Izimfihlo ezingu-7 ezikhangayo zezinombolo ezicatshangelwayo

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1. Umsuka kanye nokuthuthuka komlando wezinombolo ezicatshangelwayo

Ukuqalisa: Izimpande eziyisikwele zezinombolo ezinegethivu

Umlando wezinombolo ezicatshangelwayo uqala ngenkinga ebonakala ingenakwenzeka: ukuthola izixazululo zezibalo ezifaka izimpande eziyisikwele zezinombolo ezinegethivu. Ngekhulu le-16, izazi zezibalo zase-Italy zabhekana nale nselele ngenkathi zizama ukuxazulula izibalo zama-cubic.

U-Girolamo Cardano, isazi sezibalo seRenaissance yase-Italy kanye nodokotela, wayengomunye wabokuqala ukucabangela ngokucophelela lezi zimpande "ezingenakwenzeka". Encwadini yakhe ethi "Ars Magna" (1545), uCardano waba nesibindi sokusebenzisa lezi zinkulumo, nakuba wayezibheka "ezicashile futhi ezingenamsebenzi."

Ukuvela komqondo phakathi namakhulu eminyaka

Umqondo wezinombolo ezicatshangelwayo wavela kancane. URafael Bombelli, esinye isazi sezibalo sase-Italy, uthathe isinyathelo esibalulekile ngokusungula imithetho yokusebenza ngalezi zinombolo encwadini yakhe ethi "L'Algebra" (1572). Nokho, kwaze kwaba sekhulwini le-XNUMX lapho izinombolo ezicatshangelwayo zaqala ukwamukelwa kabanzi futhi zisetshenziswe.

ULeonhard Euler, isazi sezibalo saseSwitzerland esinolwazi oluningi, wethula uphawu 'i' ukuze lumele impande yesikwele ka--1 ngo-1777, okwenza kwaba lula kakhulu ukubhala kanye nokubala ngezinombolo ezicatshangelwayo. Lesi kwaba yisikhathi esibalulekile esavula indlela yokuthuthukiswa kokuhlaziywa okuyinkimbinkimbi.

2. Izisekelo zezibalo zezinombolo ezicatshangelwayo

Incazelo kanye nezakhiwo eziyisisekelo

Izinombolo ezicatshangelwayo zichazwa ngokusemthethweni njengezinombolo zefomu elithi a + bi, lapho u-'a' no-'b' kuyizinombolo zangempela, futhi 'i' iyiyunithi ecatshangelwayo, echazwa ngokuthi impande eyisikwele sika -1. Lokho kusho ukuthi:

i² = -1

Le ncazelo elula ivula umhlaba wamathuba wezibalo. Izinombolo ezicatshangelwayo ziyingxenye yesethi enkulu ebizwa ngokuthi izinombolo eziyinkimbinkimbi, ehlanganisa kokubili izinombolo zangempela nezicatshangelwayo.

Iyunithi yengqondo 'i'

Iyunithi yengqondo ethi 'i' iyinhliziyo yezinombolo ezicatshangelwayo. Ezinye zezinto zayo eziyisisekelo yilezi:

1. i² = -1
2. i³ = -i
3. ngi = 1

Lezi zakhiwo zenza kube lula ukwenza izibalo eziyinkimbinkimbi futhi kuxazululwe izibalo ebezibonakala zingenakwenzeka ngaphambili.

3. Ukumelwa okungokomfanekiso kwezinombolo ezicatshangelwayo

Indiza eyinkimbinkimbi

Izinombolo ezicatshangelwayo zimelelwe ngezithombe endizeni eyinkimbinkimbi, eyaziwa nangokuthi indiza ye-Argand. Kule ndiza, i-eksisi evundlile imelela ingxenye yangempela, futhi i-eksisi eqondile imelela ingxenye ecatshangelwayo.

Isibonelo, inombolo eyinkimbinkimbi 3 + 2i izomelwa njengephoyinti elinezixhumanisi (3, 2) kule ndiza.

