Masamu ndi chiyani? Zofunika Kwambiri Lingaliro Lovomerezeka la masamu. bayinare masamu

masamu ndi chiyani? Pamene anthu anayamba kugwiritsa ntchito manambala ndi ntchito nawo? Kodi mizu yake mfundo tsiku ndi tsiku monga manambala, tuzigawo twa, wochotsera, Kuwonjezera ndi chulutsa, munthu amene wakonzekera mbali ya moyo wake ndi maganizo amenewo? maganizo Greek anamusilira sayansi monga masamu, masamu ndi masamu, ngati nthetemya wokongola wa kaganizidwe anthu.

Mwina masamu si akuya sayansi zina, koma zimene zidzachitike kwa iwo, anthu kuiwala matebulo pulayimale chulutsa? Bwino kuti ife tikadziwe maganizo, pogwiritsa ntchito manambala, tuzigawo twa ndi zipangizo zina kwa anthu ndi nthawi yovuta, ndipo kwa nthawi yaitali palibe makolo athu. Ndipotu, pamaso pa chitukuko cha masamu palibe malo nzeru za anthu sikunali moonadi sayansi.

Masamu - Masamu ndi zilembo

Masamu - sayansi ya manambala, ndi zomwe munthu akuyamba ubwenzi ndi zodabwitsa za masamu. Mawu a M. V. Lomonosov, masamu - ndi khomo kuphunzira ndo kwa ife Miropoznanie. Koma ndiko kulondola, ndicho kudziwa dziko akhoza anapambulwa pa chidziwitso cha makalata ndi manambala, masamu ndi kulankhula? Mwina masiku akale, koma osati mu dziko ano, pamene kukula mofulumira sayansi amapanga malamulo yake.

Mawu "masamu" (Gk. "Arifmos") chiyambi Greek, limatanthawuza "chiwerengero". Ikufotokoza chiwerengero ndi zonse zimene nawo. Izi ndi dziko la manambala: osiyanasiyana ntchito pa manambala, malamulo n'zosangalatsa kumva, ntchito kuti imakhudzana ndi chulutsa, wochotsera, ndi zina zotero ..

Ndizovomereka kuti sitepe koyamba ndi masamu Masamu ndi olimba m'munsi pa zovuta kwambiri mafelemu, monga ajebura, kusanthula masamu, masamu apamwamba T. D.

The chinthu chachikulu cha masamu

Maziko a masamu - ndi inteja, katundu ndi malamulo omwe amaonedwa masamu apamwamba kapena chiphunzitso chiwerengero. Ndipotu mmene njira yoyenera atengedwa mu kulingalira amenewa gulu laling'ono, ngati nambala zachilengedwe amadalira mphamvu ya nyumbayi - masamu.

Choncho, funso limene lili masamu, yankho lake ndi losavuta: izo ndi sayansi ya manambala. Inde, pafupifupi alipidwa zisanu ndi ziwiri, zisanu ndi zinayi, ndi onse a dera lino osiyanasiyana. Ndipo mofanana, ndi ndime kwambiri mediocre sangathe kulemba popanda zilembo zazikulu, popanda masamu sangathe kuthetsedwa ngakhale zofunika. Ndicho chifukwa sayansi onse patsogolo pokhapokha chitukuko cha masamu ndi masamu, pokhala makamaka akonzedwa a maganizo.

Masamu - sayansi Mzimu

sayansi achilengedwe kapena malodza - masamu ndi chiyani? Ndipotu, ngati wakale Agiriki anakambirana, palibe manambala, palibe kanjedza kwenikweni kulibe. Ndi chabe malodza, amene analengedwa mu malingaliro pamene kuonera chilengedwe ndi njira zake. Ndipotu, kodi chiwerengero? Paliponse padziko sitikuchiona chirichonse chonga izo angatchedwe chiwerengero, kani, chiwerengero - ndi njira kufufuza dziko la maganizo a anthu. Mwina phunziro ili tili mkati okha? Anzeruwo kukanganira za izi kwa zaka zambiri mu mzere, kotero kuti tibwezere zochuluka tilibe ntchito. Choncho, masamu nditha mwamphamvu malo awo mu dziko masiku ano palibe munthu angathe kuonedwa chikhalidwe ndinazolowera popanda chidziwitso cha maziko ake.

