Yintoni arithmetic? theorem esisiseko arithmetic. izibalo binary

Yintoni arithmetic? Xa abantu baqala ukusebenzisa amanani baze basebenze kunye nabo? na iingcambu zawo concepts zemihla ngemihla ezifana amanani, Apho amaqhezu, ukuthabatha, ukudibanisa phinda, loo mntu wenze yinxalenye yobomi bakhe sengqondo? neengqondo Greek imangaliseke zenzululwazi ezifana nemathematika, izibalo kunye nejiyometri, njengokuba symphony omhle ekuqiqeni kwabantu.

Mhlawumbi izibalo ayikho nzulu nezinye iinzululwazi, kodwa oko kuza kwenzeka kubo, abantu balibale iitheyibhile phindo eziziziqalelo? Eqhelekileyo kuthi ukucinga, esebenzisa amanani, amaqhezu, kunye nezinye izixhobo ukunika abantu ixesha nzima, kwaye ixesha elide kwaba ayifumaneki izinyanya zethu. Enyanisweni, phambi kokuba uphuhliso arithmetic ayikho indawo yolwazi yomuntu zenzululwazi ngokwenene.

Yokalo emileyo - IMathematika alfabhethi

Yokalo emileyo - inzululwazi amanani, ngazo naziphi na umntu uqalisa ekumazini nehlabathi umdla lemathematika. Xa amazwi M. V. Lomonosov, yokalo emileyo - oku isango yokufunda, evula indlela yokuba Miropoznanie. Kodwa ke loo nto, kukwazi zehlabathi lwahlulwe ulwazi oonobumba neenombolo, izibalo kunye nentetho? Mhlawumbi Ngemihla yakudala, kodwa hayi kwihlabathi mihla, apho uphuhliso olukhawulezayo inzululwazi nobugcisa yenza imithetho yaso.

Igama elithi "izibalo" (GK. "Arifmos") imvelaphi lesiGrike, lithetha "inani". Ixilonga inani kwaye yonke into kunxulumana nabo. Nguwo lowo ihlabathi lwamanani: imisebenzi ezahlukeneyo kumanani, imithetho ngamanani, imisebenzi ezihlobene ne-phindo, ukuthabatha, njalo-njalo ..

Sele iyinto eyaziwayo ukuba inyathelo lokuqala le yokalo emileyo Mathematics kunye nesiseko esiqinileyo kwisakhiwo ngakumbi kumacandelo awo, ezifana aljebra, uhlalutyo zemathematika, kwimathematika ephakamileyo kunye t. D.

Eyona yinto arithmetic

Isiseko yokalo emileyo - yinto elipheleleyo, iimpawu kunye nemithetho leyo ingqalelo yokalo emileyo eliphezulu okanye theory inombolo. Enyanisweni, indlela nenqubo eyiyo kuthathelwa ingqalelo iyunithi omncinane, njenge inani yendalo exhomekeke emandleni kwesakhiwo - imathematika.

Ngoko ke, lo mbuzo oko kukuthi arithmetic, impendulo ilula: leyo inzululwazi amanani. Ewe, malunga njengesiqhelo ezisixhenxe, ezilithoba, kwaye konke oku neyantlukwano. Kanye nje kakuhle, kwaye iivesi ezininzi laliphakathi akakwazi ukubhala ngaphandle ialfabhethi esisiseko, ngaphandle arithmetic ayikwazi ukuba kusonjululwe nokuba imisebenzi esisiseko. Yiloo nto yonke iinzululwazi baye phambili kuphela emva kokuba uphuhliso izibalo kunye nezibalo, ukuba ngokuyintloko iseti ngeenkolelo.

