U skupu realnih brojeva
R
{\displaystyle \R}
jednačina
x
2
−
1
=
0
{\displaystyle x^{2}-1=0}
x
2
−
1
=
0
=>
x
=
±
1
{\displaystyle x^{2}-1=0=>x=\pm 1}
Slična jednačina
x
2
+
1
=
0
{\displaystyle x^2+1=0}
R
{\displaystyle \R}
i
{\displaystyle i}
i
2
=
−
1
{\displaystyle i^2=-1}
x
2
+
1
=
0
=>
x
=
±
1
{\displaystyle x^{2}+1=0=>x=\pm 1}
i
2
=
−
1
,
i
3
=
−
i
,
i
4
=
−
i
⋅
i
=
−
(
−
1
)
=
1
,
i
5
=
i
,
i
6
=
−
1
,
…
{\displaystyle \displaystyle i^{2}=-1,\quad i^{3}=-i,\quad i^{4}=-i\cdot i=-(-1)=1,\quad i^{5}=i,\quad i^{6}=-1,\ldots }
Na ovaj način dobili smo skup kompleksnih brojeva.
Kompleksan broj je broj oblika
x
+
y
i
{\displaystyle x + yi\,}
gdje su
x
{\displaystyle x}
y
{\displaystyle y}
realni brojevi , a
i
{\displaystyle i}
i
2
=
−
1
{\displaystyle i^2=-1}
x
{\displaystyle x}
R
e
(
z
)
{\displaystyle Re(z)}
y
{\displaystyle y}
I
m
(
z
)
{\displaystyle Im(z)}
Skup kompleksnih brojeva možemo smatrati proširenjem skupa realnih brojeva, odnosno svaki realni broj
x
{\displaystyle x}
y
=
0
{\displaystyle y=0}
x
=
x
+
0
i
{\displaystyle x=x+0i}
Notacija [ ] Povremeno se može naići na definiciju
i
=
−
1
{\displaystyle i=\sqrt{-1}}
−
1
∗
−
1
=
1
=
1
{\displaystyle \sqrt{-1}*\sqrt {-1}=\sqrt{1}=1}
i
=
−
1
{\displaystyle i=\sqrt{-1}}
i
∗
i
=
−
1
{\displaystyle i*i=-1}
vektori ili uređeni parovi
(
x
,
y
)
{\displaystyle (x,y)}
Definicija [ ] Definiciju kompleksnih brojeva kao uređenih parova dao je William R. Hamilton , irski matematičar (1805– 1865.) Ta se definicija temelji samo na sosobini realnih brojeva, čime se izbjegava donekle nerazjašnjeni pojam broja
1
{\displaystyle {\sqrt {1}}}
z
=
x
+
y
i
{\displaystyle z = x + yi}
z
=
x
+
y
i
{\displaystyle z = x + yi}
z
=
(
x
,
y
)
{\displaystyle z=(x,y)}
Skup kompleksnih brojeva
C
{\displaystyle \mathbb{C}}
z
=
x
+
i
y
{\displaystyle z = x + iy}
x
,
y
∈
R
{\displaystyle x, y \in \R}
Posebno je
0
=
0
+
i
0
{\displaystyle 0=0+i0}
x
=
R
e
(
z
)
{\displaystyle x=\mathrm {Re} (z)}
z
{\displaystyle z}
y
=
I
m
z
{\displaystyle y=\mathrm {Im} z}
z
{\displaystyle z}
Algebarski oblik kompleksnog broja je
z
=
x
+
i
y
{\displaystyle z = x + iy}
x
,
y
∈
R
{\displaystyle x, y \in \R}
Trigonometrijski oblik kompleksnog broja je
z
=
r
(
c
o
s
θ
+
i
s
i
n
θ
)
,
r
≥
0
,
θ
∈
R
{\displaystyle z=r(cos\theta +isin\theta ),r\geq 0,\theta \in \mathbb {R} }
pri čemu je
r
=∣
z
∣
{\displaystyle r=\mid z\mid }
θ
=
a
r
g
z
{\displaystyle \theta =arg\ z}
Eksponencijalni oblik kompleksnog broja je
z
=
r
i
θ
{\displaystyle z=r^{i\theta }}
r
≥
0
,
θ
∈
R
{\displaystyle r\geq 0,\theta \in \mathbb {R} }
pri čemu je
r
=∣
z
∣
{\displaystyle r=\mid z\mid }
θ
=
a
r
g
z
{\displaystyle \theta =argz}
Dva kompleksna broja su jednaka ako su im jednaki realni i imaginarni dijelovi.
z
1
=
z
2
↔
(
Re
(
z
1
)
=
Re
(
z
2
)
∧
Im
(
z
1
)
=
Im
(
z
2
)
)
.
