Chii svomhu? Chinokosha theorem no kuti svomhu. binary svomhu

Chii svomhu? Apo vanhu vakatanga kushandisa nhamba uye kushanda navo? Zviripiko ayo nemidzi pfungwa nezuva akaita nhamba, zvikamu zviduku, kubvisa, kuwedzera uye chokuwanziridza, kuti munhu akaita chinhu chinokosha muupenyu hwake uye maonero? pfungwa Greek vanoyemurwa sayenzi zvakadai masvomhu, svomhu uye geometry, somunhu akanaka nezvimwe evanhu zvisingarambiki.

Zvichida pamasvomhu hakusi rakadzika sezvo sayenzi mumwe, asi chii chaizoitika kwavari, vanhu vanokanganwa zvokutanga chokuwanziridza pamatafura? Familiar kwatiri musoro pamafungiro, kushandisa nhamba, zvikamu zviduku, uye zvimwe zvokushandisa kupa vanhu nguva yakaoma, uye kwenguva yakareba haana kuwanikwa madzitateguru edu. Kutaura zvazviri, pamberi kukura svomhu hapana nzvimbo zivo yevanhu hakuna zvechokwadi zvesayenzi.

Svomhu - Mathematics ndiye arufabheti

Svomhu - sayenzi nhamba, izvo munhu rinotanga kuziva inonakidza nyika wemasvomhu. Mumashoko M. V. Lomonosov, svomhu - iri isuwo wedzidzo, achivhurira nzira kwatiri Miropoznanie. Asi iye Ndizvozvo, ndiko kuziva nyika inogona kuparadzaniswa kubva mabhii uye nhamba, masvomhu uye kutaura? Zvichida mumazuva akare, asi kwete ano nyika, apo kukurumidza kukura zvesainzi anoita mitemo yayo.

Shoko "svomhu" (Gk. "Arifmos") echiGiriki mavambo, rinoreva "nhamba". It anoongorora nhamba uye zvose zvinogona nechokuita nazvo. Ndiyo nyika nhamba: siyana mashandiro ari nhamba, nhamba mitemo, mabasa kuti chokuita chokuwanziridza, kubvisa, uye zvichingodaro ..

Vazhinji vakabvuma kuti pokuiswa nhano ndiye svomhu Mathematics uye rakasimba umbowo zvikuru kunzwisisa zvikamu zvaro, akadai nemasvomhu, akarongeka pachisara yepamusoro masvomhu uye t. D.

The chikuru chinhu svomhu

Hwaro svomhu - chiri integer, zvivako uye mitemo iyo vanonzi soro svomhu kana nhamba dzidziso. Kutaura zvazviri, kuti zvakarurama nzira chinobviswa mu kufunga kwakadaro duku Unit, sezvo dzinongoitika nhamba novimba nesimba chivakwa - masvomhu.

Saka, mubvunzo kuti svomhu, mhinduro iri nyore, ndiko sayenzi nhamba. Hongu, pamusoro zvatinosiita nomwe, pfumbamwe, uye zvose siyana munharaunda iyi. Uye sezvo zvakanaka, uye zvikuru mediocre ndima havagoni kunyora pasina arufabheti zvinokosha, pasina svomhu haagoni kugadziriswa kunyange inokosha mabasa. Ndicho chikonzero ose sayenzi kunge mberi chete pashure kukura svomhu uye masvomhu, kuva chikuru chine zvekungofungira.

Svomhu - science-chipoko

Chii svomhu - dzinongoitika sayenzi kana phantom? Kutaura chokwadi, sezvo vazivi vekare vechiGiriki akakurukura, hapana nhamba, hapana mukufambisira chokwadi haako. Rinongova phantom, iyo wakasikwa pfungwa dzevanhu kana kuona zvakatipoteredza uye ayo kwendangariro. Kutaura chokwadi, chii nhamba? Hapana kumativi hatidi kuona chinhu chakadaro aigona kunzi nhamba, asi, nhamba - inzira kuongorora nyika kunoita pfungwa dzevanhu. Zvichida kudzidza ichi tine mukati pachavo? Nevazivi gakava iri zhinji emakore panhurikidzwa saka kupa mhinduro dzoga tisinganeti ita. Chero nzira, iyo svomhu aigona zvakasimba saka kutora nzvimbo yavo ano hapana anogona kunzi mumagariro chakatorwa pasina ruzivo nheyo dzayo.

