Yendani kudziko losawerengeka la masamu

Ndinalemba nkhaniyi m'malo amodzi, nditatha maphunziro ndi machitidwe ku koleji ya sayansi ya makompyuta. Ndimadziteteza ku kutsutsidwa kwa ophunzira a sukuluyi, chidziwitso chawo, maganizo awo pa sayansi komanso, makamaka, luso lawo lophunzitsa. Izi...palibe amene amawaphunzitsa.

Chifukwa chiyani ndikudzitchinjiriza chotere? Pazifukwa zosavuta - ndili ndi zaka zomwe, mwina, dziko lotizungulira silinamvekebe. Mwina ndikuwaphunzitsa kumanga ndi kumasula akavalo, osati kuyendetsa galimoto? Mwina ndimawaphunzitsa kulemba ndi cholembera? Ngakhale ndili ndi malingaliro abwino a munthu, ndimadziona kuti ndine "wotsatira", koma ...

Mpaka posachedwa, kusukulu yasekondale, adalankhula za manambala ovuta. Ndipo linali Lachitatu ili pamene ndinabwera kunyumba, ndinasiya - pafupifupi palibe wophunzira yemwe adaphunzirabe kuti ndi chiyani komanso momwe mungagwiritsire ntchito manambalawa. Ena amaona masamu onse ngati tsekwe pa chitseko chopakidwa utoto. Koma ndinadabwanso kwambiri atandiuza mmene ndingaphunzire. Mwachidule, ola lililonse la nkhani ndi maola awiri a homuweki: kuwerenga buku, kuphunzira momwe mungathetsere mavuto pamutu womwe wapatsidwa, ndi zina. Pokonzekera motere, timabwera ku masewera olimbitsa thupi, komwe timakonza zonse ... Mokondweretsa, ophunzirawo, mwachiwonekere, ankaganiza kuti atakhala pa phunziro - nthawi zambiri akuyang'ana pawindo - amatsimikizira kale kulowa kwa chidziwitso m'mutu.

Imani! Zokwanira izi. Ndifotokoza yankho langa ku funso limene ndinalandira m’kalasi limodzi ndi anthu amene ali ndi maphunziro a National Children’s Fund, bungwe limene limathandiza ana aluso ochokera m’mayiko osiyanasiyana. Funso (kapena lingaliro) linali:

— Kodi mungatiuzeko kena kake ponena za manambala osakhala enieni?

“Inde,” ndinayankha. 

Zowona za manambala

"Mnzanga ndi ine wina, ubwenzi ndi chiŵerengero cha manambala 220 ndi 284," anatero Pythagoras. Mfundo apa ndi yakuti chiwerengero cha magawo 220 ndi 284, ndipo chiwerengero cha magawo 284 ndi 220:

1 + 2 + 4 + 71 + 142 = 220

1 + 2 + 4 + 5 + 10 = 11 + 20 + 22 + 44 + 55 + 110 = 284

Kukumana kwina kosangalatsa pakati pa manambala 220 ndi 284 ndi ichi: manambala khumi ndi asanu ndi awiri apamwamba kwambiri ndi 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 , ndi 59.

Chiwerengero chawo ndi 2x220, ndipo kuchuluka kwa mabwalo ndi 59x284.

Choyamba. Palibe lingaliro la "nambala yeniyeni". Zili ngati mutawerenga nkhani yokhudza njovu, mumafunsa kuti, "Tsopano tifunsa anthu omwe si njovu." Pali zonse ndi zosakwanira, zomveka komanso zopanda nzeru, koma palibe zenizeni. Makamaka: manambala omwe si enieni samatchedwa osavomerezeka. Pali mitundu yambiri ya "manambala" mu masamu, ndipo amasiyana wina ndi mzake, monga - kuyerekeza zoological - njovu ndi earthworm.