Pamene panali inteja zabwino

Inde, chinthu chachikulu chimene chigwira ntchito masamu, - zachilengedwe chiwerengero monga 1, 2, 3, 4, ..., 152 ... etc. Masamu a manambala zachilengedwe ndi zotsatira za ndalama za zinthu wamba, monga ng'ombe dambo ndi. Komabe, tanthawuzo la "kwambiri" kapena "pang'ono" pamene chinachake watha kugwira anthu, ndipo amayenera kuyambitsa zovuta kuzimvetsa kwambiri ochuluka njira.

Koma yojambula enieni anabwera pamene malingaliro a munthu wafika moti akhoza kukhala limodzi ndi anthu ngati omwewo "ziwiri" udindo ndi 2 makilogalamu, ndipo 2 njerwa ndi mbali 2. Popeza kuti m'pofunika umboni ku mitundu, makhalidwe ndi tanthauzo la zinthu, ndiye ife zingabweretse kanthu ndi zinthu izi mu mawonekedwe a integers positive. Choncho anabadwa masamu a manambala, amene inakulitsidwa ndi kunanditsegula mu agwilitse udindo anthu.

Woteroyo mozama mfundo chiwerengero, monga ziro ndi manambala zoipa, tuzigawo twa, manambala amanena kuti manambala m'njira zina, ali ndi mbiri olemera ndi chidwi chitukuko.

Masamu ndi othandiza Aigupto

Awiri wakale mnzake anthu mu kuphunzira kwa dziko ndi kuthetsa mavuto - masamu ichi ndi masamu.

Amakhulupirira kuti mbiri ya masamu ali chiyambi chake mu East Ancient: India, Egypt, Babulo ndi China. Choncho, Rhind amtunduwu anachokera ku Iguputo (otchedwa chifukwa dzina chomwecho yoperekedwa kwa mwini), chibwenzi m'mbuyo m'zaka za m'ma XX. BC, kuwonjezera deta zamtengo wapatali zikuphatikizapo patsogolo kachigawo ndi kuchuluka kwa tuzigawo twa ndi denominators osiyana ndi numerator Angafanane ndi.

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

Koma kodi tanthauzo la chotero kuwonongeka zovuta? chakuti njira Aigupto salola abstracted kuganiza za manambala M'malo mwake anapezazo anali opangidwa okha zolinga zothandiza. Ndiko kuti, Aiguputo adzakhala zikugwira ntchito monga anapezazo, basi kuti kumanga manda, mwachitsanzo. Kunali koyenera kuti kuwerengetsa zaka za dongosolo chipsyepsye, ndipo anapanga kuti munthu akhale amtunduwu. Monga mmene taonera, patsogolo Iguputo kuwerengetsera unkatchedwa, m'malo chachikulu, kumanga, osati chikondi cha sayansi.

Pachifukwa ichi anapezazo pa mipukutu, sitinganene kuti mawunikidwe pa phunziro la ting'onoting'ono. Ambiri mwina, ndi kukonzekera zothandiza, amene anathandiza kuthetsa zina mavuto ndi ting'onoting'ono. Aigupto akale sanali kudziwa tebulo chulutsa, anapanga kuwerengetsera mwachilungamo yaitali, anayala mu subtasks ambiri. Mwina ichi ndi chimodzi mwa subtasks anthu. N'zosavuta kuona kuti kuwerengetsera ndi akusowekapo awa ndi nthawi yambiri kwambiri ndi sunakhale kwambiri. Mwina pa cifukwa ichi sitikuchiona chopereka chachikulu ndi chitukuko cha masamu ku Iguputo.

