Complex xov tooj. Tus nqi thiab Evolution "xav txog tej yam tseem ceeb"

Cov zauv - qhov yooj yim xaam khoom uas yuav tsum tau rau txawv pawg thiab suav. Lub teeb ntawm tej yam ntuj tso, integer, muaj kuab thiab irrational cov qhov tseem ceeb txhais ib plurality ntawm thiaj li hu ua cov zauv tiag tiag. Tab sis muaj tseem yog heev txawv txawv qeb - lej txhais los ntawm René Descartes li "xav txog tej yam ntau." Thiab ib tug ntawm cov uas mathematicians ntawm lub xyoo pua Leonhard Euler npaj siab mus xaiv lawv cov tsab ntawv kuv los ntawm Fabkis lo lus imaginare (xav txog tej yam). Yuav ua li cas yog cov lej?

Yog li ntawd hu ua kab zauv ntawm daim ntawv no yog ib tug + lis, uas a thiab b yog cov zauv tiag tiag, thiab kuv yog ib cov qhov taw qhia tshwj xeeb nqi uas square yog -1. Operations rau complex tus xov tooj yog ua los ntawm cov cai xws li ntau yam xyuam xim ua hauj lwm nyob rau hauv polynomials. Qhov no xyuam xim qeb tsis sawv cev rau cov kev tshwm sim ntawm tej kev ntsuas los yog suav. Rau qhov no yog heev txaus cov zauv tiag tiag. Yog vim li cas, ces, ua lawv xav tau?

Complex xov tooj raws li ib tug lej tswvyim, tsim nyog vim rau qhov tseeb hais tias ib txhia sib npaug nrog tiag tiag coefficients muaj kev daws teeb meem nyob rau hauv lub teb ntawm "zoo tib yam" xov tooj. Yog li ntawd, yuav nthuav lub Scope ntawm kev daws tsis sib npaug cia li sawv qhov yuav tsum tau ua kom paub tshiab zauv pawg. Complex tooj muaj mas theoretical paub daws teeb nws tau mus daws tau cov equations raws li ² x 1 = 0. Nws yog muab sau tias, txawm nws khees formality pawg no tus xov tooj khwb khuav thiab dav siv, e.g., rau txawv tswv yim dlaws teeb meem ntawm elasticity kev tshawb xav, hluav taws xob engineering, aerodynamics thiab hydromechanics, atomic physics thiab lwm yam kev qhuab qhia.

Module thiab sib cav ntawm ib tug zauv siv nyob rau hauv qhov kev siv sij hawm. Qhov no daim ntawv ntawm kev sau ntawv hu ua trigonometric. Nyob rau hauv tas li ntawd, lub geometrical txhais ntawm cov zauv tau ntxiv qee lub Scope ntawm lawv daim ntawv thov. Nws los ua tau siv lawv rau ib tug ntau yam ntawm koj laij daim ntawv qhia.

Mathematics tau tuaj ib tug ntev txoj kev los ntawm tej yam yooj yim tej yam ntuj tso tus xov tooj rau txoj kev tshuab thiab lawv cov functions. Ntawm cov ntawv no yuav sau ib daim tutorial. Ntawm no yog peb saib xwb ib co ntawm cov evolutionary yam ntawm kev tshawb xav tooj, ua kom nws ntshiab tag nrho cov keeb kwm thiab scientific tom qab cim asterisk tso ntawm no xyuam xim qeb.

Greek mathematician suav hais tias "muaj tseeb" tsuas ntuj tus xov tooj, uas yuav siv tau los xam dab tsi. Twb tau nyob rau hauv lub thib ob xyoo txhiab BC. e. lub ancient Egyptians thiab Babylonians nyob rau hauv ib tug ntau yam ntawm cov tswv yim suav nquag siv cov zauv feem. Cov tom ntej no tseem ceeb milestone nyob rau hauv txoj kev loj hlob ntawm kev kawm txog zauv yog cov tsos ntawm cov zauv negative nyob rau hauv ancient Tuam Tshoj ob puas xyoo ua ntej peb era. Lawv kuj siv los ntawm lub ancient Greek mathematician Diophantus, uas paub cov kev cai ntawm tej yam yooj yim ua hauj lwm rau lawv. Nrog kev pab los ntawm cov zauv negative, nws tau los ua tau los piav qhia txog ntau yam kev hloov nyob rau hauv qhov tseem ceeb, tsis yog nyob rau hauv qhov zoo dav hlau.

