Kompleks taal

Det tuuptäälen faan tau kompleks taalen ${\displaystyle z_{1}=a+b\,\mathrm {i} }$ an ${\displaystyle z_{2}=c+d\,\mathrm {i} }$ gongt so:

${\displaystyle z_{1}+z_{2}=(a+c)+(b+d)\,\mathrm {i} .}$

Jüst so gongt det uftäälen faan ${\displaystyle z_{1}}$ maner ${\displaystyle z_{2}}$:

${\displaystyle z_{1}-z_{2}=(a-c)+(b-d)\,\mathrm {i} .}$

Bi't moolnemen faan tau kompleks taalen ${\displaystyle z_{1}}$ an ${\displaystyle z_{2}}$ skel dü beaachte:

${\displaystyle z_{1}\cdot z_{2}=(ac+bd\,\mathrm {i} ^{2})+(ad+bc)\,\mathrm {i} =(ac-bd)+(ad+bc)\,\mathrm {i} .}$

Bi't dialen faan en kompleks taal ${\displaystyle z_{1}}$ troch ${\displaystyle z_{2}}$ skel dü di bröök iarst ütjwidje mä det konjugiaret kompleks taal faan di dialer: ${\displaystyle {\bar {z}}_{2}=c-d\,\mathrm {i} }$. Sodenang woort di dialer reel (an as jüst det kwadroot faan ${\displaystyle c}$ an ${\displaystyle d}$):

${\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {(a+b\,\mathrm {i} )(c-d\,\mathrm {i} )}{(c+d\,\mathrm {i} )(c-d\,\mathrm {i} )}}={\frac {ac+bd}{c^{2}+d^{2}}}+{\frac {bc-ad}{c^{2}+d^{2}}}\mathrm {i} .}$

${\displaystyle (3+2\mathrm {i} )+(5+5\mathrm {i} )=(3+5)+(2+5)\mathrm {i} =8+7\mathrm {i} }$

${\displaystyle (5+5\mathrm {i} )-(3+2\mathrm {i} )=(5-3)+(5-2)\mathrm {i} =2+3\mathrm {i} }$

${\displaystyle (3+5\mathrm {i} )\cdot (4+11\mathrm {i} )=(3\cdot 4-5\cdot 11)+(3\cdot 11+5\cdot 4)\mathrm {i} =-43+53\mathrm {i} }$

${\displaystyle {{\frac {(2+5\mathrm {i} )}{(3+7\mathrm {i} )}}={\frac {(2+5\mathrm {i} )}{(3+7\mathrm {i} )}}\cdot {\frac {(3-7\mathrm {i} )}{(3-7\mathrm {i} )}}={\frac {(6+35)+(15\mathrm {i} -14\mathrm {i} )}{(9+49)+(21\mathrm {i} -21\mathrm {i} )}}={\frac {41+\mathrm {i} }{58}}={\frac {41}{58}}+{\frac {1}{58}}\cdot \mathrm {i} }}$