Sezvo paiva yakanaka integer

Chokwadi, chikuru chinhu izvo unoshanda svomhu, - zvakasikwa nhamba yakadai 1, 2, 3, 4, ..., 152 ... etc. Svomhu rokudzora nhamba imhaka tichiregeredza zvavo zvinhu, zvakadai mhou kumafuro. Kunyange zvakadaro, tsanangudzo "zvizhinji" kana "zvishoma" kana chinhu akarega kubata vanhu, uye vaifanira kugadzira yakagozha kuverengwa unyanzvi.

Asi budiriro chaiyo akauya apo pfungwa dzevanhu yasvika papfungwa kuva mumwe chete uwandu "mbiri" kunan'anura uye 2 makirogiramu, uye 2 zvidhina uye 2 zvikamu. Chokwadi kuti zvakakosha Abstract kubva nezvimiro, unhu uye zvinoreva zvinhu, ipapo tinogona kubereka zvimwe chiito pamwe izvi zvinhu nenzira yakanaka integers. Saka akaberekwa ari svomhu nenhamba, iro mamwe yakatanga uye vaikudza mune achitora chinzvimbo munzanga.

Kwakadaro zvakadzama pfungwa nhamba, sezvo razero uye kunaka nhamba, zvikamu zviduku, nhamba anoreva nhamba mune dzimwe nzira, kuva akapfuma uye zvinofadza nhoroondo kukura.

Svomhu uye rinoshanda vaEgipita

Two shamwari yekare yevanhu vari kudzidza nyika uye nekugadzirisa matambudziko ezuva nezuva - ichi svomhu uye geometry.

Zvinofungidzirwa kuti nhoroondo svomhu ine yakatangira the Ancient East: India, Ijipiti, Bhabhironi uye China. Saka, Rhind yenhokwe yokuIjipiti mavambo (ainzi saka nokuti chete zita kuva muridzi), chaiko kune XX remakore. BC, kuwedzera kune mamwe mashoko anokosha inosanganisira kuwedzera chikamu mune uwandu zvikamu vane zvakasiyana denominators uye numerator yakaenzana mumwe.

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

Asi chii chinorehwa kunzwisisa manyowa kwakadaro? Chokwadi chokuti muIjipita nzira haashiviriri abstracted achifunga nhamba nekusiyana, kuti Masvomhu vakaitwa chete zvinangwa dzinobatsira. Kureva, vaIjipiti chichange mubasa mubhizimisi zvakadai Masvomhu, kungotaurwa kuitira kuvaka guva, somuenzaniso. Zvakanga zvakakodzera kuti kuverenga urefu zenze mamiriro, uye akaita kuti munhu agare enhokwe. Sezvinogona kuoneka, muEgipita mberi Masvomhu yainzi, asi hombe, kuvaka, pane rudo sayenzi.

Nokuda kwechikonzero ichi, Masvomhu zvakawanikwa pemipumburu, haagoni kunzi zvichiratidza panyaya zvikamu zviduku. Ungangove, ndiko inoshanda gadziriro, izvo zvakabatsira kuti zvimwe matambudziko pamwe zvikamu zviduku. VaIjipiti vekare havana kuziva chokuwanziridza tafura, akabudisa Masvomhu zvakarurama yakareba, wakavhurirwa subtasks zhinji. Zvichida ichi ndicho chimwe chezvinhu subtasks ivavo. Zviri nyore kuona kuti Masvomhu vane blanks izvi ndiyo nguva yakawanda zvikuru uye kwete vaizotibata chaizvo. Zvichida nokuti chikonzero ichi zvatisingaoni mupiro guru kukura masvomhu vekare vokuIjipiti.

