Cov lej nyuaj
Cov lej sib xyaw yog ib lub tswv yim lej uas ua lub luag haujlwm tseem ceeb hauv ntau ceg ntawm kev tshawb fawb, xws li physics, engineering, economics, thiab, ntawm chav kawm, lej nws tus kheej. Raws li kev nthuav dav ntawm cov lej tiag tiag uas peb paub hauv lub neej txhua hnub, cov lej sib xyaw qhia txog qhov tshiab rau peb txoj kev nkag siab thiab qauv ntau yam xwm txheej.
Keeb Kwm ntawm Cov Naj Npawb Sib Xyaws
Cov lej nyuaj thawj zaug tshwm sim los ntawm qhov xav tau nrhiav cov kev daws teeb meem rau cov lej quadratic uas tsis muaj cov kev daws teeb meem hauv cov lej tiag. Txij li thaum ub los, cov kws lej tau ntsib teeb meem xws li cov lej quadratic \(x^2 + 1 = 0\), uas tsis muaj cov hauv paus tiag. Qhov no yog vim rau txhua tus lej tiag \(x\), \(x^2\) yeej tsis yog qhov tsis zoo, yog li \(x^2 + 1\) yeej tsis tuaj yeem yog xoom.
Kev nkag siab tob txog cov lej sib xyaw tau pib tsim kho hauv xyoo pua 16th ua tsaug rau cov haujlwm ntawm cov kws lej European xws li Girolamo Cardano, uas siv cov hauv paus xav hauv kev daws teeb meem ntawm qee qhov kev sib npaug. Hauv xyoo pua 18th thiab 19th, cov kws lej xws li Leonhard Euler thiab Carl Friedrich Gauss tau tsim cov hauv paus ntawm kev xav txog cov lej sib xyaw, muab cov lus piav qhia zoo dua thiab qhia ntau yam ntawm cov cim uas tseem siv niaj hnub no.
Cov Lus Txhais thiab Cov Lus Cim
Ib tus lej nyuaj muaj ob feem: ib feem tiag tiag thiab ib feem xav xwb. Feem ntau, ib tus lej nyuaj tuaj yeem sau ua daim ntawv \(a + bi\), qhov twg:
– \(a\) yog qhov tiag tiag.
- \(b\) yog ib feem xav xwb.
- \(i\) yog lub chav xav xwb, txhais tias yog \(\sqrt{-1}\).
Piv txwv li, nyob rau hauv tus lej nyuaj \(4 + 3i\):
- Qhov tseeb tiag yog \(4\).
- Qhov seem xav yog \(3i\).
Kauj ruam thawj zaug hauv kev nkag siab txog cov lej nyuaj yog lees txais tias \(i\) muaj ib qho khoom nthuav heev: \(i^2 = -1\).
Kev Ua Haujlwm Yooj Yim ntawm Cov Naj Npawb Sib Xyaws
Ib yam li cov lej tiag tiag, peb tuaj yeem ua ntau yam haujlwm yooj yim rau cov lej nyuaj, xws li kev ntxiv, kev rho tawm, kev sib npaug, thiab kev faib.
Kev Ntxiv thiab Kev Rho Tawm
Yuav ntxiv ob tus lej nyuaj, peb tsuas yog ntxiv lawv cov feem tiag tiag thiab xav xwb. Piv txwv li, rau ob tus lej nyuaj \(z_1 = a + bi\) thiab \(z_2 = c + di\):
\[ z_1 + z_2 = (a + c) + (b + d)i \]
Kev rho tawm yog ua tiav tib yam, uas yog rho tawm qhov tiag tiag thiab qhov xav xwb:
\[ z_1 – z_2 = (a – c) – (b – d)i \]
Kev sib npaug
Kev sib npaug ntawm cov lej nyuaj nyuaj dua me ntsis, vim peb yuav tsum sib npaug ob qho tib si cov khoom tiag tiag thiab cov khoom xav xwb, thiab kuj xav txog cov yam ntxwv ntawm \(i\). Rau ob tus lej nyuaj \(z_1 = a + bi\) thiab \(z_2 = c + di\):
\[ z_1 \cdot z_2 = (a + bi)(c + di) = ac + adi + bci + bdi^2 \]
Nco ntsoov tias \(i^2 = -1\), yog li peb tuaj yeem ua kom yooj yim rau:
\[ z_1 \cdot z_2 = (ac – bd) + (ad + bc)i \]
Kev faib tawm
Yuav faib ob tus lej nyuaj, peb siv lub tswv yim ntawm conjugates. Tus conjugate ntawm tus lej nyuaj \(a + bi\) yog \(a – bi\). Xav tias peb xav faib \(z_1 = a + bi\) los ntawm \(z_2 = c + di\):
\[ \frac{z_1}{z_2} = \frac{a + bi}{c + di} \]
Yuav kom yooj yim dua, peb muab tus lej suav thiab tus lej faib ua feem ntau los ntawm tus lej sib txuas ntawm tus lej faib ua feem:
\[ \frac{z_1}{z_2} = \frac{(a + bi)(c – di)}{(c + di)(c – di)} = \frac{(ac + bd) + (bc – ad)i}{c^2 + d^2} \]
Kev Sawv Cev Geometric
Cov lej sib xyaw kuj tseem tuaj yeem sawv cev geometrically hauv lub dav hlau sib xyaw, qhov twg lub axis kab rov tav sawv cev rau qhov tiag tiag thiab lub axis ntsug sawv cev rau qhov xav txog. Qhov no zoo ib yam li Cartesian coordinate system uas feem ntau siv hauv geometry.
