umsebenzi onzima kakhulu: ukusuka ku-+ b√-1 ukuya ku-ab√-1. Kwenzenjani? Impela, kanje: (a + b√-1) (ab√-1) = a2+b2
Yini ethokozisayo ngalokhu? Isibonelo, iqiniso lokuthi singakwazi ukwenza izinkulumo "esingazi ngaphambili." Ifomula yokuphindaphinda efushanisiwe ye2-b2 Uyayikhumbula ifomula ye2+b2 kwakungenjalo, ngoba kwakungeke kube njalo. Esizindeni sezinombolo zangempela, i-polynomial2+b2 akunakugwenywa. Ake sisho impande yesikwele ethi "yethu" ka-"minus one" ngohlamvu u-i.2= -1. "Inombolo eyinhloko" engelona iqiniso. Futhi yilokho okuchaza ukujika kwendiza okungu-90 degree. Kungani? Ngemva kwakho konke,2= -1, futhi ukuhlanganisa ukuzungezisa okungu-90-degree kanye nokunye ukuzungezisa okungama-180 degree kunikeza ukuzungezisa okungu-45-degree. Iluphi uhlobo lokuzungezisa oluchazwayo? Ngokusobala ukujika kwe-XNUMX degree. Isho ukuthini i-i? Kuyinkimbinkimbi kakhulu:
(-ngi)2 = -i × (-i) = + i2 = -1
Ngakho-ke -i futhi ngichaza ukuzungezisa okungu-90 degree, ngakolunye uhlangothi lokuzungezisa kwami. Iyiphi esele futhi iyiphi elungile? Kufanele wenze i-aphoyintimenti. Sicabanga ukuthi inombolo icacisa ukuzungezisa ukuya endaweni izazi zezibalo eziyibheka njengenhle: ngokuphambene newashi. Inombolo -i ichaza ukuzungezisa lapho izinkomba zihamba ngakhona.
Kodwa ingabe izinombolo ezifana no-i no-i zikhona? Ingabe! Sisanda kubenza baphile. Ngilalele? Ukuthi zikhona ekhanda lethu kuphela? Awu, yini ongayilindela? Zonke ezinye izinombolo zikhona kuphela engqondweni yethu. Sidinga ukubona ukuthi izinombolo zethu ezisanda kuzalwa zisaphila yini. Ngokunembayo, noma ngabe umklamo unengqondo nokuthi azoba usizo kokuthile. Ngicela ungitshele ukuthi konke kuhamba ngohlelo nokuthi lezi zinombolo ezintsha ziwusizo ngempela. Izinombolo ezifana no-3+i, 5-7i, ngokuvamile: u-a+bi ubizwa ngokuthi izinombolo eziyinkimbinkimbi. Ngikubonise ukuthi ungazithola kanjani ngokuphotha indiza. Angafakwa ngezindlela ezahlukene: njengamaphoyinti endiza, njengamapholynomial athile, njengohlobo oluthile lwezinombolo zezinombolo ... futhi isikhathi ngasinye ziyafana: isibalo x2 +1=0 asikho isici... i-hocus pocus isikhona kakade!!!! Asijabule sijabule!!!
Ukuphela kohambo
Lokhu kuphetha uhambo lwethu lokuqala ezweni lezinombolo zomgunyathi. Kwezinye izinombolo ezingekho emhlabeni, ngizophinde ngikhulume lezo ezinenombolo engapheli yamadijithi ngaphambili, futhi hhayi ngemuva (zibizwa ngokuthi i-10-adic, kithi i-p-adic ibaluleke kakhulu, lapho u-p eyinombolo eyinhloko), ngoba isibonelo X = … … … 96109004106619977392256259918212890625
Ake sibale X ngiyacela2. Ngoba? Kuthiwani uma sibala isikwele senani silandelwa inombolo engapheli yamadijithi? Asenzeni okufanayo. Siyazi ukuthi x2 = H.
Masithole enye inombolo enjalo enenombolo engapheli yamadijithi ngaphambili enelisa isibalo. Ukusikisela: isikwele senani esigcina ngesithupha siphetha ngesithupha. Isikwele senani esigcina ngo 76 siphetha ngo 76. Isikwele senani esigcina ngo 376 siphetha ngo 376. Isikwele senani esigcina ngo 9376 siphetha ngo 9376. Isikwele senani esigcina ngo XNUMX ku… Kukhona futhi izinombolo ezincane kangangokuthi, ngokuba positive, zihlala zincane kunanoma iyiphi enye inombolo ephozithivu. Mancane kakhulu kangangokuthi ngesinye isikhathi kwanele ukuwafaka isikwele ukuze uthole uziro. Kunezinombolo ezingasigculisi isimo a × b = b × a. Kukhona nezinombolo ezingapheli. Zingaki izinombolo zemvelo ezikhona? Abaningi ngokungapheli? Yebo, kodwa malini? Lokhu kungavezwa kanjani njengenombolo? Impendulo: izinombolo ezincane kunazo zonke ezingapheli; imakwe ngohlamvu oluhle: A futhi yengezwe ngenkomba enguziro A0 , aleph-zero.
Kukhona nezinombolo esingazi ukuthi zikhona... noma ungakholwa noma ungakholwa ngokuthanda kwakho. Futhi ukukhuluma ngokufana nalokhu: Ngithemba ukuthi usathanda Izinombolo Ezingezona Zangempela, Izinombolo Zezinhlobo Zephupho.
