De complex toegevoegde van een complex getal

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Uitweiding 1: Het tegengestelde van een reëel getal

Het tegengestelde $-a$ van een reëel getal $a$ is een erg eenvoudig begrip, want $-a$ is ’het getal $a$ met het omgekeerde teken’. Op de getallenrechte kan je $-a$ beschouwen als het spiegelbeeld van $a$ rond de oorsprong.

Het tegengestelde $-z$ van een complex getal kan onmiddellijk en zonder enig probleem worden gedefinieerd:als $z=a+bi$ , dan is $-z = -(a+bi) = -a -bi$ , en in het complexe vlak is $-z$ het spiegelbeeld van $z$ door de oorsprong:

Complexe getallen kan je spiegelen rond de $x$ -as, en dat zal blijken een bijzonder nuttige operatie te zijn, die geen equivalent heeft voor een reëel getal. Terwijl men bij spiegelen door de oorsprong spreekt over het tegengestelde $-z$ , noemt men de over de $x$ -as gespiegelde het complex toegevoegde $\overline {z}$ .

Complex toegevoegde van een complex getal

De complex toegevoegde van een complex getal $z=a+bi$ , genoteerd $\overline {z}$ , is het complexe getal

$\relax \fcolorbox {black}{white}{$\overline {z} \; \perdef \; \overline {a+bi} \; \perdef \; a-bi$}$

$\overline {3+4i} = 3 - 4i.$

We noemen $z$ en $\overline z$ complex toegevoegd (aan elkaar), en beide zijn elkaars spiegeling over de $x$ -as.

$\overline {3-4i} = \answer [onlineshowanswerbutton]{3+4i}$

$\overline {3+4i} = \answer [onlineshowanswerbutton]{3-4i}$

$\overline {-3-4i} = \answer [onlineshowanswerbutton]{-3+4i}$

$\overline {-3} = \answer [onlineshowanswerbutton]{-3}$

$\overline {i} = \answer [onlineshowanswerbutton]{-i}$

$\overline {-i} = \answer [onlineshowanswerbutton]{i}$

Volgende eigenschap, waarvan het bewijs een oefening is, geeft aan dat complex toevoegen compatibel is met optelling en vermenigvuldiging.

Eigenschappen van complex toevoegen Voor complexe getallen $z$ en $w$ geldt

$\relax \fcolorbox {black}{white}{$\overline {z+w} = \overline {z} + \overline {w}$}$

$\overline {(2+3i)+(4+5i)} = 6-8i = \overline {2+3i} + \overline {4+5i}$

$\relax \fcolorbox {black}{white}{$\overline {z\cdot w} = \overline {z} \cdot \overline {w}$}$

$\overline {(2+3i)\cdot (4+5i)} = -7-22i = \overline {2+3i} \cdot \overline {4+5i}$

Bewijs de vorige eigenschap.

Bereken voor een complex getal $z=a+bi \in \C$ volgende uitdrukkingen:

$z + \overline {z} = \answer [onlineshowanswerbutton]{2a}$

$z+ \overline {z} = (a+bi) + (a-bi) = 2a$

$z -\overline {z} = \answer [onlineshowanswerbutton]{2bi}$

$z- \overline {z} = (a+bi) - (a-bi) = 2bi$

$z\cdot \overline {z} = \answer [onlineshowanswerbutton]{a^2+b^2}$

$z\cdot \overline {z} = (a+bi)(a-bi) = a^2- i^2b^2 = a^2+b^2$

Hiermee zijn volgende eigenschappen aangetoond:

Eigenschappen van complex toevoegen Voor een complex getal $z=a+bi$ geldt

$\relax \fcolorbox {black}{white}{$\overline {\overline {z}} = \overline {(\,\overline {z}\,)} = z$}$

$\overline {(\overline {2+3i})} = \overline {2 - 3i} = 2+3i$

$\relax \fcolorbox {black}{white}{$z + \overline {z} = 2\Re (z)$}$

$2+3i\;+\; \overline {2+3i} = 2 + 3i + 2 - 3i = 4 = 2\Re (2+3i)$

$\relax \fcolorbox {black}{white}{$z - \overline {z} = 2\Im (z)$}$

$2+3i\;-\; \overline {2+3i} = 2 + 3i - 2 + 3i = 6i = 2i\Im (2+3i)$

$\relax \fcolorbox {black}{white}{$z\overline {z} = |z|^2$}$

$(2+3i)(2-3i) = 4 - 9i^2 = 4 +9 = 13 = |2+3i|^2$

Zowel de som als het product van een getal $z$ en zijn complex toegevoegde $\overline {z}$ zijn steeds reëel, en dat zal een manier opleveren om het inverse $z^{-1}$ te berekenen, en dus het quotiënt van twee complexe getallen.

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norm	$\|\|\blue{[?]}\|\|$
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sym_name	$\backslash\texttt{\blue{[?]}}$
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paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Uitweiding 1: Het tegengestelde van een reëel getal

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