Ancient Greece ndi masamu nthanthi

Ambiri a chidziwitso cha Ancient East anali bwinobwino katswiri Agiriki akale, kudziwika kwa mafani wa umboni, umboni ndi nzeru zaumunthu chinyezimiritso. Amachita chidwi woteroyo. koma theorists bwino ndi anzeru ndi kuchipeza. Iwo anali wabwino kwa sayansi chifukwa masamu sizingatheke kupita mwakuya, osati kuukhadzula ndi chowonadi. Kumene, n'zotheka kuchulukana ng'ombe 10 malita 100 mkaka, koma sangathe amapita.

Agiriki kuganizira kwambiri anasiya chizindikiro kwambiri mbiri, ntchito zawo abwera kwa ife:

• Euclid ndi "Zochitika".
• Pythagoras.
• Archimedes.
• Eratosthenes.
• Zenon.
• Anaxagoras.

Ndipo, ndithudi, akutembenukira nzeru zonse za Agiriki, makamaka otsatira a milandu Pythagoras anali mokhudza za manambala, amene amaona chinsinsi dziko mogwirizana. manambala akhala choncho anaphunzira ndipo ankafufuza, ena a iwo ndi mabanja awo analembedwa katundu wapadera. Mwachitsanzo:

• Wangwiro manambala - iwo amene ali Uwerenge divisors ake onse kupatula chiwerengero yokha (6 = 1 + 2 + 3).
• manambala Friendly - manambala awa, mmodzi amene ali Uwerenge divisors onse wachiwiri ndi mosemphanitsa (Pythagorean kudziwa imodzi yokha ngati awiri: 220 ndi 284).

Ahelene, amene amakhulupirira kuti sayansi ayenera kukondedwa, osati kukhala naye chifukwa cha phindu, apita patsogolo kwambiri kufufuza, kusewera ndi kuwonjezera manambala. Tikumbukenso kuti si zonse za kafukufuku wawo akhala chimagwiritsidwa ntchito kwambiri, ena a iwo okha "kukongola."

anzeru a kum'mawa kwa Zapakati

Mofananamo, Ages Middle masamu izo kukhala chitukuko kwa m'nthawi ya kum'mawa. Amwenye anatipatsa manambalawo kuti ife mwakhama ntchito chinthu monga "ziro", ndipo udindo kusintha mawerengedwe dongosolo, mwachizolowezi ano maganizo. Kuyambira Al-phala, limene m'ma 15 ntchito mu Samarkand, tinatengera kwa decimals, popanda zomwe n'zovuta kulingalira masamu ano.

Mu njira zambiri, Europe yodziwa zomwe wa East anapangidwa zikomo zotheka kuti ntchito ya sayansi Italy Leonardo Fibonacci, amene analemba bukhu "Liber Abaci", pofotokozera ndi zaluso kalelo. Iwo wakhala mwala wapakona wa chitukuko cha masamu ameneŵa ndi masamu, kafukufuku ndi zinthu sayansi Europe.

Russian masamu

Pomaliza, masamu, yapeza malo ake ndi mizu ku Ulaya, anayamba kufalitsa pa dziko Russian. Russian masamu inafalitsidwa koyamba mu 1703 - zinali zokhudza masamu Leontiya Magnitskogo. Kwa nthawi yaitali anali kokha phunziro mu masamu. Lili mphindi koyamba ajebura ndi masamu. Ziwerengero, zomwe ankagwiritsa ntchito zitsanzo za buku Russia woyamba wa masamu, Arabic. Ngakhale mawerengero Arabic kuti akomanapo, zojambula mochita kugoba ndi chibwenzi ku m'ma 17.

