Tau tūturu me ratou āhuatanga

kī Pythagoras e te tokomaha, ko te turanga o te ao i runga i te par ki te āhuatanga nui. whakapono Plato e ki kia paunatia te maha o ngā hononga i te tītohunga me te noumenon, te āwhina ki te mohio, me ki whakatau. mai tātai i te kupu "arifmos" - te maha, te timatanga i roto i te pāngarau. Ko taea ki te whakaahua i tetahi ahanoa - i timatanga ki te aporo wāhi waitara.

Hiahia rite te take whanaketanga

I roto i te wāhanga tuatahi o te whanaketanga o te hapori te hiahia o te iwi ki te akiaki i te hiahia kia mau score - .. Kotahi putea o witi, e rua putea witi, etc. Ki te mahi i tenei, te reira i tau tūturu, te huinga o e ko te raupapa mure o pai tau tōpu N.

I muri mai, te whanaketanga o pāngarau rite te pūtaiao, i reira e tika ana i roto i te mara motuhake o tau tōpū Z - ngā reira uara tōraro, me te kore. Tona ahua i te taumata kāinga, i whakapataritari ai i te meka e i te kaute tuatahi ki te whakatika te hopoia nga nama me te parekura. I runga i te taumata pūtaiao, kua hanga tau kino reira taea ki te whakaoti ohie whārite rārangi. I roto i te tahi atu mea, he mea i teie nei e taea ki te whakapakoko i te pūnaha mama taunga, ie. A. He ko he wāhi o te tohutoro.

Ko te mahi i muri mai ko te hiahia ki te tomo tau hautanga, mai kore e pūtaiao tu tonu, ka ui atu me te ake ngā āhuatanga hou i te pūtake ariā mō te tupu pana hou. Heoi i reira ko tetahi mara o tau kouma Q.

I te pae hopea, e kore te whakatau i te mau titauraa o te rationality, no te mea e rapu kitenga hou katoa tika. He ko he mara o tūturu tau R, nga mahi o incommensurability o etahi rahinga no o ratou horihori a Euclid. Ko, te mathematician Kariki onamata tūnga maha e kore anake rite te tamau, engari kia rite ki te uara waitara āhuatanga nei i te te ōwehenga o magnitudes incommensurable. Nā ki te meka i reira he tau tūturu, "kite matou i te marama" uara pērā i "pi" me "e", kahore nei kihai i taea e pāngarau hou kua tangohia wahi.

Te auaha whakamutunga ko te tau matatini C. whakahoki te reira i te raupapa o ngā pātai, me te whakaatu marôraa tomo i mua. Nā ki te whanaketanga tere o te taurangi putanga i tonutia - ki tau tūturu, kihai i taea te whakatau o ngā raruraru maha. Hei tauira, e tu ana te whakawhetai ki te tau matatini i roto i string ariā me te hepohepo whakawhānui whārite o hydrodynamics.

Whakatakotoria he Theory. Cantor

Kua meinga tonu totohe te ariā o te mutu-, rite ko e taea ki te whakamatau whakakore ranei. I roto i te horopaki o te pāngarau, whakahaeretia nei te mau marôraa mana'o tino manatoko, whakakitea ana e ia ano tino mārama, i te ake i paunatia tonu te āhuatanga minita i roto i te pūtaiao.

Heoi, i roto i te mahi o te mathematician Georg Cantor hinga wa katoa ki te wahi. whakamatauria e ia e te huinga mure ore i reira ko te huinga mure ore, a e te mara R he nui ake te N mara, kia me tokorua o ratou i kahore mutunga. I roto i te waenganui o te rau tau XIX, huaina ana whakaaro nuitia tito me te hara ki kēnana pumau puāwaitanga, engari ka hoatu wā katoa i roto i tona wahi.

