Dzidziso yenhamba yakakwana

Dzidziso yeInteger: Kuvaka Kunzwisisa Masvomhu Ekutanga

Dzidziso yenhamba izere ndeimwe yenzvimbo dzakakosha uye dzinokosha dzemasvomhu, dzinotangira kare. Nhamba izere inhamba dzine nhamba dzakanaka, nhamba dzisina kunaka, uye zero. Semuenzaniso: …, -3, -2, -1, 0, 1, 2, 3, …. Pasinei nekureruka kwazvo, nhamba izere dzine mashandisirwo akaomarara uye anonakidza muzvikamu zvakasiyana-siyana zvemasvomhu, kusanganisira algebra, geometry, uye dzidziso yenhamba. Chinyorwa chino chichakurukura pfungwa dzakasiyana-siyana dzekutanga uye dzepamusoro mudzidziso yenhamba izere.

Nhoroondo Pfupi yeInteger Theory

Kunzwisisa nhamba dzenhamba kwagara kuripo kubvira kare. Masvomhu akarukwa mutsika dzakawanda dzekare, kusanganisira dzeBhabhironi, dzeIjipita, nedzechiGiriki. Semuenzaniso, masvomhu ekare echiGiriki, kuburikidza nemabasa aPythagoras naEuclid, akapesvedzera zvikuru kukura kwenhamba dzenhamba. Euclid, mu "Elements" yake, akaunza pfungwa huru dzenhamba nekukamurwa, izvo zvichiri kushanda nanhasi.

Kunzwisisa nhamba dzenhamba dzose kwakachinja-chinja munhoroondo yese, kusvika munguva yemazuva ano. Muzana remakore rechi18 nerechi19, nyanzvi dzemasvomhu dzakadai saCarl Friedrich Gauss dzakatanga kuongorora zvakadzama hunhu hwenhamba dzese kuburikidza ne "Disquisitiones Arithmeticae" yake. Rugwaro rwake rwakava rumwe rwehwaro hwedzidziso yenhamba yakachena.

Hunhu Hwekutanga hweIntegers

Nhamba dzese dzine hunhu hwekutanga uye mashandiro ayo ari hwaro hwakakosha mumasvomhu:

1. Kuwedzerwa uye Kuwanda kweZviratidzo: Kuwedzerwa kwenhamba dzese nhamba i0, ukuwo kuwanda kwenhamba i1. Izvi zvinoreva:
– a + 0 = a yenhamba dzese a.
– a × 1 = a yenhamba dzese a.

2. Kuwedzeredza Kupesana: Nhamba yese ine chinowedzera chakapesana (kana kuti chakapesana). Kune nhamba a, chinopesana ndi -a. Saka, a + (-a) = 0.

3. Kuchinja-chinja uye Kubatana mukuwedzera nekuwedzera:
– Kuwedzerwa nekuwanda kwenhamba dzese dzinoenderana, kureva a + b = b + a uye a × b = b × a.
– Kusanganisa nekuwanda kwenhamba dzese zvinorevawo kubatana, zvinoti (a + b) + c = a + (b + c) uye (a × b) × c = a × (b × c).

4. Kugoverwa kweKuwanza pamusoro peKuwedzera: Kuitwa kwekuwanza pamusoro pekuwedzera kunogovera, kureva a × (b + c) = a × b + a × c.

Dzidziso dzePamusoro muInteger Theory

Dzimwe dzidziso dzakakosha mudzidziso ye integer ndedzinotevera:

1. Dzidziso yeKupatsanurana:
– Papeya yega yega yenhamba a na b (ine b ≠ 0), pane nhamba q (quotient) na r (yasara) zvekuti a = bq + r, apo 0 ≤ r < |b|. 2. Theorem yaEuclid: - Nhamba yega yega yakakura kupfuura 1 inogona kupatsanurwa kuita chigadzirwa chakasiyana chenhamba dze prime (mune chero kurongeka). Maitiro aya ekupatsanurwa anonzi prime factorization.