{\displaystyle z_{1}=z_{2}\,\,\leftrightarrow \,\,(\operatorname {Re} (z_{1})=\operatorname {Re} (z_{2})\,\land \,\operatorname {Im} (z_{1})=\operatorname {Im} (z_{2})).}
Konjugirano kompleksni broj broja
z
=
x
+
i
y
{\displaystyle z = x + iy}
z
¯
=
x
−
i
y
{\displaystyle \bar z=x-iy}
Modul ili apsolutna vrijednost kompleksnog broja
z
{\displaystyle z}
r
=
|
z
|
=
x
2
+
y
2
{\displaystyle r=\vert z\vert ={\sqrt {x^{2}+y^{2}}}}
Sabiranje i oduzimanje kompleksnih brojeva [ ]
sabiranje .
Neka su
z
1
=
x
1
+
i
y
1
{\displaystyle z_{1} = x_{1} + iy_{1}}
z
2
=
x
2
+
i
y
2
{\displaystyle z_{2} = x_{2} +iy_{2}}
z
1
+
z
2
=
x
1
+
x
2
+
i
(
y
1
+
y
2
)
{\displaystyle \displaystyle z_{1}+z_{2}\displaystyle =x_{1}+x_{2}+i(y_{1}+y_{2})}
i oduzimanje
z
1
−
z
2
=
x
1
−
x
2
+
i
(
y
1
−
y
2
)
{\displaystyle \displaystyle z_{1}-z_{2}\displaystyle =x_{1}-x_{2}+i(y_{1}-y_{2})}
Osobine sabiranja kompleksnih brojeva [ ]
(
z
1
+
z
2
=
z
2
+
z
1
{\displaystyle (z_{1}+z_{2}=z_{2}+z_{1}}
∀
z
1
,
z
2
∈
C
{\displaystyle \forall z_{1},z_{2}\in \mathbb {C} }
komutativnost sabiranja
z
1
+
(
z
2
+
z
3
)
=
(
z
1
+
z
2
)
+
z
3
,
{\displaystyle z_{1}+(z_{2}+z_{3})=(z_{1}+z_{2})+z_{3},}
∀
z
1
,
z
2
,
z
3
∈
C
{\displaystyle \forall z_{1},z_{2},z_{3}\in \mathbb {C} }
asocijativnost sabiranja
∃
0
∈
C
z
+
0
=
z
{\displaystyle \exists 0\in \mathbb {C} z+0=z}
∀
z
∈
C
{\displaystyle \forall z\in \mathbb {C} }
Kompleksni broj
0
=
(
0
,
0
)
=
0
+
0
i
{\displaystyle 0=(0,0)=0+0i}
(
∀
z
∈
C
)
(
∃
(
−
z
)
∈
C
z
+
(
−
z
)
=
0
{\displaystyle (\forall z\in \mathbb {C} )(\exists (-z)\in \mathbb {C} z+(-z)=0}
Kompleksni broj
−
z
=
(
−
x
,
−
y
)
=
−
x
−
y
i
{\displaystyle -z=(-x,-y)=-x-yi}
Množenje kompleksnih brojeva [ ] Neka su
z
1
=
x
1
+
i
y
1
{\displaystyle z_{1} = x_{1} + iy_{1}}
z
2
=
x
2
+
i
y
2
{\displaystyle z_{2} = x_{2} +iy_{2}}
U skupu kompleksnih brojeva definisano je množenje
z
1
⋅
z
2
=
(
x
1
+
i
y
1
)
(
x
2
+
i
y
2
)
=
x
1
x
2
+
i
y
1
x
2
+
i
x
1
y
2
+
i
2
y
1
y
2
=
x
1
x
2
−
y
1
y
2
+
i
(
x
1
y
2
+
x
2
y
1
)
{\displaystyle \displaystyle z_{1}\cdot z_{2}\displaystyle =(x_{1}+iy_{1})(x_{2}+iy_{2})=x_{1}x_{2}+iy_{1}x_{2}+ix_{1}y_{2}+i^{2}y_{1}y_{2}\displaystyle =x_{1}x_{2}-y_{1}y_{2}+i(x_{1}y_{2}+x_{2}y_{1})}
Osobine množenja kompleksnih brojeva [ ]
(
z
1
∗