Ukuma kwe-polar kanye ne-rectangular

Izinombolo eziyinkimbinkimbi zingavezwa ngezindlela ezimbili eziyinhloko:

1. Umumo ongunxande: a + b
2. Ifomu le-polar: r(cos θ + i sin θ), lapho u-r engubukhulu futhi θ iyi-engeli ene-eksisi yangempela.

Ifomu le-polar liwusizo ikakhulukazi ekuphindaphindeni nasekuhlukaniseni, kuyilapho ifomu elingunxande lilungele ukuhlanganisa nokukhipha.

4. Ukusebenza ngezinombolo ezicatshangelwayo

Faka ukhiphe

Ukwengeza nokukhipha izinombolo ezicatshangelwayo kulula. Ungeza noma ukhiphe izingxenye zangempela nezicatshangelwayo ngokwehlukana.

(a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) – (c + di) = (a – c) + (b – d)i

Ukuphindaphinda nokuhlukanisa

Ukuphindaphindeka kwezinombolo ezicatshangelwayo kusebenzisa indawo i² = -1:

(a + bi)(c + di) = (ac – bd) + (ad + bc)i

Ukuhlukaniswa kuyinkimbinkimbi kakhulu futhi kusebenzisa inqubo yokuphindaphinda nge-conjugate eyinkimbinkimbi:

5. Ukusetshenziswa okungokoqobo kwezinombolo ezicatshangelwayo

Kwezobunjiniyela bukagesi

Izinombolo ezicatshangelwayo zibalulekile kubunjiniyela bakagesi, ikakhulukazi ekuhlaziyeni amasekhethi ashintshashintshayo amanje (AC). Asetshenziselwa ukumela i-impedance, ehlanganisa ukumelana nokusabela kube inombolo eyinkimbinkimbi eyodwa.

Isibonelo, kumjikelezo we-RLC (resistance-inductance-capacitance), i-impedance ivezwa kanje:

Z = R + i(ωL – 1/ωC)

Lapho u-R ewukumelana, L inductance, C amandla, kanye ω imvamisa ye-angular.

Ku-quantum physics

Ku-quantum mechanics, izinombolo ezicatshangelwayo zidlala indima ebalulekile emsebenzini wegagasi we-Schrödinger, ochaza isimo se-quantum sesistimu. I-equation ye-Schrödinger ithi:

iℏ∂ψ/∂t = Ĥψ

Lapho ℏ kuwukuncishiswa kwe-Planck okungaguquki, ψ kuwumsebenzi wamagagasi, futhi u-Ĥ ungu-opharetha we-Hamiltonian.

6. Izinombolo ezicatshangelwayo ekuxazululeni izibalo

Izibalo ze-quadratic kanye ne-cubic

Izinombolo ezicatshangelwayo zisivumela ukuthi sixazulule izibalo ze-quadratic ezingenazo izixazululo zangempela. Isibonelo, isibalo x² + 1 = 0 sinezixazululo x = ±i.

Endabeni yezibalo zama-cubic, izinombolo ezicatshangwayo zibalulekile ukuveza zonke izixazululo ezingenzeka, ngisho nalapho zonke zingokoqobo.

I-Cardano Formula

Ifomula ye-Cardano yokuxazulula i-cubic equations ngokuvamile ihlanganisa izinombolo ezicatshangelwayo, ngisho nalapho izixazululo zokugcina zingokoqobo. Lesi senzakalo, esaziwa ngokuthi "casus irreducibilis", sasingenye yezinkomba zokuqala zokubaluleka kwezinombolo ezicatshangelwayo ku-algebra.