Yokalo emileyo - isayensi-ghost

Yintoni arithmetic - nenzululwazi yendalo okanye phantom? Enyanisweni, njengoko zobulumko zamandulo zamaGrike waqiqa nabo, akukho manani, akukho manani enyanisweni ayikho. Yinto nje phantom, eyenziwe ingcamango yabantu xa ujonga okusingqongileyo kunye neenkqubo zayo. Enyanisweni, yintoni na inani? ukubona ndawo ke musa into efana okunokuthi kubizwa inani, kunoko, inani - le yindlela ukuvavanya ihlabathi ngayo ingqondo yomntu. Mhlawumbi le sifundo siye ngaphakathi ngokwabo? Zobulumko ukuphika kangangeenkulungwane ezininzi mkrozweni mnye, ngoko ukunika impendulo musa ukwenza thina alupheli. Noba indlela, lo yokalo emileyo ngoko ngokuqinileyo ukuthatha indawo abo ehlabathini mihla akukho namnye unako kuqwalaselwa entlalweni zilungiswe ngaphandle ulwazi iziseko zayo.

Njengoko bekukho nenombolo evumayo

Kakade ke, eyona njongo leyo isebenza arithmetic, - inani zendalo ezifana 1, 2, 3, 4, ..., 152 ... njl Lokubala eliguqukayo amanani zendalo kusisiphumo indleko bezinto eziqhelekileyo, ezifana iinkomo edlelweni. Sekunjalo, nenkcazo "eninzi" okanye "kancinane" xa into uyekile ukubamba abantu, kwaye kwafuneka ukuba zizakhele nobunzima kakhulu ukubala buciko.

Kodwa ke impumelelo yokwenene xa ingqondo yomntu lifikelele kwinqanaba ukuba ube mnye kunye nenani elilinganayo "ezimbini" libhekisela 2 kg, no-2 isitena, amalungu ama-2. Inyaniso ukuba kuyimfuneko ezingabonakaliyo kwezo fom, iimpawu kunye nentsingiselo izinto, ngoko sinako ukuvelisa isenzo ezinye ezi izinto ngokohlobo integers ezintle. Ngoko ke wazalwa le lokubala eliguqukayo amanani, nto leyo ziphuhliswa ngaphezulu kwaye banzi xa ahleli indawo eluntwini.

Ngolo olunzulu ingqiqo yenani, njengoko zero kunye amanani akhabayo, amaqhezu, amanani zithetha yamanani kwezinye iindlela, abe ubutyebi bembali kunye umdla lophuhliso.

Izibalo and practical amaYiputa

Ezimbini iqabane yamandulo yabantu isifundo lwehlabathi kunye nokusombulula iingxaki zemihla ngemihla - eli izibalo kunye nejiyometri.

Kukholelwa ukuba kwimbali arithmetic imvelaphi yayo kwi-East yaMandulo: India, Egypt, iBhabhiloni kunye neTshayina. Ngoko ke, Rhind ngqangi imvelaphi zaseYiputa (ngoko igama kuba igama elifanayo besive kumnini), imvelaphi yayo elandwa kwinkulungwane XX. BC, ukongeza kweminye data zexabiso kuquka ukwandiswa xiphemu xa imali amaqhezu anedinomineyitha ezahlukeneyo kwinani esilingana omnye.

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

Kodwa yintoni na intsingiselo yokubola ezintsonkothileyo? Isibakala sokuba le ndlela waseYiputa akayi kuyinyamezela atsalwe ecinga amanani, phezu koko, Ukubala kwenziwa kuphela iinjongo ezithile. Oko kukuthi, amaYiputa aya eluntwini ezifana ishishini njengoko izibalo, kuphela ukwenzela ukwakha engcwabeni, umzekelo. Kwafuneka ukuba ukubala ubude ngesakhiwo fin, kwaye wenza ukuba umntu ahlale papyrus. Njengoko sibonile, inkqubela engumYiputa izibalo kwathiwa, kunokuba omkhulu, ukwakha, kunokuba uthando yezenzululwazi.

Ngenxa yesi sizathu, izibalo afumaneka ngqangi, asikwazi ngokuba izibhengezo ngesihloko amaqhekeza. Kungenzeka, le ukulungiselela ngokoqobo, nto leyo yanceda ukuba ukusombulula ngakumbi iingxaki ngamaqhezu. AmaYiputa amandulo babengazi itafile phindo, luvelise izibalo ngokungenamkhethe ende, zasasazeka kulo yayo ezininzi. Mhlawumbi oku yenye yezo yayo. Kulula ukuba ukuqaphela ukuba izibalo kunye kwezi zikhewu ixesha elide kakhulu yaye hayi ethembisa kakhulu. Mhlawumbi ngenxa yesi sizathu asikaziboni igalelo elikhulu kuphuhliso lwemathematika ngamaYiputa amandulo.