Ancient Greece uye svomhu nouzivi

Vakawanda zivo Ancient East vakanga vakabudirira Inoziva nevaGiriki vekare, aizivikanwa mafeni kushandisa, Abstract nouzivi kufungisisa. Dzidzira navo kufarira chinhu zvishoma asi yakanakisisa theorists nevafungi zvakaoma kuwana. Zvakanga zvakanaka kuti sayenzi nekuti pamasvomhu hazvibviri kuenda rakadzika, kwete nokuikwachura sezvazviri. Zvechokwadi, zvinokwanisika kuti vawande 10 mhou uye 100 litres mukaka, asi kwete kukwanisa kufamba kure.

VaGiriki nemafungiro chaizvo kusiya chinokosha chiratidzo munhau, uye mabasa avo vauya kwatiri:

• Euclid uye "Elements".
• Achibvuma pfungwa yaPythagoras.
• Archimedes.
• Eratosthenes.
• Zenon.
• Anaxagoras.

Uye, chokwadi, inodzora zvose muuzivi hwevaGiriki, uye kunyanya vateveri Achibvuma pfungwa yaPythagoras nyaya dzaiva saka vaneshungu pamusoro nhamba, izvo vaiona kwavari maererano chakavanzika nyika. Nhamba vave saka akadzidza uye kuongororwa, kuti vamwe vavo uye dzavo vakaroorana raibva chaiwo ehupfumi. Somuenzaniso:

• Perfect nhamba - avo vari Verengai vanhu vose divisors zvayo kunze nhamba pacharo (6 = 1 + 2 + 3).
• Friendly nhamba - nhamba idzi, rimwe riri uwandu vose divisors rechipiri ndiEzekieri (Pythagorean kuziva mumwe chete akadai peya: 220 uye 284).

VaGiriki, uyo aidavira kuti sayenzi anofanira kudiwa, kwete kuva naye nekuda pfuma, ndakaita kufambira mberi kukuru, kuongorora, achiridza uye achiwedzera nhamba. Zvinofanira kucherechedzwa kuti havasi vose zvavo tsvakurudzo zvave nevakawanda, vamwe vavo vaiva chete "kuti runako."

Eastern nevafungi of Ages

Saizvozvowo, kuMiddle Ages svomhu nayo inotumidzwa kukura kwayo okumabvazuva vomuzuva. MaIndia akatipa nhamba kuti isu nokushingaira kushandisa chinhu chakadai "razero", uye nzvimbo kushanduka kukarukureta hurongwa, zvatinosiita ano maonero. Kubva Al-Porridge, izvo remakore rechi15 vaishanda Samarkand, hatina nhaka decimals, pasina iro zvakaoma kufungidzira svomhu ano.

Munzira dzakawanda, Europe vakajairana nebudiriro East rakaitwa zvibvire anoonga basa musayendisiti Italian Leonardo Fibonacci, uyo akanyora bhuku "Liber Abaci", ajairane pamwe yokuMabvazuva zvakagadzirwa. It wekona ari kukura nemasvomhu uye svomhu, kutsvakurudza uye mabasa dzesayenzi mu Europe.

Russian svomhu

Pakupedzisira, svomhu wakawanei nzvimbo yaro uye midzi Europe, vakatanga kupararira panyika Russian. Russian rokutanga svomhu rakabudiswa muna 1703 - waiva bhuku pamusoro svomhu Leontiya Magnitskogo. Kwenguva refu yaiva chete Tutorial mu masvomhu. Rine kutanga nguva nemasvomhu uye geometry. Nhamba, izvo zvaishandiswa mienzaniso reRussia bhuku rokutanga svomhu, Arabic. Kunyange zvazvo sezviverengo Arabic akasangana kare, ari sokunyorwa chaiko kune remakore rechi17.