Nhamba huru muInteger Theory Nhamba huru dzinoita basa rakakosha muInteger theory. Nhamba huru inhamba huru kupfuura 1 isina positive divisors kunze kwe1 uye pachayo. Mienzaniso yenhamba huru ndeiyi 2, 3, 5, 7, zvichingodaro. Chimwe chinhu chakakosha chenhamba huru ndechekuti hadzigone kugoverwa neimwe nhamba huru kunze kwe1 uye ivo pachavo pasina kusiya imwe yasara. Izvi zvinoita kuti nhamba huru dzive "zvivharo zvekuvaka" zvenhamba huru, zvichibvumira factorization yakasarudzika. Fundamental Theorem yeArithmetic Fundamental Theorem yeArithmetic inotaura kuti nhamba huru yega yega n (n > 1) inogona kupatsanurwa zvakasiyana kuita chigadzirwa chenhamba huru. Iyi theorem ndiyo hwaro hwezvidzidzo zvakawanda zvepamusoro mudzidziso yenhamba. Semuenzaniso, kune integer 30, dzidziso huru yearithmetic inoti 30 inogona kupatsanurwa se 2 × 3 × 5, uye iyi factorization yakasiyana zvisinei nekurongeka kwadzinoitwa.

Mashandisirwo edzidziso yeInteger

Dzidziso yenhamba yakakwana inowanikwa muzvikamu zvakawanda zvemasvomhu uye muhupenyu hwezuva nezuva. Heano mamwe mashandisirwo edzidziso yenhamba yakakwana:

1. Kunyora manhamba: Manhamba ePrime anoshandiswa zvakanyanya mumasystem akasiyana-siyana emazuva ano e cryptographic, akadai seRSA (Rivest-Shamir-Adleman). Kuchengetedzwa kwemasystem mazhinji e encryption kwakavakirwa pakuoma kwekubatanidza manhamba makuru muzvinhu zvavo zveprime.

2. Magadzirirwo eKombuta: Magadzirirwo emakombiyuta anowanzo shandiswa pakushandisa nhamba dzese, dzakadai sekutsvaga, kurongedza, uye kugadzirisa data.

3. Combinatorics neGraph Theory: Kushandiswa kwenhamba dzese kwakakosha muchikamu ichi, kunyanya pakuverenga maumbirwo akasiyana uye kuongorora hunhu hwegirafu.

4. Zvehupfumi neMari: Nhamba dzose dzinoshandiswa mukuverenga purofiti, kuongorora nhamba uye mamodheru emasvomhu anotsanangura kukura kwehupfumi.

Kushandiswa muMasvomhu Kudzidza

Kudzidza dzidziso yenhamba dzese kunosimbisa hwaro hwemasvomhu hunobatsira pakunzwisisa pfungwa dzakaoma. Semuenzaniso, kunzwisisa maturusi ekutanga akadai sekushanda kwenhamba dzese, maitiro ekubatanidza, ekubatanidza, uye ekuparadzira kwakakosha pakugadzirisa maequation e algebraic.

Mhedziso

Dzidziso yeInteger ndiyo hwaro hwemasvomhu akafara uye inoita basa rakakosha mukusimudzira pfungwa dzakawanda dzepamusoro dzemasvomhu. Kunyange zvazvo dzichiita sedziri nyore, dzidziso iyi ine kuoma kwayo kunoibvumira kushandiswa mumhando dzakasiyana-siyana dzemashandisirwo, kubva pakunyora cryptography kusvika kuma algorithms emakombiyuta. Kunzwisisa kwakasimba kwepfungwa dzakakosha uye dzidziso dziri mudzidziso iyi kwakakosha kwete chete kumasvomhu edzidziso asiwo pakushandiswa kwayo kwakawanda musainzi netekinoroji.