Real nhamba vavo zvivako

Achibvuma pfungwa yaPythagoras aidzidzisa kuti nhamba iri pakuvambwa kwenyika iri ndima chete makuru okunze. Plato aidavira kuti uwandu rinobatanidza Chiitiko uye noumenon, kubatsira kuziva, kuti pachikero uye kuti vabudise mhedziso. Svomhu rinobva pashoko "arifmos" - nhamba, pokutangira mu masvomhu. Zvinokwanisika kutsanangura chero chinhu - kubva chepuraimari kusvikira yemaapuro chaihwoihwo dzezvimwe.

Inoda somunhu kukura chinhu

In kwokutanga nzendo kukura vanhu zvido vanhu aparidze kudikanwa mamakisi - .. One bhegi zviyo, zviyo bhegi maviri, etc. Kuti aite izvi, yakanga dzinongoitika nhamba, iyo yakatarwa chiri haaperi kutevedzana rekuva integers N.

Gare gare, kukura wemasvomhu somunhu sainzi, zvakanga zvakakodzera mune chaiyo mumunda integers Z - zvinosanganisira kunaka tsika uye razero. Kuonekwa kwake mumhuri vanochiva yakanga kushatiriswa chokwadi kuti kwokutanga chizvidavirire aifanira zvimwe kugadzirisa zvikwereti uye kurasikirwa. On masayendisiti sandara, kunaka nhamba vakaita zvinoita kugadzirisa nyore Linear equations. Pakati pezvimwe zvinhu, zviri ikozvino zvinogoneka mufananidzo chiduku batanidza ezvinhu, kureva. A. Paiva achitandadza bhuku.

Danho rakatevera raiva kudiwa kupinda fractional nhamba, sezvo sainzi hakuiti mirai, zvakawanda zvakawanwa matsva vakakumbira yokungotaura hwaro itsva kusunda kukura. Saka paiva munda kuna kodzera nhamba Q.

Pakupedzisira, asisiri ienzane. rationality, nekuti zvose Zvakawanikwa zvitsva zvinoda kururamiswa. Paiva munda chaiyo nhamba R, mabasa Euclid kwakaita incommensurability vamwe kwazvo nokuda irrationality kwavo. Ndiko, vekare yemasvomhu papi kwete chete nhamba sechinhu dzose, asi sezvo husingaoneki ukoshi iyo kunoratidzwa nhamba incommensurable magnitudes. Nokuti pane chaidzo nhamba, "takaona chiedza" tsika dzakadai se "Pi" uye "e", pasina iro masvomhu ano vasingagoni zvaitika.

Kwokupedzisira utsanzi aiva kunzwisisa nhamba C. It akapindura mibvunzo yakatevedzana uye akaramba kare akapinda postulates. Nokuda kukurumidza kukura nemasvomhu Zvakazoitika oneka - pamwe chaiko nhamba, kusarudza matambudziko akawanda zvakanga zvisingaiti. Somuenzaniso, nemhaka zvakaoma nhamba tambo dzidziso uye nyonganyonga kuwedzera equations pamusoro hydrodynamics vakamira panze.

Isa pfungwa. Cantor

Pfungwa kusaverengeka hwagara kwakakonzera gakava, sezvo zvakanga zvakaoma kuratidza kana kusaratidza. Munguva masvomhu, iro vakavhiyiwa zvachose riitike postulates, kunoratidzwa pacharo zvikuru pachena, zvikuru kuti edzidziso chinoumba ichiri zvakayerwa zvesayenzi.

Zvisinei, kuburikidza nebasa yemasvomhu Georg Cantor nguva yose yakawira panzvimbo. Akaratidza kuti magumo rinovira pane magumo yakatarwa, uye kuti munda R mukuru munda N, regai vaviri uye haatongogumi. In pakati pezana XIX remakore, pfungwa dzake kunzi pachena maturo uye mhosva dzenhoroondo chisingaperi canons, asi nguva pandichaisa zvose panzvimbo yayo.