7. Izinombolo ezicatshangelwayo ekuhlaziyeni okuyinkimbinkimbi

Imisebenzi eyinkimbinkimbi

Ukuhlaziya okuyinkimbinkimbi, okucwaninga imisebenzi yezinto eziguquguqukayo eziyinkimbinkimbi, kuwumkhakha ocebile ngamathiyori amahle kanye nokusetshenziswa okungokoqobo. Imisebenzi efana no-e^z, sin(z), kanye nelogi(z) inwebeka ngokwemvelo endizeni eyinkimbinkimbi, iveze izici ezimangalisayo.

Isibonelo, umsebenzi we-exponential oyinkimbinkimbi uhlobene ne-trigonometry nama-exponentials ngefomula ka-Euler:

e^(iθ) = cos θ + ngona θ

I-Basic theorem ye-algebra

Omunye wemiphumela ebaluleke kakhulu ehlanganisa izinombolo eziyinkimbinkimbi i-theorem eyisisekelo ye-algebra, ethi yonke i-polynomial ye-degree n inezimpande eziyinkimbinkimbi zika-n (ukubala ukuphindaphindeka). Le theory, yafakazelwa okokuqala ngu-Carl Friedrich Gauss, ikhombisa ukuthi izinombolo eziyinkimbinkimbi ziqeda kanjani uhlelo lwezinombolo ngendlela izinombolo zangempela ezingakwazi ngayo.

8. Ilukuluku nezindida ezihlobene nezinombolo ezicatshangelwayo

Izinombolo ezicatshangelwayo ziye zadala ilukuluku lezibalo ezimbalwa kanye nezingqinamba ezisobala. Enye edume kakhulu ubunikazi buka-Euler:

e^(iπ) + 1 = 0

Le fomula, echazwe isazi sefiziksi uRichard Feynman “njengendlela ephawuleka kakhulu yezibalo,” ixhumanisa izinto ezinhlanu eziyisisekelo ngendlela elula ngendlela emangalisayo.

Enye ilukuluku ukungqubuzana okusobala kwenkulumo ethi:

i^i = e^(-π/2) ≈ 0.207879576…

Inombolo ecatshangelwayo ephakanyiselwe emandleni acatshangelwayo iphumela enambeni yangempela!

9. Ikusasa lezinombolo ezicatshangelwayo kwizibalo nesayensi

Njengoba singena ekhulwini lama-21, izinombolo ezicatshangelwayo ziyaqhubeka nokuba yithuluzi elibalulekile kwizibalo ezithuthukile kanye ne-theoretical physics. Ukusetshenziswa kwazo emikhakheni efana ne-string theory kanye ne-quantum computing kusikisela ukuthi zizohlala zibalulekile ekuqondeni kwethu indawo yonke.

Esikhathini sobuhlakani bokwenziwa kanye nokufunda komshini, izinombolo eziyinkimbinkimbi zithola izinhlelo zokusebenza ezintsha ekucubungulweni kwesignali kanye nokwenza ngcono i-algorithm.

Isiphetho

Izinombolo ezicatshangelwayo, kunokuba zibe nje ilukuluku lezibalo, seziphenduke ithuluzi elibaluleke kakhulu kwizibalo, i-physics kanye nobunjiniyela. Kusukela emisuka yazo eyimpikiswano kuya ekusetshenzisweni kwazo okuphambili kusayensi yesimanje, lezi zinombolo zisivumele ukuba sandise ukuqonda kwethu umhlaba futhi sixazulule izinkinga ezake zabonakala zingenakwenzeka.

Njengoba siqhubeka nokuhlola imingcele yezibalo nefiziksi, izinombolo ezicatshangwayo zizohlala ziyingxenye ebalulekile kuzikhali zethu zobuhlakani. Zisikhumbuza ukuthi ezibalweni nasesayensini, ngezinye izikhathi kumelwe sibeke inselele emicabangweni yethu ukuze sithole amaqiniso ajulile ngendawo yonke.

Ingabe uye wakhangwa yilolu hambo emhlabeni wezinombolo ezicatshangelwayo? Yabelana ngalesi sihloko nabangani bakho kanye nozakwenu ukuze nabo bathole umlingo walezi zinhlangano zezibalo!