Yamandulo Greece izibalo yentanda

Uninzi kukwazi East yaMandulo nenziwe yi amaGrike amandulo, owaziwa kubalandeli bokubonisa abstract, abstract kunye bulumko ngempumelelo. Sizisebenzise abanomdla nto ngaphantsi kodwa nkolelo zibhetele iinkcuba kunzima ukufumana. Kwaba kuhle ukuba isayensi kuba izibalo akwenzeki ukuya nzulu, hayi lelokuchitha ngayo nenyani. Kakade ke, kusenokwenzeka ukuba zande iinkomo-10 ne-100 iilitha zobisi, kodwa kuba nako ukuya kude.

KwanakumaGrike ukucabanga washiya ngokunzulu uphawu kakhulu kwimbali, nemisebenzi yabo, afikile kuthi:

• Euclid and "Elements".
• KaPythagoras.
• Archimedes.
• Eratosthenes.
• Zenon.
• Anaxagoras.

Kwaye, kakade, ujika zonke-bulumko yamaGrike, ingakumbi abalandeli amatyala kaPythagoras baba akawuvali gqitha amanani, leyo ingqalelo kubo ngokuvumelana eliyimfihlakalo ehlabathini. Amanani ziye ngoko bafunda zaphandwa, ukuba abathile kubo kunye ezitshatileyo zabo zabulawa iipropati ezizodwa. Umzekelo:

• amanani Perfect - ezo sum of bonke angawona zayo ngaphandle inani ngokwalo (6 = 1 + 2 + 3).
• amanani Friendly - la manani, enye yazo na udibaniso zonke angawona ze isininzi yesibini kunye nosekela (kaPythagoras bayazi enye kuphela isibini onjalo: 220 284).

AmaGrike, ababekholelwa ukuba inzululwazi kufuneka ukuthandwa, ukuba abe naye ngenxa yenzuzo, ithathe amanyathelo amakhulu, ukuhlola, edlala kunye nokongeza amanani. Kufuneka kuqatshelwe ukuba ayizizo zonke yophando lwabo ziye zasetyenziswa kakhulu, abanye babo baba kuphela "ngenxa yobuhle."

iinkcuba eMpumalanga zamaXesha Aphakathi

Ngokufanayo, kumaXesha Aphakathi izibalo ukuba unetyala uphuhliso kwayo empumalanga awayephila. I AmaNdiya yasinika amanani ukuba basebenzise ngenkuthalo into efana "kiswano", yaye indawo ukwahluka kwindlela ukubala, imbono yesiqhelo mihla. Ukusuka Al-isidudu, nto leyo kwinkulungwane ye-15 wasebenza Samarkand, siye ilifa ngenani leshumi, ngaphandle apho kunzima ukucinga izibalo mihla.

Ngeendlela ezininzi, eYurophu wayeqhelene eziyimpumelelo East wenziwa Bulelani kunokwenzeka ukuba umsebenzi sisazinzulu Italian uLeonardo Fibonacci, owabhala incwadi "Book Abaci", ekuqhelaniseni kunye ezintsha yaseMpuma. Kuye laba yintloko yembombo wophuhliso algebra nasekubaleni, uphando kunye nemisebenzi lwenzululwazi eYurophu.

izibalo Russian

Ekugqibeleni, arithmetic, uye wafumana indawo yayo kwaye isekelwe eYurophu, waqalisa ukusasaza kumhlaba Russian. Russian arithmetic yokuqala yapapashwa ngo-1703 - kwaba incwadi malunga arithmetic Leontiya Magnitskogo. Ukuze ixesha elide kwaba ngofundiso kuphela kwimathematika. Uqulathe eyona zihlandlo kokuqala algebra kunye nejiyometri. Amanani, leyo ezazisetyenziswa kwimizekelo textbook yokuqala eRashiya ngayo arithmetic, Arabhu. Nangona Amanani Arabic baye badibana ngaphambili, xa nemikrolo imvelaphi yayo elandwa kwinkulungwane ye-17.

