Los amanani okwenyani yenza i iseti yamanani asetyenziswa kakhulu kwizibalo nakubomi bemihla ngemihla. Ziquka amanani anengqondo nangenangqiqo kwaye ziqala kumanani esiwasebenzisayo ekubaleni, ekulinganiseni, okanye ekuhlawuleni, ukuya kulawo avela kwiingcamango eziphambili ezifana ne-calculus okanye i-physics. Naliphi na inani elinokumelwa kwi-a umgca-manani oqhubekayo, nokuba yi-whole, i-fractional okanye inee-decimals ezingenasiphelo, yinxalenye yamanani okwenyani.
Eli qela lavela ngokwembali ngenxa yesidingo sokuba chaza ngokuchanekileyo Ubungakanani obungenakubonakaliswa ngamanani awayesaziwa ngelo xesha. Amagama angacacanga afana nokuthi "mancinci kakhulu" okanye "phantse abe yi-zero" angqinekanga anganelanga kuphuhliso oluqinileyo lohlalutyo lwezibalo, nto leyo eyakhokelela ekumiselweni kweengcamango zomda kunye nenani lokwenyani. Iingcali zembali ezininzi zibeka iinkqubo zokuphucula kunye nokwenza ingqiqo ibe sesikweni phakathi Inkulungwane ye-15 neye-17nangona inkcazo yanamhlanje nengqongqo yahlanganiswa kamva.
Nangona iimpucuko zamandulo ezifana nezaseYiputa zazisele zisebenzisa amaqhezuYayingamaGrike afunda ingcamango "yenani" ngendlela yefilosofi ngakumbi. Isikolo sikaPythagore sathi "yonke into yinani," kwaye xa bezama ukuveza ubude obuthile (njengomgca othe tyaba wesikwere) bafumanisa ukuba Ayizizo zonke izixa ezinokubhalwa njengeqhezu leenombolo ezipheleleyo.Kulapho kuvela khona amanani angenangqondo, aza kuthi kamva agqibezele iseti yamanani okwenyani.
Athini amanani okwenyani kwaye amelwe njani?
Amanani okwenyani achazwa ngolu hlobo onke amanani ahambelana nendawo emgceni wamanani wokwenyaniLo mgca udlula ngaphandle komda uye ekhohlo (amaxabiso angalunganga) nasekunene (amaxabiso amahle), kuquka u-zero, amaqhezu, iidesimali ezilinganiselweyo, kunye needesimali eziphindaphindayo nezingapheliyo ezingaphindiyo.
Le seti idla ngokuchazwa ngoonobumba R okanye isimboli ℝNgokwesiko, iseti yamanani okwenyani ingachazwa njengomanyano lwee-subsets ezimbini eziphambili: amanani aqiqileyo (Q) kunye namanani aqiqileyo (I). Koko kukuthi, ℝ = Q ∪ I.
Imizekelo yamanani okwenyani yile: 5, 0, −9, 3/4, −7/2, 3,45, 0,333… (1/3), √2, √10, π, ephakathi kwezinye ezininzi. Zonke zinokufumaneka kumgca wamanani wokwenyani ngokusebenzisa inqaku elichazwe kakuhle.
Ngaphezu koko, amanani okwenyani icandelo eliphantsi lamanani antsonkothileyoAmanani antsonkothileyo amelwa njengo-a + bi, apho u-a kunye no-b zingamanani okwenyani kwaye u-ei yiyunithi ecingelwayo (ingcambu yesikwere ka-−1). Xa u-b = 0, inani elintsonkothileyo u-a + 0i lihambelana nenani lokwenyani, ngoko ke inani ngalinye lokwenyani lingabonwa njengenani elintsonkothileyo elinenxalenye ecingelwayo engu-zero.
Ukwahlulahlulwa kwamanani okwenyani ngokohlobo lwawo
Ukuhlelwa kwamanani okwenyani kudla ngokuhlelwa ngokwamaqela aliqela adibeneyo. Awona aqhelekileyo ngala amanani endalo, iinombolo, isengqiqweni e engenangqiqoKwinqanaba elikhulu, ngaphakathi kwe-ℝ sifumana amaqela amabini amakhulu: isengqiqweni e engenangqiqoKwaye ngaphakathi kwamanani aqikelelweyo, kukho amanani endalo, amanani apheleleyo, kunye namanani aqhezukileyo.