z
2
=
z
2
∗
z
1
{\displaystyle (z_{1}*z_{2}=z_{2}*z_{1}}
∀
z
1
,
z
2
∈
C
{\displaystyle \forall z_{1},z_{2}\in \mathbb {C} }
z
1
∗
(
z
2
+
z
3
)
=
(
z
1
+
z
2
)
+
z
3
{\displaystyle z_{1}*(z_{2}+z_{3})=(z_{1}+z_{2})+z_{3}}
∀
z
1
,
z
2
,
z
3
∈
C
{\displaystyle \forall z_{1},z_{2},z_{3}\in \mathbb {C} }
∃
1
∈
C
z
∗
1
=
z
{\displaystyle \exists 1\in \mathbb {C} z*1=z}
∀
z
∈
C
{\displaystyle \forall z\in \mathbb {C} }
1
{\displaystyle 1}
∀
z
∈
C
)
(
z
≠
0
)
(
∃
z
′
∈
C
z
∗
(
−
z
)
=
1
{\displaystyle \forall z\in \mathbb {C} )(z\neq 0)(\exists z'\in \mathbb {C} z*(-z)=1}
z
1
∗
(
z
2
+
z
3
)
=
z
1
∗
z
2
+
z
1
∗
z
3
{\displaystyle z_{1}*(z_{2}+z_{3})=z_{1}*z_{2}+z_{1}*z_{3}}
∀
z
1
,
z
2
,
z
3
∈
C
{\displaystyle \forall z_{1},z_{2},z_{3}\in \mathbb {C} }
Realan proizvod dva kompleksna broja [ ] U skupu kompleksnih brojeva skalarnom proizvodu vektora odgovara pojam realnog proizvoda kompleksnih brojeva koji je skalarni proizvod vektora koji su određeni kompleksnim brojevima koji se množe.
Definicija
Realan proizvod kompleksnih brojeva
a
{\displaystyle a}
b
{\displaystyle b}
a
∘
b
{\displaystyle a\circ b}
a
∘
b
=
1
2
(
a
¯
b
+
a
b
¯
)
{\displaystyle a\circ b={\frac {1}{2}}({\overline {a}}b+a{\overline {b}})}
Neka su A i B tačke određene kompleksnim brojevima
a
=∣
a
∣
(
c
o
s
φ
+
i
s
i
n
φ
)
{\displaystyle a=\mid a\mid (cos\varphi +isin\varphi )}
b
=∣
b
∣
(
c
o
s
ψ
+
i
s
i
n
ψ
)
{\displaystyle b=\mid b\mid (cos\psi +isin\psi )}
a
∘
b
=∣
a
∣∣
b
∣
(
c
o
s
φ
+
i
s
i
n
ψ
)
=∣
O
A
∣∣
A
B
∣
c
o
s
A
O
B
^
{\displaystyle a\circ b=\mid a\mid \mid b\mid (cos\varphi +isin\psi )=\mid OA\mid \mid AB\mid cos{\widehat {AOB}}}
Osobine realnog proizvoda dva kompleksna broja
a
∘
a
=∣
a
∣
2
{\displaystyle a\circ a=\mid a\mid ^{2}}
a
∘
b
=
b
∘
a
{\displaystyle a\circ b = b\circ a}
a
∘
b
¯
=
a
∘
b
{\displaystyle {\overline {a\circ b}}=a\circ b}
(
α
a
)
∘
b
=
α
(
a
∘
b
)
=
a
∘
(
α
b
)
{\displaystyle (\alpha a)\circ b=\alpha (a\circ b)=a\circ (\alpha b)}
(
a
z
)
)
b
z
)
=∣
z
∣
2
(
a
∘
b
)
{\displaystyle (az))bz)=\mid z\mid ^{2}(a\circ b)}
a
∘
b
=
0
<=>
O
A
⊥
O
B
{\displaystyle a\circ b=0<=>OA\perp OB}
a
{\displaystyle a}
b
{\displaystyle b}
Realan proizvod kompleksnih brojeva
a
{\displaystyle a}
b
{\displaystyle b}
O
{\displaystyle O}
A
B
{\displaystyle AB}
