Chii yakanaka integer? History, kukura, unhu

Math aparadzana mukuru Uzivi pamusoro remakore rechitanhatu BC. e., uye kubva panguva iyoyo chakatanga guta rworukundo munyika. Danho rimwe nerimwe kukura vakauya chinhu chitsva - ane rinodudza nhoroondo kumhuka, tinoshandurirwa differential uye chinokosha calculus, alternated remakore, nzira akatonyanya kuvhiringidza, uye kuuya nguva apo "kutanga akaoma zvikuru pamasvomhu -. nayo anyangarika dzose nhamba" Asi chii akarara shure?

The pokutangira

The dzinongoitika nhamba dzaiva pamusoro ndima yokutanga kwemasvomhu mashandiro. Kamwe shure, vaviri shure, matatu musana ... Ivo akazviratidza nemhaka musayendisiti Indian uyo pakutanga akauya positional nhamba hurongwa. Shoko "positional" zvinoreva kuti nzvimbo mumwe digit munyika dzinoverengeka zvachose kunorondedzerwa uye anoenderana ayo woga. Somuenzaniso, nhamba 784 uye 487 - nhamba ndiwo chete, asi nhamba havasi kufanana wokutanga unosanganisira 7 zana, asi wechipiri - 4. chete Innovation maIndia akanhonga navaArabhu, wakabudisa uwandu kwemarudzi kuti tinoziva Now.

Munguva dzekare, nhamba zvakabatanidzwa usingaonekwi kukosha, mukuru yemasvomhu Achibvuma pfungwa yaPythagoras aidavira kuti nhamba iri pakutanga mwoyo rezvisikwa iri ndima chete zvinokosha - moto, mvura, pasi, mhepo. Kana tikafunga zvose chete kwemasvomhu divi, zvino ndiyo yakanaka integer? Munda dzinongoitika nhamba runotaurwa sezvo N uye ane haaperi nhevedzano zvinove zvakanaka integers uye 1, 2, 3, ... + ∞. Zero Kwabviswa. Ainyanya kushandiswa kuverenga zvinhu uye kutaridzwa hurongwa.

Ndechipi dzinongoitika nhamba iri masvomhu? axioms pamusoro Peano

Field N ari mujinga iro anoisa zvokutanga masvomhu. Nokufamba kwenguva, mashoko yoga munda integers, zvakaringana nhamba, dzakaoma nhamba.

Basa Italian yemasvomhu Dzhuzeppe Peano zvibvire mberi yokuronga patsva svomhu, vakaita make chokungoitawo uye vakagadzirira pasi mamwe mhedziso kuti vaende mhiri munda nharaunda N. Chii chiri wechisikigo nhamba, hazvina rawanikwa kare mumutauro nyore, zvinotevera ichakurukurwa maererano munhu akarongeka tsanangudzo Peano axioms.

• Unit anoonekwa somunhu dzinongoitika nhamba.
• Nhamba inotevera yepanyama nhamba, munhu wechisikigo.
• Pamberi chikwata chiri rwokuzvarwa nhamba.
• Kana nhamba b anofanira kuva zvose nhamba c, uye uwandu D, ipapo munenge = d.
• The axiom pamusoro induction, izvo zvinozoita anoratidza kuti munhu wechisikigo nhamba, kana mashoko kunotsamira pane parameter ichokwadi kuti nhamba 1, ipapo taifunga kuti anoshandira N nhamba yeminda inongoitika nhamba N. Ipapo kutaura ichokwadi kuti N = 1 kubva kumunda dzinongoitika nhamba N.

Basic akavhiyiwa kuti munda rwokuzvarwa nhamba

Sezvo munda N ndiye akatanga masvomhu, zviri kubatwa sezvo umambo tsanangudzo, uye nharaunda pazasi nhamba kwema tsika. Vari yakavharwa uye kwete. Musiyano chikuru ndechokuti oparesheni anosimbiswa kusiya namahwindo mugumisiro mukati yakatarwa N, pasinei nhamba zvinobatanidzwa. Zvaringana kuti vari dzinongoitika. Zvaizobuda yasara kusanganiswa kusakarongeka hakusi zvakarurama uye zvinoenderana kuti kune avo anobatanidzwa okuti, sezvo zvingava zvinopesana inokosha tsananguro. Nokudaro, akadzivira akavhiyiwa:

• Kuwedzera - X + Y = z, apo X, y, z kubva kumunda N;
• chokuwanziridza - X * Y = z, apo X, y, z kubva kumunda N;
• Exponentiation - X ^Y, apo X, y anobva N. Field

The akavhiyiwa asara, mugumisiro risingagoni zviripo kutsunga akapoteredza "ndiko kuti dzinongoitika nhamba" sezvinotevera:

• Kubvisa - X - Y = z. Field dzinongoitika nhamba wavatendera kana refu X and chete;
• Kupesana - X / and = z. Field dzinongoitika nhamba anosiya chete kana z yakakamurwa ja pasina vakasara, i.e. zvakaenzana.

Properties nenhamba, kuva munda N

All mberi kwemasvomhu pfungwa hwuchavapano pasi ehupfumi izvi, zvikuru sezvisingakoshi, asi hapana kudukupiswa.

• Commutative pfuma Kuwedzera - X + Y = y + X, umo nhamba X, y yaisanganisira mubhokisi N. Kana inonyatsozivikanwa "kubva kwemahofisi Verengai haina kushandurwa."
• Commutative pfuma chokuwanziridza - X * Y = y * X, umo nhamba X y anobva N. Field
• Associative pfuma Uyezve - (x + Y) + z = X + (Y + z), apo X, y, z anobva N. Field
• Associative pfuma chokuwanziridza - (x * Y) * z = X * (and * z), umo nhamba X, y, z anobva N. Field
• distributive pfuma - X (Y + z) = X * and + X * z, umo nhamba X, y, z anobva N. Field

Zviri Achibvuma pfungwa yaPythagoras

Mumwe wokutanga matanho mukuziva vadzidzi yose zvokutanga wemasvomhu hurongwa mushure kunzwisisa pachavo chii nhamba dzinonzi dzinongoitika, ndiro tafura Achibvuma pfungwa yaPythagoras. Rinogona kunzi kwete chete kubva nemaonero sayenzi, asiwo sechinhu chinokosha shongwe nesayenzi.

Izvi tafura chokuwanziridza kwave kuchiita nhamba kuchinja nokufamba kwenguva: zvakanzi kubviswa zero, uye nhamba kubva 1 kusvika 10 kumiririra pachavo, dzakakudzingai mirairo unokosha (vamazana, zviuru ...). Zviri tafura umo mazita okuremekedza mitsara uye mbiru - nhamba uye zviri masero mharadzano akaenzana chigadzirwa yavo.

In Tsika kurovedza okupedzisira emakore mashoma pakanga kudiwa kudzidza Pythagorean tafura "kuitira", kureva kutanga vakaenda nomusoro. Chokuwanziridza 1 akanga Haatauri sezvo mugumisiro akaenzana 1 kana mukuru chikonzero. Zvichakadaro, vari tafura inogona kuonekwa chinoabatsira muenzaniso: zvinobikwa nhamba kwokuwedzera nechimwe danho, izvo zvakaenzana rokuremekedza tambo. Saka, chinhu chechipiri inotiratidza kangani unofanira kutora rokutanga, kuti vawane achida chigadzirwa. ino akasiyana yamunenge uya kwaiitwa kuMiddle Ages: kunyange muchiziva kuti chinhu yakanaka integer, uye sei zviri sezvisingakoshi, vanhu vakakwanisa omesa pachako nezuva nokushandisa hurongwa kuti kwaibva madhigirii maviri.

A subset sezvo paucheche wemasvomhu

Panguva iyoyo, mumunda dzinongoitika nhamba N ichaongororwa somumwe subsets ari kunzwisisa nhamba chete, asi hazvina kuita navo zvishoma zvinokosha zvesayenzi. Natural nhamba - chokutanga kuti mwana anodzidza nokudzidza pachedu uye nyika yakatipoteredza. Kamwe munwe, vaviri chigunwe ... Nemhaka kwaari, murume inoumbwa musoro kufunga, uyewo kukwanisa kuona chikonzero uye mugumisiro goho, tichigadzira nzira huru matongo.