Bhuku racho pacharo riri akashongedzwa nemifananidzo Archimedes uye Achibvuma pfungwa yaPythagoras, uye pazuva rokutanga papeji - mufananidzo svomhu somukadzi. Iye anogara pachigaro choushe, pasi makanyorwa rechiHebheru nokuda zita raMwari, uye pamatanho kuti kukonzera paaritari, akanyorwa shoko "kupesana", "kuwedzera", "kuwedzera", uye zvichingodaro. D. Mumwe anogona chete kufungidzira ukoshi akazomupandukira chokwadi akadaro, izvo zvava kunzi hakuzivani.

Bhuku pamapeji 600 inorondedzera sezvo hwaro yakafanana Kuwedzera uye chokuwanziridza pamatafura, uye zvikumbiro sayenzi kufamba mugungwa.

Hazvishamisi, munyori akasarudza mufananidzo Greek nevafungi yake bhuku, nekuti iye akanga Ugare kubudikidza nokunaka svomhu, achiti, "svomhu ave chislitelnitsa ipapo wakanaka, nezavistnoe ...". Kutungamirirwa svomhu zvakanaka nheyo, nokuti wokugamuchirwa wayo kwakapararira anogona kunzi kutanga nokukurumidza kukura kwepfungwa nesayenzi muRussia uye zvayo dzidzo.

tsamwa of times

Prime nhamba - zviri wechisikigo nhamba, inova 2 chete akanaka divisors: 1 uye pacharo. zvimwe zvose nhamba, kunze 1 anonzi ematongerwo. Mienzaniso inonyanya nhamba: 2, 3, 5, 7, 11, uye vamwe vose kuti havasi vamwe pane 1 divisors uye nhamba pacharo.

Kana nhamba 1, zviri shoma - vanobvumirana kuti zvinofanira kufungwa kana nyore kana akakura. Simple Pakutanga, nyore nhamba anovanza unsolved zvakavanzika zhinji mukati mavo.

Euclid kwakaita theorem no rinoti isingaperi uwandu of times, uye Eratosthenes akasvika rinokosha svomhu "nomusero", iyo anobvisa yakaoma nhamba, vachisiya nyore chete.

Chaizvoizvo Its ndiko kusimbisa wokutanga undelete nhamba yacho, uye iri unozotevera unoshamisa vaya vari multiples nawo. Tikapfuudza nzira iyi kakawanda - uye kuwana tafura inonyanya nhamba.

Chinokosha theorem no kuti svomhu

Pakati zvaanorumbidza pamusoro chikuru nhamba vanofanira zvikuru kutaura inokosha svomhu theorem no.

Basic svomhu theorem no inoti chero integer kupfuura 1, kana iri nyore kana unogona yaora kuva chigadzirwa inonyanya nhamba kusvikira hurongwa yokudzokorora zvinhu, chete nzira.

Chinokosha theorem no kuti svomhu akaratidza zvikuru yainetsa uye nokunzwisisa haina kufanana kungoita.

Pakutanga, mukuru nhamba - chepuraimari pfungwa, asi hazvisi. Physics uyewo dzaimboonekwa chepuraimari atomu, kusvikira akawana mukati ari muchadenga. Of times akatsaurira yakanaka nyaya yemasvomhu Don Zagier "wokutanga mukuru nhamba mamiriyoni makumi mashanu."

Kubva "maapuro vatatu" mitemo deductive

Kuti zvechokwadi anogona kunzi Akazadzisa hwaro zvose sayenzi - mitemo svomhu. Kunyange kuva mwana vose svomhu chiso, kudzidza nhamba makumbo uye maoko ari nezvidhori, nhamba cubes, maapuro uye zvichingodaro. D. Saka isu kudzidza svomhu, iyo zvino mberi mu mitemo yakawanda kunzwisisa.