1. Amanani asengqiqweni
Kubizwa ngokuba amanani anengqiqo kubo bonke abo banokumelwa njenge i-quotient yeenombolo ezimbiniOko kukuthi, njengeqhezu p/q apho u-p no-q beyi-integers kunye no-q ≠ 0. Le seti imelwe ngunobumba QAmanani aqikelelweyo aquka amanani amahle, amanani angalunganga, kunye no-zeroNgenxa yoko, zigubungela iintlobo ngeentlobo zobukhulu.
Inani eliqikelelweyo lingabhalwa njengeqhezu, kodwa lingabonakala njenge idesimali echanekileyo (umzekelo 3,5), idesimali ephindaphindayo ecocekileyo (0,7777…) okanye idesimali ephindaphindayo exutyiweyo (2,58333…). Naliphi na kwezi meko lihlala livuma ukubonakaliswa njengeqhezu leenombolo ezipheleleyo.
Amanani aqikelelweyo aquka zombini iinombolo njengaye iqhekezaNgoko ke, nayiphi na inombolo epheleleyo (−3, 0, 5…) nayo inengqondo, kuba ingabhalwa njengo-p/1. Oku kuthetha ukuba U-ℤ yinxalenye esezantsi ye-Q.
Amanani anengqondo asivumela ukuba senze imisebenzi ngaphandle kokushiya iseti ukudibanisa, ukuthabatha, ukuphindaphinda kunye nokwahlula (ngaphandle kokwahlulahlula ngo-zero). Ke ngoko, kuthiwa u-Q ivaliwe kwezi zenzo.
a) Amanani apheleleyo
Los ii-enteros ezininzi ngaba iseti yenziwe yi amanani endalo, yakho izinto ezichaseneyo nezingalunganga kunye zeroBamelwa yile leta Z kwaye ziquka amaxabiso anje nge …, −3, −2, −1, 0, 1, 2, 3, …
Kumgca-manani, amanani apheleleyo abonakala kwi ngasekunene kwe-zero, u-zero uhlala kwi indawo esembindini kwaye ii-negative integers zibekwe kwi- ngasekhohloOlu lungiselelo luvumela ukuthelekisa okulula kobukhulu babo: okuya phambili ngasekunene, kokukhona inani lilikhulu.
- Babizwa amanani endalo esiyisebenzisela yona izinto zokubala okanye bonisa ulandelelwano (1, 2, 3, …). Zii-integers ezi-positive kwaye zihlala zibonakaliswa ngunobumba u-N.
- El zero imele a ixabiso elingentoI-zero ayongezi xabiso xa imi yodwa, kodwa indawo yayo kwinani ilitshintsha ngokupheleleyo ixabiso layo. I-zero ngasekunene kwedijithi iphinda-phinda ixabiso layo ngeshumi (2 iba ngama-20), ngelixa ngasekhohlo ingatshintshi inani (02 ilingana no-2).
- Los amanani apheleleyo Zimela iimeko ezichasene nezendalo, ezinje amatyala, amaqondo obushushu angaphantsi kwe-zero o amanqanaba angaphantsi kwesalathisoUkuze bazibize ngamagama, igama elithi "minus" libekwa phambi kwenani: "minus four" ibhalwa ngo-−4.
Iinombolo ezipheleleyo zivaliwe phantsi ukongeza, ukuthabatha, kunye nokuphindaphindaUkusebenza phakathi kwee-integer ezimbini kusoloko kubangela enye i-integer. Nangona kunjalo, ukwahlulahlula ii-integer ezimbini kunokubangela inani elingeyiyo i-integer (umzekelo, 3/4), ngoko ke azivalwanga phantsi kokwahlulahlula.
b) Ukuqhekeka
Amanani aqikelelweyo aquka amanani angamaqhezu, eyaqala ngenxa sombulula iingxaki zokusasazwa xa ukwahlulwahlulwa kwamanani endalo kunganikanga siphumo senani elipheleleyo.
Inani eliqhezukileyo ligama elibonisa ukwahlulwa kwenani elinye ngomnye. Ibandakanya a amanani (isixa-mali siyahlulwahlulwa) kunye idinomineyitha (ibe ziindawo ezingaphi ezahlulwe ngazo), zahlulwe ngebha ethe tye okanye ethe tyaba.
