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Komplexkonjugatet av en summa

Komplexkonjugatet av en summa kan skrivas som summan av konjugaten.

Det här kan visas genom att skriva talen på rektangulär form, t.ex.

z_{1} = a + b i och z_{2} = c + d i,

och utveckla vänsterledet. Notera att med dessa val blir konjugaten

\overset{z}{ˉ}_{1} = a - b i

och

\overset{z}{ˉ}_{2} = c - d i .

z1+z2

z1=a+bi, z2=c+di

a+bi+c+di

\OT

a+c+bi+di

Bryt ut i

a+c+i(b+d)

a+bi=a−bi

a+c−i(b+d)

Multiplicera in i

a+c−(bi+di)

Ta bort parentes & byt tecken

a+c−bi−di

\OT

a−bi+c−di

$a - b i = \overset{z}{ˉ}_{1}$ , $c - d i = \overset{z}{ˉ}_{2}$

\overset{z}{ˉ}_{1} + \overset{z}{ˉ}_{2}

Alltså är

\overline{z_{1} + z_{2}} = \overset{z}{ˉ}_{1} + \overset{z}{ˉ}_{2}

Q.E.D.

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