Buku wokha sukhalira chokongoletsedwa ndi zithunzi za Archimedes ndi Pythagoras, ndipo pa tsiku loyamba tsamba - masamu fano ngati mkazi. Iye akukhala pa mpando wachifumu, pansi kwalembedwa mawu achiheberi dzina la Mulungu, ndipo pa masitepe kuti atsogolere ku guwa, lolembedwa ndi mawu akuti "kugawanika", "zoonjezera", "Kuwonjezera", ndi zina zotero. D. wina Kodi mukuganiza kodi phindu adampereka choonadicho, amene tsopano amaonedwa wamba.

Buku la masamba 600 limafotokoza maziko a Kuwonjezera ngati ndi matebulo chulutsa, ndi ofunsira sayansi panyanja.

N'zosadabwitsa kuti wolemba wasankha fano la anzeru achigiriki buku lake, chifukwa iye anali chidwi ndi kukongola kwa masamu, kunena, "masamu ali chislitelnitsa kumeneko ndinu okongola, nezavistnoe ...". Njira imeneyi ndi masamu bwino linakhazikitsidwa, chifukwa ndi umwana wake ambiri tinganene kuti chiyambi cha kukula mofulumira lingaliro sayansi Russia ndi maphunziro ambiri.

primes amakhumudwa

chiwerengero Yaikulu - ndi nambala zachilengedwe, zomwe ndi 2 okha divisors zabwino: 1 ndi palokha. Onse manambala ena, kupatula 1 akutchedwa gulu. Zitsanzo za manambala yaikulu: 2, 3, 5, 7, 11, ndi ena onse amene sali divisors ena kuposa 1 Chiwerengero yokha.

Pamene chiwerengero 1, ndi pa yambiri - pali pangano kuti tiyenera kuziganizira ngakhale yosavuta kapena pawiri. Zambiri pa Koyamba, nambala yosavuta Wobisa ambiri sanathetsedwe zinsinsi mwa iwo okha.

Lingaliro Lovomerezeka Euclid amanena kuti chiwerengero wopandamalire wa primes, ndi Eratosthenes anabwera ndi wapadera masamu "sieve" imene idzathetsa manambala zovuta kusiya yosavuta yekha.

Kwenikweni ake ndi kutsindika woyamba chiwerengero undelete, ndi chidwi wotsatira iwo amene ali multiples za izo. Tikufotokozera njirayi kangapo - ndi kupeza tebulo la manambala yaikulu.

Zofunika Kwambiri Lingaliro Lovomerezeka la masamu

Mwa kufotokoza zimene manambala yaikulu ayenera mwapadera kutchula zofunika masamu Lingaliro Lovomerezeka.

Kuweta masamu Lingaliro Lovomerezeka limanena kuti inteja uliwonse woposa 1, kapena zophweka kapena akhoza decomposed mu mankhwala a manambala yaikulu mpaka dongosolo la zinthu kubwereza, njira yokha.

Zofunika Kwambiri Lingaliro Lovomerezeka la masamu anatsimikizira wolemera kwambiri, ndi kumvetsa si lofanana chabe ndizosowa.

Koyamba, manambala yaikulu - pulayimale lingaliro, koma izo siziri. Physics komanso ankaganiza pulayimale atomu, mpaka anapeza mkati chilengedwe a. Primes odzipereka wokongola nkhani masamu Don Zagier "The woyamba miliyoni makumi asanu manambala yaikulu."

Kuchokera "maapulo atatu" malamulo deductive

Umenewu angatchedwe ndi kumakula maziko a sayansi yonse - malamulo a masamu. Ngakhale ndili mwana masamu onse nkhope, kuphunzira chiwerengero cha miyendo ndi manja pa zidole, chiwerengero cha cubes, maapulo ndi zina zotero. D. Kotero ife kuphunzira masamu, kenako m'kati mwa malamulo ovuta kwambiri.

moyo wathu wonse kutiuza ndi malamulo a masamu, amene anali munthu wamba kwambiri zothandiza wa sayansi kuti zonse amapereka. Kuphunzira manambala - ndi "masamu-mwana" imene anayambitsanso munthu ku dziko la manambala ngati manambala ana aang'ono.