Our upenyu hwose kutiudza nemitemo svomhu, izvo zvaiva zvavo murume kupfuura anobatsira zvose sainzi anopa. Kudzidza nhamba - ndiye "svomhu-mwana", izvo otaura munhu munyika nhamba sezvo digits mu muduku.

Higher svomhu - deductive nesayenzi kuti kudzidza mitemo svomhu. Vazhinji vavo tinoziva, zvichida hatizivi mashoko avo chaiwo.

Murairo Kuwedzera uye chokuwanziridza

integers Chero vaviri uye, b kungaratidzwa sezvo Verengai imwe b, + riri dzinongoitika nhamba. Ichitaura Uyezve, zvinotevera mitemo:

• Commutative, kuti rinoti permutation ari mashoko anoisa mari haachinji, kana munhu + b = b + mumwe.
• Associative kuti akati mari hakunei nzira kuti vanhu muzvikwata zvisina tsarukano okuti munzvimbo, kana munhu + (b + c) = (a + b) + c.

Mitemo svomhu, akadai Uyezve - mumwe inokosha, asi vari kushandiswa ose sayenzi, tisingatauri upenyu hwezuva nezuva.

Chero maviri integers ane nokuva b kungaratidzwa iri chigadzirwa kana B * imwe * b, riri dzinongoitika nhamba. Kushandisa chigadzirwa kune commutative uye associative mitemo kufanana kuwedzera:

• mumwe * b = B * ane;
• mumwe * (b * c) = (a * b) * c.

Zvinofadza kuti pane mutemo, izvo rinobatanidza kuwedzera uye chokuwanziridza, aizivikanwawo somunhu kuparadzira kana mutemo distributive:

munhu (b + c) = AB + č

mutemo uyu anotidzidzisa kushanda mumabhuraketi, ivo kuvhura, saka tinogona kare kushanda Masvomhu zvikuru kunzwisisa. Ndivo mitemo achatitungamirira kuburikidza sokunofadza asi kunzwisisa nyika nemasvomhu.

Law svomhu hurongwa

pamusoro mitemo yevanhu pfungwa inoshandisa rimwe zuva, vachiongorora wachi yake uye kuverenga yacho. Uye, zvisinei, uye inofanira kuitwa mumutauro chaiwo.

Kana tine integers maviri zvakanaka uye b, ipapo pfungwa dzinotevera:

• munhu akaenzana akundinamata, kana = b;
• imwe asingasviki b, kana
• mumwe mukuru b, kana> b.

Pakati zvitatu chete kunogona chete. The Basic Law, izvo anodzora muitiro, akati: Kana

Panewo mitemo kusunga zvaiitwa murayiro Kuwedzera uye chokuwanziridza: kana munhu

Mitemo svomhu akatidzidzisa kushanda nhamba, nezviratidzo mumabhuraketi vanoshandura zvose kuva nezvimwe rinopindirana nhamba.

Positional uye nonpositional vaverengwa hurongwa

Tinogona kutaura kuti nhamba - ndiwo mutauro wemasvomhu, kubva zvivanakire iyo kunoenderana zvinhu zvakawanda. Kune zvakawanda mugadziriro kukarukureta, izvo, kufanana bumbiro emitauro yakasiyana siyana anosiyana.

Funga nhamba hurongwa kubva pfungwa kunokanganisa zvinzvimbo pamusoro quantitative kukosha digit panzvimbo ino. Somuenzaniso, Roman kwava nonpositional apo nhamba mumwe nomumwe encoded neboka chaicho akagadzirwa chinokosha kutaurwa: I / V / X / L / C / D / M. Ivo ukuwo, nhamba 1/5/10/50/100/500 / 1000. Munyika ino, chirungamutauro haachinji quantitative kutsunga kwayo, zvichienderana nenguva chii woiisa kuita: .. The wekutanga, wechipiri, etc. Kuti dzimwe nhamba, zvakakodzera kuisa pasi pechingoro. Somuenzaniso:

• DCC = 700.
• CCM = 800.