āhuatanga Basic o te parae R

tau tūturu kore i anake te taua āhuatanga rite te podmozhestva e ngā ratou, engari e kō e te tahi atu masshabnosti i te viivii ore o ona huānga:

• Zero R. vai me no ki te mara c + = c 0 mo tetahi c o R.
• Kore te vai a no ki te mara R. c x 0 = 0 mo tetahi c o R.
• Ko te c ōwehenga: d ka d ≠ 0 te vai me he tika hoki tetahi c, d o R.
• Field R whakahau, i.e. ki te c ≤ d, d ≤ c, ka c = d mo tetahi c, d o R.
• Addition i roto i te mara R he āpiti, i.e. c + d = d + c, mo tetahi c, d o R.
• Whakarea i roto i te mara R he āpiti, i.e. x c x d = d c mo c katoa, d o R.
• Addition i roto i te mara R he herekore i.e. (c + d) + f = c + (d + f) mo tetahi c, d, f o R.
• Whakarea i roto i te mara R he herekore i.e. (c x d) x f = c x (d x f) mo tetahi c, d, f o R.
• Hoki ia maha o te mara R ritenga ki reira i reira, penei e c + (-c) = 0, te wahi c, -c i R.
• Hoki te vai ia maha o mara R tona kōaro, pērā e c x c ^-1 = 1 te wahi c, c ^-1 o R.
• vai Unit me no ki R, kia e te c x 1 = c, hoki tetahi c o R.
• He te tohatoha ture mana, na taua c x (d + f) = c x c x f d +, mo tetahi c, d, f o R.
• Ko te R mara ko kore ko te kore e rite ki te kotahitanga.
• Field R Ko transitive: ki te c ≤ d, d ≤ f, ka c ≤ f mo tetahi c, d, f o R.
• I roto i te R me tua atu kia, e toaati: ki te c ≤ d, ka c + f ≤ d + f mo c katoa, d, f o R.
• I roto i te tikanga o te R me whakareatanga hono: ki te 0 ≤ c, 0 ≤ d, ka 0 ≤ c x d mo tetahi c, d o R.
• Ka rite ki he tonu kino, me te pai tau tūturu, i.e., mo tetahi c, d o R f, i reira te vai i R, e c ≤ f ≤ d.

Kōwae mara R

Ko te tūturu tau ngā he mea pēnei i te kōwae. Whakaritea te reira rite te | f | hoki tetahi f i R. | f | = F, ki te 0 ≤ f me | f | = -f, ki te 0> f. Ki te whakaaro tatou i te kōwae rite te uara āhuahanga, he mea he tawhiti - e kore e mea faufaa te reira, "haere" koe rite kore i roto i te kino ki te pai whakamua ranei.

tau Matatini me te tūturu. He aha e nga ritenga me ngā rerekētanga?

Na roto i te a nui, matatini, me te mau tau - he ratou tetahi me te taua, ki te kahore e piri te tuatahi te kōwae pohewa i, te tapawha o e he rite ki te -1. Elements āpure R me taea te māngai C i te tātai e whai ake nei:

• c = d + f x i, ai d, f no ki te mara R, me i - wae pohewa.

Ki te tiki i te c o R f i roto i tenei take riro noa ki kia kore, arā, i reira he anake te wahi tūturu o te tau. No te mea te mara o tau matatini he te āhuatanga taua whakaturia rite te mara o tūturu, f x i = 0 ki te f = 0.

Ki te pā ngā rerekētanga mahi, hei tauira i roto i te mara R whārite pūrua e kore e taea te whakatika ki te he tōraro te discriminant, i kore e te pouaka C tangohia tenei whāititanga i te whakauru i te wae pohewa i.

hua

"Pereki" o axioms me mau marôraa i runga nei ki te pāngarau turanga, e kore e huri. I runga i etahi o ratou e tika ana ki te hua o te kōrero, me te whakataki o ariā hou whakanohoia te "pereki" e whai ake nei, i roto i te heke mai kia meinga te pūtake mo te taahiraa i muri mai. Hei tauira, tau māori, ahakoa te meka e he ratou he huinga o te R mara tūturu, e kore e ngaro tona hāngaitanga. Ko reira ki a ratou te pūtake o te tauhanga pūmotu katoa, e haamata ki te matauranga o te tangata o te rangimarie.

Mai i te wāhi mahi o te tirohanga, te tau tūturu titiro rite te rārangi tika. Ko taea ki te whiriwhiri i te aronga, ki te tautuhi i te takenga te ware me te. Kei o te tau mure ore o ngā, ia o nei hāngai ki te tau tūturu kotahi, ahakoa o ahakoa kahore ranei kouma Direct. Mai i te whakaahuatanga ko reira mārama e korero tatou e pā ana ki te ariā, e hāngai ana nei te pāngarau i roto i whānui, me te tātari pāngarau i roto i ngā.