Nyob rau hauv lub xya xyoo pua AD, nws twb kom meej meej tsim hais tias tus square cov hauv paus hniav ntawm zoo naj npawb ib txwm muaj ob tug tseem ceeb - nyob rau hauv tas li ntawd mus zoo, kuj tsis zoo. Los ntawm lub caij nyoog kawg mus extract lub square hauv paus ntawm lub li ib txwm algebraic txoj kev uas lub sij hawm nws twb xav tias tsis yooj yim sua: muaj yog tsis muaj tej nqi ntawm x rau x ² = ─ 9. Rau ib tug ntev lub sij hawm nws tsis ua cas. Nws tsuas yog nyob rau hauv lub thib kaum rau xyoo pua, thaum muaj thiab tau tau nquag kawm nyhav sib npaug, qhov yuav tsum tau mus rau extract lub square hauv paus ntawm cov zauv negative, raws li nyob rau hauv cov mis rau hauv kev daws ntawm cov kab zauv muaj tsis tau tsuas yog cov thawv muaj rau sab, tab sis kuj cov square hauv paus hniav.

Qhov no mis yog robust, yog hais tias tus kab zauv muaj rau ntawm cov ib tiag tiag paus. Nyob rau hauv rooj plaub ntawm lub xub ntiag nyob rau hauv kab zauv ntawm peb tiag tiag cov hauv paus hniav rau lawv kho twb tau nrog tus xov tooj tsis zoo nqi. Nws hloov tawm hais tias txoj kev mus rau rov qab sau los ntawm peb cov hauv paus hniav ntawm tus tsis yooj yim sua los ntawm lub standpoint ntawm kev kawm txog zauv ntawm lub lag luam lub sij hawm.

Rau ib tug piav qhia txog cov uas ua paradox Italian algebraists J. Cardano twb npaj siab yuav ua kom paub ib tug tshiab qeb ntawm qhov txawv qhov xwm ntawm tus xov tooj, uas yog hu ua complex. Kuv xav li cas nws Cardano lawv xav tias lawv puas thiab ua txhua yam kom tsis txhob muaj kev thov kom lawv lub tswv yim ua zauv pawg. Tab sis twb nyob rau hauv 1572 ib phau ntawv nyob lwm Italian algebraist Bombelli, uas twb ncauj lus kom ntxaws cai rau kev ua hauj lwm nyob rau hauv cov lej nyuaj.

Thoob plaws hauv lub thib kaum xya xyoo pua txuas ntxiv qhov kev sib tham ntawm cov zauv xwm ntawm cov ntaub ntawv tus xov tooj thiab muaj peev xwm ntawm lawv cov geometric txhais lus. Tsis tas li ntawd maj tsim thiab paub cov txheej txheem ntawm kev ua hauj lwm nrog lawv. Thiab nyob rau lub lem ntawm lub 17th thiab 18th centuries, cov kev tshawb xav ntawm complex numbers raug tsim. Ib tug enormous pab mus rau txoj kev loj hlob thiab kev txhim kho ntawm lub hom phiaj ntawm kev khiav dej num ntawm txoj variables twb qhia Lavxias teb sab thiab Soviet zaum. N. I. Muskhelishvili koom nyob rau hauv nws daim ntawv thov rau cov teeb meem ntawm lub hom phiaj ntawm elasticity, Keldysh thiab Lavrentiev lej tau siv nyob rau hauv lub teb ntawm hydro- thiab aerodynamics, thiab Vladimir Bogolyubov - nyob rau hauv quantum teb ziag no.