Chachiwiri, tikhala tikuchita zinthu zomwe mukudziwa kale kuti ndizoletsedwa: kutenga masikweya mizu ya manambala olakwika. Chabwino, masamu adzagonjetsa zopinga zoterozo. Koma zikumveka? Mu masamu, monga mu sayansi ina iliyonse, kaya chiphunzitso chimalowa kosatha m'nkhokwe ya chidziwitso zimatengera ... Ngati zilibe ntchito, ndiye kuti zimathera mu zinyalala, ndiye mu zinyalala zina za mbiri ya chidziwitso. Popanda manambala omwe ndikunena kumapeto kwa nkhaniyi, ndizosatheka kupanga masamu. Koma tiyeni tiyambe ndi zinthu zing’onozing’ono. Kodi manambala enieni ndi chiyani, mukudziwa. Amadzaza mzere wambiri komanso wopanda mipata. Mumadziwanso kuti manambala achilengedwe ndi chiyani: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …….. - zonse sizingagwirizane. kukumbukira ngakhale wamkulu. Amakhalanso ndi dzina lokongola: zachilengedwe. Ali ndi zinthu zambiri zosangalatsa. Mumakonda bwanji izi:

1 + 15 + 42 + 98 + 123 + 179 + 206 + 220 = 3 + 11 + 46 + 92 + 129 + 175 + 210 + 218

1² + 15² + 42² + 98² + 123² + 179² + 206² + 220² = 3² + 11² + 46² + 92² + 129² + 175² + 210² + 218²

1³ + 15³ + 42³ + 98³ + 123³ + 179³ + 206³ + 220³ = 3³ + 11³ + 46³ + 92³ + 129³ + 175³ + 210³ + 218³

1⁴ + 15⁴ + 42⁴ + 98⁴ + 123⁴ + 179⁴ + 206⁴ + 220⁴ = 3⁴ + 11⁴ + 46⁴ + 92⁴ + 129⁴ + 175⁴ + 210⁴ + 218⁴

1⁵ + 15⁵ + 42⁵ + 98⁵ + 123⁵ + 179⁵ + 206⁵ + 220⁵ = 3⁵ + 11⁵ + 46⁵ + 92⁵ + 129⁵ + 175⁵ + 210⁵ + 218⁵

1⁶ + 15⁶ + 42⁶ + 98³ + 123⁶ + 179⁶ + 206⁶ + 220⁶ = 3⁶ + 11⁶ + 46⁶ + 92⁶ + 129⁶ + 175⁶ + 210⁶ + 218⁶

1⁷ + 15⁷ + 42⁷ + 98³ + 123⁷ + 179⁷ + 206⁷ + 220⁷ = 3⁷ + 11⁷ + 46⁷ + 92⁷ + 129⁷ + 175⁷ + 210⁷ + 218⁷

“Nkwachibadwa kukhala wokondweretsedwa ndi ziŵerengero zachibadwa,” anatero Karl Lindenholm, ndi Leopold Kronecker (1823–1891) akulongosola mosapita m’mbali kuti: “Mulungu analenga ziŵerengero zachibadwa—china chilichonse ndi ntchito ya munthu! Magawo (otchedwa manambala omveka ndi akatswiri a masamu) alinso ndi zinthu zodabwitsa:

ndi kufanana:

mungathe, kuyambira kumanzere, kupukuta ma pluses ndikuwasintha ndi zizindikiro zochulukitsa - ndipo kufanana kudzakhalabe koona:

Ndipo kotero.

Monga mukudziwa, pamagawo a/b, pomwe a ndi b ndi manambala, ndi b ≠ 0, amati nambala yomveka. Koma m’Chipolishi mokha amadzitcha zimenezo. Amalankhula Chingerezi, Chifalansa, Chijeremani ndi Chirasha. nambala yomveka. Mu Chingerezi: manambala omveka. Manambala opanda nzeru ndi zopanda nzeru, zopanda nzeru. Timalankhulanso Chipolishi za malingaliro opanda nzeru, malingaliro ndi zochita - izi ndi misala, zongoganizira, zosamvetsetseka. Amati akazi amawopa mbewa - sizinthu zopanda nzeru?