Nangona yonke inombolo inokuthathwa njengeqhezu eline-denominator engu-1, kweli candelo kukho umahluko okhethekileyo phakathi kwe- amaqhezu afanelekileyo nangafanelekanga:
- Las amaqhezu afanelekileyo Ezi zezo apho i-numerator ikhona khona mncinci kune-denominator. Zimela ubungakanani incinci kunenye, umzekelo 3/5.
- Las amaqhezu angafanelekanga yiba nenombolo inkulu okanye ilingana ne kune-denominator, ebonisa ubungakanani inkulu okanye ilingana nenye, njenge-7/4 okanye i-5/5.
Ngaphezu koko, amanani amaninzi aqikelelweyo anokubhalwa ngolu hlobo iidesimali ezichanekileyo okanye eziphindaphindayoNgoko ke, inani elifana no-0,25 lilingana no-1/4 (echanekileyo idesimali), ngelixa u-0,333… lilingana no-1/3 (idesimali ecocekileyo ephindaphindwayo).
2. Amanani angenantsingiselo
Los amanani angenangqiqo Zizo ezo Azinakubonakaliswa njengeqhezu leenombolo ezimbini ezipheleleyoUkubonakaliswa kwayo kwedesimali kuhlala kunjalo engenasiphelo kwaye engekho ngamaxesha athile: iidesimali ziyaqhubeka ngaphandle kokuphinda kwipateni emiselweyo.
Imizekelo yakudala yile inani π (ubudlelwane phakathi kobude bomda kunye nobubanzi bawo), inombolo e (isiseko see-logarithms zendalo), umlinganiselo wegolide φ okanye iingcambu zamanani aphambili angengawo ama-square agqibeleleyo, afana no-√2, √3, √5, √7, njl.
Ngokwembali, amanani angenangqondo avela xa umfundi kaPythagoras wazama ukuveza i-diagonal yesikwere esinobude becala 1 njengeqhezu, efumanisa ukuba Bekungekho zimbini ii-integers u-p no-q kangangokuba u-p/q = √2Nangona isikolo sasePythagorean saqala ukuchasa, olu phando lubonise ukuba iseti yamanani aqikelelweyo ayanelanga ukuchaza zonke iijiyometri.
Izidalwa ezingenangqondo zinokubonwa njenge ukongeza kwamanani aqikelelweyo ngaphakathi kwamanani okwenyaniOko kukuthi, ukuba sibiza u-Q ngokuthi yi-seti yamanani aqikelelweyo kunye no-ℝ yi-seti yamanani okwenyani, iseti yamanani aqikelelweyo ingachazwa ngolu hlobo ℝ − Q: onke amanani okwenyani angengowengqiqo.
Ngaphezu koko, iintlobo ezimbini ezibalulekileyo zamanani angaqhelekanga zahlulwe: algebra y ogqithileyo.
- Los amanani e-algebra zezo zisisombululo i-equation ethile ye-algebra kunye nee-integers ezipheleleyo. Umzekelo, i-√2 ayilunganga kwaye i-algebraic, kuba sisisombululo se-x² − 2 = 0.
- Los amanani angaphaya kwamandla Azinakufunyanwa njengezisombululo kuyo nayiphi na i-algebraic equation ene-integer coefficients. Azivezwa linani eliqingqiweyo leengcambu kwaye iidesimali zayo azilandeli naluphi na uhlobo lwepatheni olubonakalayoEzi ziquka u-π kunye no-e.
Iimpawu ezisisiseko zamanani okwenyani
Iseti yamanani okwenyani isenza sikwazi ukwenza imisebenzi ukongeza kunye nokuphindaphinda ukufezekisa uthotho lweempawu ezenza kube lula ukubala kunye nokuqiqa ngezibalo. Phakathi kwezona zibalulekileyo zezi tshixa, la uthungelwano, la ubudlelwane, la ubukho bezinto ezingathathi cala kunye ubukho bezinto ezichaseneyo.
Tshixa
Ipropati ye tshixa ibonisa ukuba isixa okanye imveliso yamanani amabini okwenyani isoloko iyinto eyahlukileyo inani lokwenyaniUkuba u-a kunye no-b bangabakwa-ℝ, ngoko u-a + bya·b nabo bangabakwa-ℝ. Oku kusenza sikwazi ukusebenza ngaphandle kokushiya iseti, into ebalulekileyo kuphuhliso lwe-algebra kunye nohlalutyo.