Apamwamba masamu - deductive sayansi kuti amaphunzira malamulo a masamu. Ambiri a iwo tikudziwa, ngakhale mwina ife sitikudziwa awo kalembedwe kenikeni.

Lamulo la Kuwonjezera ndi chulutsa

Aliyense integers awiri ndi b tingasonyezere monga Uwerenge b +, amenenso nambala achilengedwe. Ponena za Komanso, malamulo otsatirawa:

• Commutative, amene amanena kuti permutation mawu malo kuchuluka sasintha, kapena + B = b + a.
• Associative kuti anati ndalama sikudalira njira grouping mawu mu malo, kapena + (b + c) = (a + b) + c.

Malamulo a masamu, monga Komanso, - imodzi yakuti, koma ntchito sayansi onse, osanenapo tsiku ndi tsiku.

Aliyense integers awiri ndi b tingasonyezere mankhwala kapena b * ndi * b, amenenso nambala achilengedwe. Kugwiritsa ntchito mankhwala yemweyo malamulo commutative ndi associative kuti Kuwonjezera:

• ndi * B = b * ndi;
• ndi * (b * c) = (a * b) * c.

N'zochititsa chidwi kuti pali malamulo, chomwe chili Kuwonjezera ndi chulutsa, amatchedwanso yogawa kapena malamulo distributive:

ndi (b + c) = AB + Mac

lamulo limeneli likutiphunzitsa ntchito ndi m'mabokosi, kutsegula potero tikhoza kale ntchito ndi mitunduyi zovuta zambiri. Amenewa ndiwo malamulo amene atitsogolera mwa takambiranazi koma zovuta dziko la ajebura.

Chilamulo masamu kuti

za malamulo maganizo anthu limagwiritsa ntchito tsiku lililonse, afufuze maso ake ndi kuwerengera ngongole. Ndipo, koma, ndipo izo ziyenera kukhala chilankhulo chinachake.

Ngati tili integers awiri HIV ndi b, ndiye mungachite zotsatirazi:

• ndi wofanana B, kapena = b;
• ndi zosakwana b, kapena
• ndi wamkulu kuposa b, kapena> b.

Pa njira zitatu basi mmodzi yekha. Chilamulo Basic, omwe amalamulira ndondomeko, anati: ngati

Palinso malamulo amene wamanga zochita za dongosolo la Kuwonjezera ndi chulutsa: ngati

Malamulo a masamu anatiphunzitsa ntchito ndi manambala, zizindikiro ndi m'mabokosi, kutembenuzira zonse mu nthetemya amafunika manambala.

Pamalo ndi nonpositional dongosolo manambala

Tikhoza kunena kuti chiwerengero - ichi ndi chinenero cha masamu ku mayiko potsata zinthu zambiri. Pali kachitidwe zambiri mawerengedwe, amene, ngati alifabeti a zinenero zosiyanasiyana amasiyana.

Taonani dongosolo chiwerengero mfundo udindo zimakhudza mtengo kachulukidwe ya manambala mu udindo uwu. Mwachitsanzo, kachitidwe ka Chiroma ndi nonpositional pamene nambala aliyense ndiyotengera ndi akonzedwa yeniyeni ya zilembo lapadera: I / V / X / L / C / D / M. Iwo, motero, chiwerengero 1/5/10/50/100/500 / 1000. Mu dongosolo ili, chithunzi sikusintha mtima wake kachulukidwe, malingana pa malo kuyenera: .. The choyamba, chachiwiri, etc. Kuti manambala ena, m'pofunika kuyala maziko. Mwachitsanzo:

• DCC = 700.
• CCM = 800.