Kale, manambala anali ndi mzimu. Chilichonse chinkatanthauza chinachake, chilichonse chinkaimira chinachake, chilichonse chinkasonyeza kachigawo kakang’ono ka kugwirizana kwa Chilengedwe, ndiko kuti, m’Chigiriki, Cosmos. Mawu omwewo "cosmos" amatanthauza ndendende "dongosolo, dongosolo". Zofunika kwambiri zinali zisanu ndi chimodzi (nambala yabwino kwambiri) ndi khumi, chiwerengero cha manambala otsatizana 1 + 2 + 3 + 4, opangidwa ndi manambala ena omwe chizindikiro chawo chakhalapo mpaka lero. Choncho Pythagoras anaphunzitsa kuti manambala ndi chiyambi ndi gwero la chirichonse, ndi kungotulukira manambala opanda nzeru adatembenuza gulu la Pythagorean kupita ku geometry. Ife tikudziwa zolingalira zochokera kusukulu zimenezo

√2 ndi nambala yopanda nzeru

Tiyerekeze kuti pali: ndipo gawo ili silingachepe. Makamaka, onse p ndi q ndi osamvetseka. Tiyeni tione: 2q²=p². Nambala p singakhale yosamvetseka, kuyambira pamenepo p² zikanakhalanso, ndipo mbali yakumanzere ya kufanana ndi kuchulukitsa kwa 2. Choncho, p ndi even, i.e., p = 2r, choncho p.²= 4r². Timachepetsa equation 2q²= 4r² mwa 2. Timapeza q²= 2r² ndipo tikuwona kuti q iyeneranso kukhala yofanana, zomwe timaganiza kuti sizili choncho. Kutsutsana komweku kumamaliza umboni - fomula iyi nthawi zambiri imapezeka m'mabuku aliwonse a masamu. Umboni wokhazikika uwu ndi chinyengo chomwe amachikonda kwambiri a sophists.

Kukula uku sikunamvetsetsedwe ndi a Pythagoras. Chilichonse chiyenera kufotokozedwa ndi manambala, ndi diagonal ya square, yomwe aliyense angathe kukoka ndi ndodo pamchenga, alibe, ndiko kuti, kutalika, kutalika. “Chikhulupiriro chathu chinali chachabechabe,” zikuoneka kuti a Pythagoras akutero. Mwanjira yanji? Ziri ngati ... zopanda nzeru. Union inayesetsa kudzipulumutsa mwa njira zamagulu. Aliyense amene angayerekeze kuwulula kukhalapo kwawo manambala opanda nzeru, anayenera kulangidwa ndi imfa, ndipo, mwachiwonekere, chiweruzo choyamba chinaperekedwa ndi mbuye mwiniyo.

Koma "lingalirolo linadutsa popanda vuto." Nthawi yagolide yafika. Agiriki anagonjetsa Aperisi (Marathon 490, Block 479). Demokalase idalimbikitsidwa, malo atsopano amalingaliro afilosofi ndi masukulu atsopano adawuka. A Pythagoras anali akulimbanabe ndi manambala opanda nzeru. Ena analalikira, kuti, sitidzazindikira chinsinsi ichi; tikhoza kulingalira ndi kudabwa ndi Uncharted. Zotsirizirazi zinali za pragmatic kwambiri ndipo sankalemekeza Chinsinsi. Panthaŵiyo, panali mipangidwe iŵiri yamaganizo imene inatheketsa kumvetsetsa manambala opanda nzeru. Mfundo yakuti timawamvetsa bwino masiku ano ndi a Eudoxus (zaka za m'ma XNUMX BC), ndipo kunali kumapeto kwa zaka za m'ma XNUMX pamene katswiri wa masamu wa ku Germany Richard Dedekind anapereka chiphunzitso cha Eudoxus chitukuko choyenera malinga ndi zofunikira za kulimba mtima. masamu logic.

Kuchuluka kwa ziwerengero kapena kuzunzidwa

Kodi mungakhale opanda manambala? Ngakhale moyo ukanakhala wotani... Tinkayenera kupita kusitolo kukagula nsapato ndi ndodo, zomwe poyamba tinkayesa kutalika kwa phazi. "Ndikufuna maapulo, aa, awa!" - tikuwonetsa ogulitsa pamsika. "Kodi kuli kutali bwanji kuchokera ku Modlin kupita ku Nowy Dwur Mazowiecki"? "Pafupi kwambiri!"