Le ncwadi ngokwayo Zazihonjiswe ngemifanekiso kaArchimedes kunye Pythagoras, kwaye kwiphepha lokuqala - umfanekiso izibalo, njengoko umfazi. Yena uhleli etroneni, ngaphantsi kubhaliwe kwathiwa igama lesiHebhere igama likaThixo, yaye phezu amanyathelo ezikhokelela esibingelelweni, sibhalwe elithi "division", "ukwanda", "ukongeza", njalo njalo. D. One unokuthelekelela kuphela yintoni ixabiso angcatshwa ukuwafunda, ezo ngoku eziqhelekileyo.

Le ncwadi inamaphepha 600 ichaza njengesiseko zokudibanisa efana kunye neetheyibhile phindo, kunye izicelo Sciences olwandle.

Akumangalisi ke, umbhali ukhethe umfanekiso yawona lesiGrike kwincwadi yakhe, kuba yena ngokwakhe uyotywe yi ubuhle arithmetic, esithi: "Izibalo kuye chislitelnitsa apho uyinzwakazi, nezavistnoe ...". Le ndlela arithmetic lusekelwe kakuhle, ngenxa yokuba ukwamkelwa yayo ngokubanzi ingaqatshelwa ekuqaleni uphuhliso olukhawulezayo ingcamango yezenzululwazi eRashiya kunye nemfundo ngokubanzi.

etsala nzima

inani Prime - kuyinto inani yendalo, nto leyo 2 kuphela angawona HIV: 1 ngokwayo. Zonke ezinye amanani, ngaphandle 1 kuthiwa zizonke. Imizekelo yamanani yinkulumbuso: 2, 3, 5, 7, 11, yaye zonke ezinye abangengabo angawona ngaphandle kwe-1 kunye nenombolo ngokwayo.

Njengoko inani 1, kunjalo ngokomrhumo - kukho isivumelwano sokuba kufuneka ziqwalaselwe akukho ezilula okanye ezimbaxa. Simple xa uqala, inani elula uyayigubungela iimfihlelo ezininzi engekasonjululwa phakathi kwabo.

theorem Euclid ithi ukuba inani olungenasiphelo of etsala, kunye Eratosthenes weza izibalo "ngesihlungulo" ezizodwa, apho akubikho amanani enzima, eshiya elula kuphela.

kubume bayo ukuba igxininiswe inani lokuqala ukubuyisa, kwaye ekuxabeleni olulandelayo ngaphandle ezo iziphindwa nto. Siphindaphinda le nkqubo izihlandlo ezininzi - kwaye ufumane itheyibhile yamanani yinkulumbuso.

theorem elikhulu arithmetic

Phakathi baphawule amanani prime kufuneka ukukhankanya ngokukodwa le theorem basic arithmetic.

theorem Basic arithmetic ithi naliphi na inani elipheleleyo ngaphezu kwe-1, okanye elula okanye nga ibolile ibe imveliso yamanani yinkulumbuso ukuya ngokomyalelo imiba phindo, ekuphela kwendlela.

theorem esisiseko arithmetic yaba kakhulu ngumsina, nokuqonda alifani nje ezisisiseko.

Xa uqala kuqala, amanani inkulumbuso - ingqiqo samabanga aphantsi, kodwa ayikho. Physics benifudula ingqalelo atom samabanga aphantsi, de wafumana ngaphakathi nendalo. Etsala lanikezela ibali yezibalo omhle Don Zagier "Izigidi eziyi amahlanu zokuqala amanani yinkulumbuso."

Ukususela "apile amathathu" nemithetho aqale

Oko ngokwenene kungenziwa ngokuba isiseko balomeleza zonke inzululwazi - imithetho arithmetic. Njengokuba umntwana bonke ubuso arithmetic, ukufunda inani imilenze neengalo kwi onodoli, inani iityhubhu, ama-apile kunye njalo njalo. D. Ngoko sifunda arithmetic, ngoko inkqubela zibe imithetho entsonkothileyo kakhulu.