Kunyatsoziva kwatiri nhamba hurongwa kushandisa sezviverengo Arabic ndiyo positional. Zvakadaro gadziriro nhamba pakuyerera rinorondedzera uwandu digits Somuenzaniso,-digit matatu nhamba: 333, 567, etc. Kurema chero zvinoyerera kunotsamira chinzvimbo uri nhamba ndeimwe kana nezvimwe, semuenzaniso mufananidzo 8 yechipiri nzvimbo ine ukoshi 80. Zviri kwese kuti grade hurongwa, pane dzimwe positional maitiro akadai binary.

binary svomhu

Tiri munodziziva grade gadziriro, zvinosanganisira chete-ikakuruma uye multi-ikakuruma nhamba. The mufananidzo kuruboshwe iri digit nhamba iri kagumi mukuru kukosha kune imwe kurudyi. Saka, taiwanzokuudzai verenga 2, 17, 467, uye zvichingodaro. D. Zviri yakasiyana pfungwa uye nzira chikamu, unonzi "binary svomhu." Izvi hazvishamisi, nokuti binary svomhu haina kusikirwa munhu pfungwa, uye nokuda kombiyuta. Kana svomhu nenhamba rakabva kuverengwa, mamwe abstracted kubva pasi pfuma kuti "kupfeka" svomhu, ipapo ichi hakushande pamwe kombiyuta yako. Kuti vakwanise kugoverana ruzivo rwavo pamwe kombiyuta, murume aifanira wekuvamba muenzaniso kukarukureta.

Binary svomhu anoshanda pamwe binary arufabheti randinogara chete 0 uye 1. Uye kushandisa arufabheti ichi inonzi binary hurongwa.

Kusiyana binary svomhu grade kuti kukosha nzvimbo kuruboshwe havasisirizve 10, uye 2 nguva. Binary nhamba vari muchimiro 111, 1001 uye zvichingodaro. D. Tinofanira kunzwisisa nhamba idzi? Saka, isu funga nhamba 1100

1. The digit yokutanga kuruboshwe - 1 * 8 = 8, uchiyeuka kuti digit wechina, izvo zvinoreva kuti rinofanira kane 2, tinowana 8 chinzvimbo.
2. Second digit 1 * 4 = 4 (nzvimbo 4).
3. Wechitatu digit 0 * 2 = 0 (nzvimbo 2).
4. Wechina digit 0 * 1 = 0 (nzvimbo 1).
5. Saka nhamba yedu 1100 = 8 + 4 + 0 + 0 = 12.

Ndiko, kuchinja kuti muchikwata mutsva kuruboshwe kukosha kwawo munyika binary ino zvikawanda ne 2 uye grade - kusvika 10. akadaro hurongwa ane drawback: ndiko kukura zvakare guru matomu kuti vanofanira kunyora nhamba. Mienzaniso grade nhamba dvochinyh sezvo zvinogona kuonekwa zvinotevera tafura.

Grade nhamba anomiririrwa mu binary fomu pazasi.

Rinoshandiswawo octal, uye hexadecimal chiverengero gadziriro.

This chisinganzwisisike svomhu

Chii svomhu, "mbiri kuwedzera mbiri" kana unexplored zvakavanzika nhamba? Sezvaunogona kuona, svomhu, anogona, uye zvinoita sokuti pakutanga aitarire nyore, asi hazvina pachena kunonyengera nyore. Zvinokwanisika kudzidza vana, uye pamwe Tete Owl kubva katuni "svomhu-mwana", uye iwe unogona svetuka kupinda yakadzika kwesayenzi kuti anenge youzivi. Munhau nayo chabva kuverenga zvinhu kunamata kunaka nhamba. Chimwe chinhu ndechokuti vamwe: pakatangwa inokosha postulates kuti svomhu, sayenzi zvose zvinogona kuvimba kwake simba pafudzi.