A
{\displaystyle A}
B
{\displaystyle B}
a
{\displaystyle a}
b
{\displaystyle b}
Tačka
M
{\displaystyle M}
a
+
b
2
{\displaystyle \frac{a+b}{2}}
O
{\displaystyle O}
M
{\displaystyle M}
r
=
a
−
b
2
=
∣
a
−
b
∣
2
{\displaystyle r={\frac {a-b}{2}}={\frac {\mid a-b\mid }{2}}}
O
M
2
−
r
2
=∣
a
+
b
2
∣
−
∣
a
−
b
2
∣=
(
a
+
b
)
(
a
¯
+
b
¯
4
−
(
a
−
b
)
(
a
¯
−
b
¯
)
4
=
a
∘
b
{\displaystyle OM^{2}-r^{2}=\mid {\frac {a+b}{2}}\mid -\mid {\frac {a-b}{2}}\mid ={\frac {(a+b)({\overline {a}}+{\overline {b}}}{4}}-{\frac {(a-b)({\overline {a}}-{\overline {b}})}{4}}=a\circ b}
Neka su tačke
A
{\displaystyle A}
B
{\displaystyle B}
C
{\displaystyle C}
D
{\displaystyle D}
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
d
{\displaystyle d}
A
B
⊥
C
D
{\displaystyle AB\perp CD}
(
a
+
b
)
∘
(
c
+
d
)
=
0
{\displaystyle (a+b)\circ (c+d)=0}
b
−
a
d
−
c
∈
i
R
{
0
}
{\displaystyle {\frac {b-a}{d-c}}\in i\mathbb {R} \ {\begin{Bmatrix}0\end{Bmatrix}}}
R
e
(
b
−
a
d
−
c
)
=
0
{\displaystyle Re({\frac {b-a}{d-c}})=0}
Središte kružnice opisane oko trougla
A
B
C
{\displaystyle ABC}
A
{\displaystyle A}
B
{\displaystyle B}
C
{\displaystyle C}
A
B
C
{\displaystyle ABC}
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
H
{\displaystyle H}
h
=
a
+
b
+
c
{\displaystyle h=a+b+c}
Kompleksan proizvod dva kompleksna broja [ ] Kompleksan proizvod dva kompleksna broja je analogan vektorskom proizvodu vektora.
Definicija Kompleksan broj
a
×
b
=
a
¯
b
−
a
b
¯
2
{\displaystyle a\times b={\frac {{\overline {a}}b-a{\overline {b}}}{2}}}
a
{\displaystyle a}
b
{\displaystyle b}
Neka su
A
{\displaystyle A}
B
{\displaystyle B}
a
=∣
a
∣
(
c
o
s
φ
+
i
s
i
n
φ
)
{\displaystyle a=\mid a\mid (cos\varphi +isin\varphi )}
a
=∣
b
∣
(
c
o
s
ψ
+
i
s
i
n
ψ
)
{\displaystyle a=\mid b\mid (cos\psi +isin\psi )}
∣
a
×
b
∣=∣
a
∣∣
b
∣
s
i
n
(
φ
−
ψ
)
=∣
O
A
∣∣
A
B
∣
s
i
n
A
O
B
^
=
2
P
A
O
B
{\displaystyle \mid a\times b\mid =\mid a\mid \mid b\mid sin(\varphi -\psi )=\mid OA\mid \mid AB\mid sin{\widehat {AOB}}=2P_{AOB}}
Neka su
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
a
×
b
¯
=
−
a
×
b
{\displaystyle {\overline {a\times b}}=-a\times b}
a
×
b
=
0
<=>
a
=
0
∨
b
=
0
∨
a
=
λ
b
{\displaystyle a\times b=0<=>a=0\lor b=0\lor a=\lambda b}
λ
∈
R
{
0
}