Ubomi bethu bonke esazisa ukuba imigaqo arithmetic, ezibe kuba indoda leyo eqhelekileyo ukuba iluncedo kakhulu kuzo zonke inzululwazi unika. Uphononongo yamanani - kuba "Izibalo-sana", nto leyo esazisa umntu ehlabathini kwamanani njengamanani kwabantwana abancinane.

Izibalo Higher - inzululwazi aqale ukuba ufunda imithetho arithmetic. Abaninzi kubo siyazi, nangona mhlawumbi asazi amagama abo ngqo.

Umthetho kokudibanisa nokuphinda

Naziphi na integers ezimbini kunye b lungabonakaliswa njengoko udibaniso b, nto leyo nayo inani yendalo +. Ngokuphathelele koku, le mithetho ilandelayo:

• Aluguquki, ukuba uthi lobeko lwe amagama ubeka imali ayitshintshi, okanye a + b = b + a.
• Ezokunxulumana uthe mali komntu akuxhomekekanga ekuzithandeleni indlela ngokwamaqela imigaqo kwiindawo, okanye + (b + c) = (a + b) + c.

Imithetho arithmetic, ezifana Ukongezelela, - omnye esisiseko, kodwa zisetyenziswa zonke zenzululwazi, bangabi sakhunjulwa ubomi bemihla ngemihla.

Nayiphi na i integers ezimbini b ingavezwa imveliso okanye b * * b, nto leyo nayo inani yendalo. Ukuze ube nokufaka isicelo imveliso imithetho efanayo properties kwaye associative ukuba kongezwe:

• a * b = b * a;
• i * (b * c) = (a * b) * c.

Okubangel 'umdla kukuba kukho umthetho, okuhlanganisa ukudibanisa phindo, eyaziwa njenge ulwabiwo okanye umthetho ngayinye:

a (b + c) = ab ac +

Lo mthetho lisifundisa ukuba sisebenze izibiyeli, evula kuzo, ngaloo ndlela sinako sele ukusebenza kunye neefomula ezimbaxa ngakumbi. Yiyo leyo imithetho iya kuthi ikhokelele ngokusebenzisa ihlabathi akaginywa entsokothileyo algebra.

UMthetho ukuze arithmetic

malunga nemithetho ekuqiqeni yabantu isebenzisa ngosuku ngalunye, ebheka iwashi nokubala iindleko. Kwaye, kodwa, kwaye oko kufuneka kwenziwe kwi ulwimi oluthile.

Ukuba une integers ezimbini HIV kwaye b, ngoko khetho lulandelayo:

• a 'ulingana b, okanye = b;
• a ngaphantsi kwe b, okanye
• a mkhulu b, okanye> b.

Kwezi ndlela zintathu nje abe mnye kuphela. UMthetho esiSiseko, olawula loo nkqubo, wathi: ukuba

Kukho imithetho yekeyeke iintshukumo umyalelo kokudibanisa uphinda-phindo: ukuba

Imithetho arithmetic wafundisa ukusebenza ngamanani, iimpawu kunye kwizibiyeli, ajike yonke into ibe symphony evumelanayo amanani thina.

inkqubo inani ngokokuma kunye nonpositional

Sinokuthi ukuba amanani - oku ulwimi lwemathematika, ukususela lula leyo kuxhomekeke kwizinto ezininzi. Kukho iindlela ezininzi yokubala, leyo, efana nobumba iilwimi ezahlukeneyo zahlukile.

Qwalasela inkqubo inani ukusuka kwindawo kwizikhundla igalelo kwixabiso ubungakanani lika kwesi sikhundla. Umzekelo, inkqubo eRoma nonpositional apho inani ngalinye ufakwe yi iseti ethile ababasebenzisi: I / V / X / L / C / D / M. Ba, ngokulandelelana, amanani 1/5/10/50/100/500 / 1000. Kule nkqubo, i-nani ayitshintshi umiselo lwayo oluninzi, kuxhomekeke kwi yintoni efaneleko: .. kuqala, okwesibini, njl Ukuze ufumane amanye amanani, kuyimfuneko ukuba abeke phantsi isiseko na. Umzekelo:

• DCC = 700.
• CCM = 800.