Basic ehupfumi dzesango R

Actual nhamba kwete chete kuva chete ehupfumi sezvo podmozhestva kuti zvinosanganisira, asi vari vaisanganisa nezvimwe masshabnosti pamusaka zvinhu zvayo;

• Zero R. ariko uye ndezvaiye kusango munenge + = munenge 0 chipi C R.
• Zero ariko uye ndezvaiye kusango R. C X 0 = 0 chipi C R.
• The reshiyo d: D kana D ≠ 0 ariko uye chinoshanda upi c, D pamusoro R.
• Field R akarayira, i.e. kana munenge ≤ D, D ≤ c, ipapo munenge = D upi c, D pamusoro R.
• Uyezve mumunda R riri commutative, i.e. C + D = D + c, nokuti chipi C, D pamusoro R.
• Chokuwanziridza mumunda R riri commutative, i.e. X C X D = D C nokuda munenge zvose, D ose R.
• Uyezve mumunda R ndiyo associative i.e. (c + d) + add = C + (D + f) upi c, D, f pamusoro R.
• Chokuwanziridza mumunda R ndiyo associative i.e. (c X d) X F = C X (D X f) upi c, D, f pamusoro R.
• Nokuti mumwe nhamba yomunda R rwakatarisana ipapo, zvakadai c + (-c) = 0, apo c, -c kubva R.
• Nokuti nhamba nechimwe munda R chiripo Ivan Kuchin yayo, zvakadai c X C ^-1 = 1 apo c, munenge ^-1 pakati R.
• Unit chiripo uye ndezvaiye R, kuti munenge x 1 = c, nokuti chipi C R.
• Rine kuparadzira simba mutemo, saka c X (D + f) = C X D + C X f, upi C, D, F pamusoro R.
• The R Munda razero haana kuenzana kubatana.
• Field R ndiye transitive: kana munenge ≤ D, D ≤ f, ipapo munenge ≤ add upi c, D, f pamusoro R.
• Munguva R uye kuwedzera kuti vari interconnected: kana munenge ≤ D, ipapo munenge + add ≤ D + add kuti munenge zvose, D, f pamusoro R.
• In murayiro R uye chokuwanziridza kwakabatana: kana 0 ≤ c, 0 ≤ D, ipapo 0 ≤ C X D upi c, D pamusoro R.
• Sezvo akaipa uye zvakanaka chaizvo nhamba dziri chinoenderera, i.e., upi C, D pamusoro R f, kune kubva R, c ≤ F ≤ d.

Module munda R

The chaiyo nhamba anosanganisira chinhu chakadai somunhu module. Akasarudzwa sezvamakaudzwa | F | chero add mu R. | F | = F, kana 0 ≤ add uye | f | = -f, kana 0> f. Kana tikafunga module somunhu geometric ukoshi, ndiko kure - hazvina basa, "akapfuura" iwe se razero ari akaipa kune zvakanaka kana mberi.

Complex uye chaiyo nhamba. Ndezvipi zvakafanana uye kusiyana?

Kazhinji, kunzwisisa uye chaizvo nhamba - ndivo mumwe chete, asi kuti wekutanga akabatana zvekufungidzira Unit i, yaiungana izvo akaenzana -1. Zvinhu minda R uye C kunogona anomiririrwa mutoo:

• C = D + add X i, sezvamakaropafadzwa D, f ndezvaJehovha kusango R, uye ini - zvekufungidzira chikwata.

Kuti C R add kana ichi kureva kungoti vaifungidzira kuti zero,, panongova chikamu chaiye nhamba. Nekuti munda zvakaoma nhamba ane chete chikabviswa wakaiswa sezvo kusango chaiye, f X 0 Ini = kana F = 0.

With maererano dzinoshanda kusiyana, somuenzaniso mumunda R quadratic mukana wokuti vanhu vanga haagoni kugadziriswa kana discriminant akaipa, asi C bhokisi asingadi kusangomanikidzira kukamurwa iyi kudzidzisa zvekufungidzira chikwata i.

zvabuda

"Zvidhinha" pamusoro axioms uye postulates pamusoro izvo chawo masvomhu handishanduki. Vamwe vavo nokuda zvibereko mashoko uye pakatanga itsva dzidziso aiisa zvinotevera "zvidhinha", izvo mune ramangwana ave hwaro danho rinotevera. Somuenzaniso, njodzi dzinongoitika nhamba, pasinei vari subset chaidzo munda R, usingatangi rirasikirwe kwayo. Zviri kwavari hwaro zvose zvokutanga svomhu, unotanga nokuziva munhu worugare.

Kubva inoshanda nemaonero, chaiye nhamba Dzakafanana achirurama mutsetse. Zvinokwanisika kusarudza zvokuita, kuziva mavambo nenamo. Direct ine isingaperi nhamba pfungwa, chimwe nechimwe unoenderana chete nhamba chaiyo, zvisinei kana kodzera. Kubva tsananguro zviri pachena kuti tiri kutaura pfungwa, iyo yakavakirwa wemasvomhu zvavo, uye pamusoro kwemasvomhu kuongorora zvikuru.