Manambala amagwiritsidwa ntchito poyeza. Ndi chithandizo chawo, timafotokozeranso mfundo zina zambiri. Mwachitsanzo, kukula kwa mapu kukuwonetsa momwe dera ladzikolo lacheperachepera. Sikelo ya ziwiri mpaka imodzi, kapena kuti 2 chabe, imasonyeza mfundo yakuti chinthu chachulukidwa kaŵiri mu ukulu wake. Tinene masamu: homogeneity iliyonse imafanana ndi nambala - sikelo yake.

Ntchito. Tinapanga kope la xerographic, kukulitsa chithunzicho kangapo. Kenako chidutswa chokulitsacho chinakulitsidwanso nthawi za b. Kodi general magnification sikelo ndi chiyani? Yankho: a × b kuchulukitsa ndi b. Mamba awa ayenera kuchulukitsidwa. Nambala ya "minus one", -1, imagwirizana ndi kulondola kumodzi komwe kumakhazikika, mwachitsanzo, kuzungulira madigiri 180. Ndi nambala iti yomwe ikufanana ndi kutembenuka kwa madigiri 90? Palibe nambala yoteroyo. Ndi, ndi…kapena kani, posachedwapa. Kodi mwakonzeka kuzunzidwa mwamakhalidwe? Limbani mtima ndikutenga gawo lalikulu la kuchotsera wani. Ndikumvera? Simungathe kuchita chiyani? Pajatu ndinakuuzani kuti mulimbe mtima. Kokani! Hei, chabwino, kukoka, kukoka... Ndithandiza... Apa: -1 Tsopano popeza tiri nayo, tiyeni tiyese kuigwiritsa ntchito... chitsanzo.:

√-4 = 2√-1,-16 = 4√-1

"Mosasamala kanthu za kupsinjika maganizo komwe kumaphatikizapo." Izi ndi zomwe Girolamo Cardano adalemba mu 1539, kuyesera kuthana ndi zovuta zamaganizo zomwe zimagwirizanitsidwa nazo - monga momwe adatchulidwira posachedwa - zongoyerekeza. Anaganiza izi ...

...Ntchito. Gawani 10 mu magawo awiri, zomwe zimapangidwa ndi 40. Ndikukumbukira kuti kuchokera ku gawo lapitalo iye analemba motere: Ndithudi zosatheka. Komabe, tiyeni tichite izi: gawani 10 mu magawo awiri ofanana, aliyense wofanana ndi 5. Muchulukitse iwo - zinapezeka 25. Kuchokera pa 25, tsopano chotsani 40, ngati mukufuna, ndipo mupeza -15. Tsopano yang'anani: √-15 kuwonjezeredwa ndi kuchotsedwa pa 5 kumakupatsani zotuluka 40. Izi ndi manambala 5-√-15 ndi 5 + √-15. Kutsimikizira zotsatira zake kunachitika ndi Cardano motere:

“Mosasamala kanthu za kuwawa kwa mtima, chulukitsani 5 + √-15 ndi 5-√-15. Timapeza 25 - (-15), yomwe ili yofanana ndi 25 + 15. Choncho, mankhwala ndi 40 .... Ndizovuta kwambiri."

Chabwino, ndi zingati: (1 + √-1) (1-√-1)? Tiyeni tichuluke. Kumbukirani kuti √-1 × √-1 = -1. Zabwino. Tsopano ntchito yovuta kwambiri: kuchokera ku + b√-1 mpaka ab√-1. Chinachitika ndi chiyani? Ndithudi, motere: (a + b√-1) (ab√-1) = a²+b²

Chosangalatsa ndi chiyani pa izi? Mwachitsanzo, mfundo yakuti tikhoza factorize mawu kuti "sitinkadziwa kale." Chidule cha njira yochulukitsira ya²-b² Mukukumbukira fomula²+b² izo sizinali, chifukwa izo sizikanakhoza kukhala. Pamalo a manambala enieni, polynomial²+b² sichingalephereke. Tiyeni titchule "zathu" masikweya mizu ya "minus one" ndi chilembo i.²= -1. Ndi nambala yoyamba "yopanda zenizeni". Ndipo ndizomwe zimafotokozera kutembenuka kwa 90 digiri ya ndege. Chifukwa chiyani? Izi zili choncho,²= -1, ndi kuphatikiza kusinthasintha kwa madigiri 90 ndi kusinthasintha kwina kwa digirii 180 kumapereka kusinthasintha kwa digirii 45. Ndi kasinthasintha wamtundu wanji womwe ukufotokozedwa? Mwachiwonekere kutembenuka kwa madigiri XNUMX. Kodi-ndikutanthauza chiyani? Ndizovuta pang'ono:

(-ine)² = -i × (-i) = +i² = -1

Chifukwa chake -ndimafotokozanso kuzungulira kwa digirii 90, mosiyana ndi kuzungulira kwa ine. Ndi iti yomwe yatsala ndi iti yomwe ili yolondola? Muyenera kupangana nthawi. Timaganiza kuti nambala i imatchula kuzungulira komwe akatswiri a masamu amawona kuti ndi abwino: motsatana ndi koloko. Nambala -i imalongosola kuzungulira komwe zisonyezo zikuyenda.

Koma kodi manambala monga ine ndi -ine alipo? Ndi! Tinangowabweretsa kumoyo. Ndikumvera? Kuti alipo m'mutu mwathu okha? Chabwino tiyembekezere chiyani? Ziwerengero zina zonse ziliponso m'maganizo mwathu. Tiyenera kuwona ngati manambala athu obadwa kumene apulumuka. Zowonjezereka, kaya mapangidwewo ndi omveka komanso ngati angakhale othandiza pa chinachake. Chonde tengerani mawu anga kuti zonse zili bwino komanso kuti manambala atsopanowa ndiwothandiza kwambiri. Manambala ngati 3+i, 5-7i, makamaka: a+bi amatchedwa manambala ovuta. Ndinakuwonetsani momwe mungawapeze pozungulira ndege. Akhoza kulowetsedwa m'njira zosiyanasiyana: monga mfundo mu ndege, monga ma polynomials ena, monga mtundu wina wa mawerengero ... ndipo nthawi iliyonse ali ofanana: equation x² +1=0 palibe element... hocus pocus ilipo kale!!!! Tiyeni tisangalale ndi kusangalala !!!

Mapeto a ulendo

Izi zikumaliza ulendo wathu woyamba mdziko muno wa manambala abodza. Mwa manambala ena osadziwika, nditchulanso omwe ali ndi manambala osawerengeka kutsogolo, osati kumbuyo (amatchedwa 10-adic, kwa ife p-adic ndi yofunika kwambiri, pomwe p ndi nambala yayikulu), chifukwa chitsanzo X = … … … 96109004106619977392256259918212890625

Tiyeni tiwerenge X chonde². Monga? Nanga bwanji ngati tiwerengetsa masikweya a nambala kutsatiridwa ndi nambala yopanda malire ya manambala? Chabwino, tiyeni tichite zomwezo. Tikudziwa kuti x² = H.

Tiyeni tipeze nambala ina yotero yokhala ndi manambala osawerengeka kutsogolo omwe amakwaniritsa equation. Langizo: sikweya ya nambala yomwe imathera sikisi imatheranso sikisi. Sikweya ya nambala yomwe imathera mu 76 imatheranso mu 76. Sikweya ya nambala yomwe imathera mu 376 imatheranso mu 376. 9376 pa… Palinso manambala ang'onoang'ono kwambiri kotero kuti, pokhala abwino, amakhalabe ochepa kuposa nambala ina iliyonse. Ndi ang'ono kwambiri moti nthawi zina kumakhala kokwanira kuti apeze ziro. Pali manambala omwe samakwaniritsa chikhalidwe a × b = b × a. Palinso manambala opanda malire. Kodi manambala achilengedwe alipo angati? Zopanda malire zambiri? Inde, koma zingati? Kodi izi zingafotokozedwe bwanji ngati nambala? Yankho: yocheperako mwa manambala opanda malire; imalembedwa ndi chilembo chokongola: A ndikuwonjezeredwa ndi ziro index A₀ , aleph-zero.

Palinso manambala omwe sitikudziwa kuti alipo ... kapena kuti mutha kukhulupirira kapena kusakhulupirira momwe mukufunira. Ndipo kuyankhula za izi: Ndikukhulupirira kuti mumakondabe Nambala Zosawerengeka, Nambala Zamitundu Yongopeka.