Okunye wazana kuthi inkqubo yesi usebenzisa Amanani Arabic yi ngokokuma. Xa nkqubo inani obhobhozayo uchaza linani lamasuntswana, umzekelo, amanani ama ezintathu-: 333, 567, njl Ubunzima nayiphi na obhobhozayo kuxhomekeke kwindawo apho nani enye okanye enye, umz inani 8 kwindawo yesibini unalo ixabiso 80. Kuyinto eqhelekileyo ukuba inkqubo wokugqibela, kukho ezinye inkqubo ngokokuma ezifana binary.

izibalo binary

Thina inkqubo eqhelekileyo lokugqibela, eziquka single-bit kunye multi-bit amanani. Inani kwi ngasekhohlo inani idijithi izihlandlo ezilishumi olukhulu kubalulekile kulowo ngasekunene. Ngoko ke, sasidla eyesi-2, 17, 467, njalo njalo. D. It is a logic indlela icandelo ezahlukeneyo, leyo ibizwa ngokuba "arithmetic yokubini." Akumangalisi oku, kuba arithmetic lokubini akadalwanga ukucinga oluntu, nangenxa computer. Ukuba lokubala eliguqukayo amanani isuka ekubaleni, nto leyo kutsalwa ukusuka ku "ze" izibalo sepropati, ngoko oku akusebenzi ne khompyutha yakho. Ukuze ukwazi ukwabelana ngolwazi lwabo computer, indoda kwafuneka ukuba zizakhele ukubala model.

arithmetic Binary isebenza oonobumba yokubini, nezibabandakanya kuphela 0 no 1 Ke nokusetyenziswa kwale alfabhethi kuthiwa inkqubo yokubini.

Ngokungafaniyo ngeshumi yokubini izibalo ukuthi ukubaluleka indawo yasekhohlo akukho kuphinda-10, kunye namaxesha-2. amanani yokubini yefom 111, 1001 njalo njalo. D. Sifanele siyiqonde njani la manani? Ngenxa yoko, sicinga inombolo 1100

1. Le yokuqala ngasekhohlo - 1 * 8 = 8, ekhumbula ukuba umvo yesine, nto leyo ithetha ukuba kufuneka liphindaphindwe-2, siye sifumane indawo 8.
2. digit Second 1 * 4 = 4 (indawo 4).
3. Umvo yesithathu 0 * 2 = 0 (isikhundla 2).
4. Umvo yesine 0 * 1 = 0 (isikhundla 1).
5. Ngoko ke inani lethu 1100 = 8 + 4 + 0 + 0 = 12.

Oko kukuthi, akulula xa uqeqeshelwa ukuba udidi olutsha ukuya ekhohlo intsingiselo yayo kwinkqubo yokubini yande ngo-2 kunye lokugqibela - ukuya kwi-10 Loo nkqubo ethile mnye: kuba amasuntswana ukukhula enkulu kakhulu ukuba kufuneka ukuba ubhale amanani. Imizekelo amanani ngeshumi dvochinyh njengoko kunokubonwa kolu luhlu lulandelayo.

amanani Decimal zimelwe ngohlobo yokubini ngezantsi.

Iyasetyenziswa elinekona ezintandathu, kunye nenkqubo hexadecimal inani.

Le izibalo eliyimfihlakalo

Yintoni arithmetic, "uzawudibanisa isibini nesinye isibini" okanye iimfihlelo unexplored lwamanani? Njengoko ubona, arithmetic, uyakwazi, kwaye kubonakala xa uqala elula, kodwa akukho lula ezicacileyo ekhohlisayo. Kuyenzeka ukuba ukufunda abantwana, yaye kunye uMakazi Owl ukusuka khathuni "Izibalo-sana", kwaye singene uphando nzulu lwenzululwazi ukuze phantse bulumko. Embalini uphumile ukusuka nokubala izinto ukunqula ubuhle lwamanani. Inye into esiqiniseke ngayo yile: kunye nokusekwa postulates ezisisiseko arithmetic, bonke inzululwazi thembela egxalabeni lakhe olomeleleyo.