Cov ces kaum thiab qhov ntev hauv daim ntawv no kuj muaj kev txhais lus. Qhov ntev lossis modulus ntawm tus lej nyuaj \(z = a + bi\) yog qhov deb ntawm qhov ntawd mus rau qhov pib (0,0), thiab tuaj yeem suav nrog siv Pythagorean theorem:
\[ |z| = \sqrt{a^2 + b^2} \]
Lub caij no, lub kaum sab xis lossis kev sib cav ntawm tus lej nyuaj yog lub kaum sab xis uas tsim los ntawm txoj kab txuas qhov taw tes mus rau keeb kwm nrog lub axis tiag tiag, uas tau qhia ua radians.
Cov Kev Siv Tus Naj Npawb Sib Xyaws
Cov lej sib xyaw muaj ntau yam kev siv tau, txij li kev tsim kho vaj tse mus txog rau quantum physics. Qee qhov piv txwv ntawm kev siv cov lej sib xyaw suav nrog:
Kev Tsim Kho Hluav Taws Xob thiab Khoom Siv Hluav Taws Xob
Hauv kev tshuaj xyuas AC (Alternating Current) circuit, cov lej sib xyaw yog siv los sawv cev rau impedance, voltage, thiab tam sim no. Impedance hauv cov ntsiab lus no yog qhov ntsuas nyuaj ntawm kev tiv thaiv uas suav nrog tsis yog tsuas yog kev tiv thaiv ntshiab tab sis kuj tseem muaj reactance.
Kev Tshawb Fawb Txog Quantum
Hauv quantum physics, lub wave function uas piav qhia txog lub xeev ntawm ib qho subatomic particle feem ntau yog qhia ua tus lej complex. Lub wave function no ua lub luag haujlwm tseem ceeb hauv kev txiav txim siab qhov probability ntawm ib qho particle qhov chaw thiab lub zog uas nws muaj nyob hauv ib lub system.
Kev Ua Haujlwm Teeb Liab
Hauv kev ua cov teeb liab, Fourier Transform yog ib qho cuab yeej tseem ceeb uas siv cov lej nyuaj. Fourier Transform rhuav cov teeb liab lub sijhawm ua cov khoom sib xyaw zaus uas tuaj yeem tshuaj xyuas thiab hloov kho sib cais.
Kev Siv Tshuab Ua Kua thiab Aerodynamics
Hauv kev kho tshuab kua, cov lej nyuaj siv los daws ntau yam teeb meem uas cuam tshuam nrog kev ntws ob sab. Txoj kev ua haujlwm muaj peev xwm nyuaj pab txiav txim siab cov qauv ntws thiab siv cov tswv yim aerodynamic.
Xaus
Cov lej sib xyaw yog ib lub tswv yim muaj zog thiab siv tau ntau yam hauv kev lej. Txawm hais tias thaum xub thawj lawv zoo li tsis muaj tseeb thiab tsis nyob hauv qhov tseeb txhua hnub, lawv cov kev siv hauv ntau yam kev tshawb fawb qhia txog qhov tseem ceeb ntawm kev nkag siab thiab kev ua tus tswv ntawm lub tswv yim no.
Nrog keeb kwm nplua nuj thiab kev siv dav dav, cov lej sib xyaw tsis yog tsuas yog nthuav dav kev suav lej xwb tab sis kuj tau tsim txoj hauv kev rau ntau yam kev tsim kho tshiab thiab kev tshawb pom hauv kev tshawb fawb thiab thev naus laus zis. Raws li kev nthuav dav ntawm lub kaw lus lej tiag tiag, cov lej sib xyaw muab cov khoom tseem ceeb rau kev tshuaj xyuas thiab kev daws teeb meem nyuaj dua hauv lub neej tiag tiag.