Ipropathi yokutshintsha
La Ipropathi yokutshintshiselana Ithi isiphumo sokongeza okanye ukuphinda-phinda amanani amabini okwenyani Akuxhomekeke kwiodolo apho umsebenzi wenziwa khona. Oko kukuthi, u-a + b = b + aya·b = b·a kuwo onke amanani okwenyani u-a kunye no-b. Le mpawu yenza kube lula kakhulu ukubala kunye nokubhala amabinzana.
ipropati yonxulumano
La impahla edibeneyo ibonisa ukuba, xa kuhlanganiswa okanye kuphinda-phinda amanani okwenyani amathathu nangaphezuluIndlela ezibekwe ngayo ngamaqela ayichaphazeli isiphumo. Kwiimpawu: (a + b) + c = a + (b + c) kunye no (a·b)·c = a·(b·c). Ngenxa yoku, imisebenzi emide inokuhlelwa kwakhona ukuze kube lula ukuyisombulula.
Into engathath'icala
Kumanani okwenyani kukho ezimbini izinto ezingathathi cala esisiseko:
- El zero yiyo isongezo esingathathi calakuba ukuyongeza nakweyiphi na inani lokwenyani akutshintshi ixabiso layo: a + 0 = a.
- El enye yiyo ubuwena obuphindaphindeneyo, ekubeni ukuyiphinda-phinda nangaliphi na inani lokwenyani kunika inani elifanayo: a·1 = a.
I-inversion eyongezelelweyo kunye ne-multiplicative inverse
Kwinani ngalinye lokwenyani kukho inani inguqu eyongezelelweyo kwaye, ngaphandle kwe-zero, i- inguquko ephindaphindayo:
- El inguqu eyongezelelweyo Isazisi sokongeza senani u-a ngu-−a, kuba xa zidityaniswa kunye, isazisi sokongeza sifunyanwa: a + (−a) = 0.
- El inguquko ephindaphindayo o ngokubuyisela yenani u-≠ 0 ngu-1/a, kuba u-a·(1/a) = 1.
Amanani okwenyani kumgca wamanani nakubomi bemihla ngemihla
Inani lokwenyani ngalinye lingamelwa yi inqaku kumgca mananiInqaku ngalinye kuloo mgca lihambelana nenani lokwenyani elahlukileyo. Olu nxibelelwano lomntu ngamnye lusenza sikwazi ukubona imisebenzi efana nokudibanisa (ukutshintsha uye ngasekunene okanye ngasekhohlo), ukuthabatha, ukungalingani, kunye nemigama.
Kumgca wamanani wokwenyani, ulandelelwano lwamanani imiselwa yindawo ekuyo: indawo ekude ngasekunene yileyo, Okukhona inani lilikhulu ezinxulumeneyo; phambili ngasekhohlo, Okukhona incinci iya kubaAkukho nani lokwenyani "lokugqibela" elilungileyo okanye elingalunganga, kuba iseti ℝ iqulethe izinto ezingenamkhawulo kuzo zombini iindlela.
Kubomi bemihla ngemihla, amanani okwenyani asetyenziswa rhoqo: kuba ubude bokulinganisa (iimitha, iisentimitha), amaqondo obushushu akhawulezayo (iidigri ezilungileyo nezingalunganga), bala amaxesha, lawula imali (ibhalansi, amatyala, inzala), thelekisa iishedyuli o hlalutya idatha kwizibalo kunye nezoqoqosho.
Kwicandelo lesayensi netekhnoloji, amanani okwenyani asisiseko icalculus eyahlukileyo kunye nendibaniselwano, la ifiziksi yakudala neyanamhlanje, la engineering, la IT kunye nezinye iinkalo ezininzi. Ubungakanani obufana nesantya, ukukhawulezisa, amandla, okanye ubukhali bubonakaliswa ngamanani okwenyani, kwaye ukuphathwa kwawo ngokuchanekileyo kuvumela ukwenziwa kwemodeli yezinto ezintsonkothileyo.
Ukuqonda ukuhlelwa kwamanani okwenyani, ii-subsets zawo, kunye neempawu akupheleli nje ekuncedeni ukufunda izibalo, kodwa kukwanceda Iyaqinisa ingqiqo ecacileyo nengacacanga.Inceda ekwakheni indlela yokucinga kwaye iphucula ubuchule bokusombulula iingxaki kwiimeko ezahlukeneyo kakhulu.