Bwino kwa ife dongosolo numeral ntchito manambala Arabic ndi Pamalo. Mu dongosolo chotero chiwerengero cha kumaliseche imalongosola chiwerengero cha manambala, mwachitsanzo, atatu manambala manambala: 333, 567, etc. Kulemera kwa aliyense wa kumaliseche chimadalira pa malo amene chithunzi ndi chimodzi kapena chimzake, mwachitsanzo chithunzi 8 mu malo chachiwiri ali ndi phindu 80. Ndi mmene kwa dongosolo decimal, pali ena Pamalo dongosolo monga bayinare.

bayinare masamu

Ndife bwino dongosolo decimal, wopangidwa mwa manambala osakwatiwa-bit ndi Mipikisano pang'ono. Chithunzi kumanzere mu chiwerengero manambala ndi kakhumi kwambiri kufunika kwa wina kumanja. Kotero, ife tinkakonda kuwerenga 2, 17, 467, ndi zina zotero D. ndi osiyana logic ndi njira gawo wotchedwa. "Bayinare masamu." Zimenezi n'zosadabwitsa, chifukwa masamu bayinare ndi sanalengedwa chifukwa mfundo anthu, ndi kompyuta. Ngati masamu a manambala linachokera ku ochuluka, amene zina abstracted ku katundu phunziro kwa "maliseche" masamu, ndiye chithe ndi kompyuta. Kuti athe kugawana nzeru zawo ndi kompyuta, munthu amayenera kuyambitsa chitsanzo mawerengedwe.

Bayinare masamu ntchito ndi zilembo bayinare, chomwe yekha 0 ndipo 1. Ndipo ntchito zilembo izi dongosolo bayinare.

Mosiyana bayinare masamu decimal kuti kufunika kwa malo a kumanzere salinso 10, ndipo nthawi 2. manambala bayinare ndi wa mawonekedwe 111, 1001 ndi zina zotero. D. Kodi tiyenera kumvetsetsa manambala awa? Motero, tikambirana chiwerengero cha 1100

1. The manambala woyamba kumanzere - 1 * 8 = 8, pokumbukira kuti manambala chachinayi, kutanthauza kuti ayenera kuchulukitsa ndi 2, timachotsa 8 malo.
2. Chachiwiri manambala 1 * 4 = 4 (udindo 4).
3. The manambala lachitatu 0 * 2 = 0 (udindo 2).
4. The manambala chachinayi 0 * 1 = 0 (malo 1).
5. Kotero chiwerengero wathu 1100 = 8 + 4 + 0 + 0 = 12.

Ndiko kuti, kusintha kwa gulu latsopano kumanzere kwa tanthauzo lake mu dongosolo bayinare ndi kuchulukitsa ndi 2 ndi decimal - 10 dongosolo koteroko drawback chimodzi: izo ali wamkulu kwambiri Tinthu kukula ayenera kulemba manambala. Zitsanzo manambala decimal dvochinyh monga Tingaone mu tebulo zotsatirazi.

manambala Decimal awa akuimira mu mawonekedwe bayinare pansipa.

Amagwiritsidwanso ntchito octal, ndi dongosolo hexadecimal manambala.

Izi masamu chachinsinsi

masamu, "ziwiri kuphatikiza ziwiri" kapena unexplored zinsinsi za manambala ndi chiyani? Monga mukuonera, masamu, mungathe, ndipo zikuoneka pa koyamba losavuta, koma si zoonekeratu chinyengo mosavuta. N'zotheka kuphunzira ana, ndi pamodzi ndi Aunt Kadzidzi ku zojambula "masamu-mwana", ndipo mukhoza kuyenda pansi pa madzi kwakuya kafukufuku wa sayansi pafupifupi nthanthi dongosolo. Mu mbiriyakale zachoka kuwerengera zinthu kulambira kukongola kwa manambala. Chinthu chimodzi chili: ndi kukhazikitsidwa kwa postulates zofunika masamu, sayansi zonse tingadalire mapewa amphamvu.