C
{\displaystyle \mathbb {C} }
Kompleks taalen (mengde
C
{\displaystyle \mathbb {C} }
reel taalen uk jodiaren, diar dü ei üüb en taalstrual
Uun det mengde faan a reel taalen jaft at nian liasang för
x
{\displaystyle x}
liknang
x
2
+
1
=
0
{\displaystyle x^{2}+1=0}
x
=
−
1
{\displaystyle x={\sqrt {-1}}}
Diaram as det nuadag, en nei taal iintufeeren, det imagineer taal
i
{\displaystyle \mathrm {i} }
i
2
=
−
1
{\displaystyle \mathrm {i} ^{2}=-1}
i
{\displaystyle \mathrm {i} }
imagineer ianhaid .
Kompleks taalen kön uun det furem
a
+
b
⋅
i
{\displaystyle a+b\cdot \mathrm {i} }
a
{\displaystyle a}
b
{\displaystyle b}
i
{\displaystyle \mathrm {i} }
Tuuptäälen faan tau kompleks taalen uun't kompleks fial (det Gauß -fial Det tuuptäälen faan tau kompleks taalen
z
1
=
a
+
b
i
{\displaystyle z_{1}=a+b\,\mathrm {i} }
z
2
=
c
+
d
i
{\displaystyle z_{2}=c+d\,\mathrm {i} }
z
1
+
z
2
=
(
a
+
c
)
+
(
b
+
d
)
i
.
{\displaystyle z_{1}+z_{2}=(a+c)+(b+d)\,\mathrm {i} .}
Jüst so gongt det uftäälen faan
z
1
{\displaystyle z_{1}}
z
2
{\displaystyle z_{2}}
z
1
−
z
2
=
(
a
−
c
)
+
(
b
−
d
)
i
.
{\displaystyle z_{1}-z_{2}=(a-c)+(b-d)\,\mathrm {i} .}
Bi't moolnemen faan tau kompleks taalen
z
1
{\displaystyle z_{1}}
z
2
{\displaystyle z_{2}}
z
1
⋅
z
2
=
(
a
c
+
b
d
i
2
)
+
(
a
d
+
b
c
)
i
=
(
a
c
−
b
d
)
+
(
a
d
+
b
c
)
i
.
{\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
z
1
{\displaystyle z_{1}}
z
2
{\displaystyle z_{2}}
konjugiaret kompleks taal faan di dialer:
z
¯
2
=
c
−
d
i
{\displaystyle {\bar {z}}_{2}=c-d\,\mathrm {i} }
c
{\displaystyle c}
d
{\displaystyle d}
z
1
z
2
=
(
a
+
b
i
)
(
c
−
d
i
)
(
c
+
d
i
)
(
c
−
d
i
)
=
a
c
+
b
d
c
2
+
d
2
+
b
c
−
a
d
c
2
+
d
2
i
.
{\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} .}
(
3
+
2
i
)
+
(
5
+
5
i
)
=
(
3
+
5
)
+
(
2
+
5
)
i
=
8
+
7
i
{\displaystyle (3+2\mathrm {i} )+(5+5\mathrm {i} )=(3+5)+(2+5)\mathrm {i} =8+7\mathrm {i} }
(
5
+
5
i
)
−
(
3
+
2
i
)
=
(
5
−
3
)
+
(
5
−
2
)
i
=
2
+
3
i
{\displaystyle (5+5\mathrm {i} )-(3+2\mathrm {i} )=(5-3)+(5-2)\mathrm {i} =2+3\mathrm {i} }
(
3
+
5
i
)
⋅
(
4
+
11
i
)
=
(
3
⋅
4
−
5
⋅
11
)
+
(
3
⋅
11
+
5
⋅
4
)
i
=
−
43
+
53
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} }
(
2
+
5
i
)
(
3
+
7
i
)
=
(
2
+
5
i
)
(
3
+
7
i
)
⋅
(
3
−
7
i
)
(
3
−
7
i
)
=
(
6
+
35
)
+
(
15
i
−
14
i
)
(
9
+
49
)
+
(
21
i
−
21
i
)
=
41
+
i
58
=
41
58
+
1
58
⋅
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} }}
Wikibooks: Iinfeerang tu kompleks taalen Wikibooks: Kompleks taalen
Tekst üüb Öömrang
Commonskategorii: Kompleks taalen – Saamlang faan bilen of filmer
Wikibooks: Iinfeerang tu kompleks taalen (sjiisk)