{\displaystyle \lambda \in \mathbb {R} \ {\begin{Bmatrix}0\end{Bmatrix}}}
a
×
b
=
−
b
×
a
{\displaystyle a\times b=-b\times a}
α
(
a
×
b
)
=
(
α
a
)
×
b
=
a
×
(
α
b
)
{\displaystyle \alpha (a\times b)=(\alpha a)\times b=a\times (\alpha b)}
∀
α
∈
R
{\displaystyle \forall\alpha\in\mathbb{R}}
Ako su
A
(
a
)
{\displaystyle A(a)}
B
(
b
)
{\displaystyle B(b)}
O
(
0
)
{\displaystyle O(0)}
a
×
b
=
0
{\displaystyle a\times b=0}
O
{\displaystyle O}
A
{\displaystyle A}
B
{\displaystyle B}
Neka su
A
(
a
{\displaystyle A(a}
B
(
b
{\displaystyle B(b}
a
{\displaystyle a}
b
{\displaystyle b}
a
×
b
=
{
2
i
P
A
O
B
z
a
t
r
o
u
g
a
o
A
O
B
p
o
z
i
t
i
v
n
o
o
r
i
j
e
n
t
i
s
a
n
−
2
i
P
A
O
B
z
a
t
r
o
u
g
a
o
A
O
B
n
e
g
a
t
i
v
n
o
o
r
i
j
e
n
t
i
s
a
n
{\displaystyle a\times b={\begin{cases}2iP_{AOB}\ za\ trougao\ AOB\ pozitivno\ orijentisan\\-2iP_{AOB}\ za\ trougao\ AOB\ negativno\ orijentisan\end{cases}}}
A
(
a
)
{\displaystyle A(a)}
B
(
b
)
{\displaystyle B(b)}
C
(
c
)
{\displaystyle C(c)}
P
A
B
C
=
{
1
2
(
a
×
b
+
b
×
c
+
c
×
a
)
a
k
o
j
e
A
B
C
p
o
z
i
t
i
v
n
o
o
r
j
e
n
t
i
s
a
n
1
2
(
a
×
b
+
b
×
c
+
c
×
a
)
a
k
o
j
e
A
B
C
n
e
g
a
t
i
v
n
o
o
r
j
e
n
t
i
s
a
n
{\displaystyle P_{ABC}={\begin{cases}{\frac {1}{2}}(a\times b+b\times c+c\times a)\ ako\ je\ ABC\ pozitivno\ orjentisan\\{\frac {1}{2}}(a\times b+b\times c+c\times a)\ ako\ je\ ABC\ negativno\ orjentisan\end{cases}}}
Neka su
A
(
a
)
{\displaystyle A(a)}
B
(
b
)
{\displaystyle B(b)}
C
(
c
)
{\displaystyle C(c)}
Tačke
A
{\displaystyle A}
B
{\displaystyle B}
C
{\displaystyle C}
(
b
−
a
)
×
(
c
−
a
)
=
0
{\displaystyle (b-a)\times (c-a)=0}
a
×
b
+
b
×
c
+
c
×
a
=
0
{\displaystyle a\times b+b\times c+c\times a=0}
Neka su
A
(
a
)
{\displaystyle A(a)}
B
(
b
)
{\displaystyle B(b)}
C
(
c
)
{\displaystyle C(c)}
D
(
d
)
{\displaystyle D(d)}
A
B
∥
C
D
{\displaystyle AB \parallel CD }
(
b
−
a
)
×
(
d
−
c
)
=
0
{\displaystyle (b-a)\times (d-c)=0}
Dijeljenje kompleksnih brojeva [ ]
z
1
z
2
=
x
1
+
i
y
1
x
2
+
i
y
2
⋅
x
2
−
i
y
2
x
2
−
i
y
2
=
x
1
x
2
+
y
1
y
2
x
2
2
+
y
2
2
+
i
y
1
x
2
−
x
1
y
2
x
2
2
+
y
2
2
,
za
z
2
≠
0
{\displaystyle \displaystyle {\frac {z_{1}}{z_{2}}}\displaystyle ={\frac {x_{1}+iy_{1}}{x_{2}+iy_{2}}}\cdot {\frac {x_{2}-iy_{2}}{x_{2}-iy_{2}}}\displaystyle ={\frac {x_{1}x_{2}+y_{1}y_{2}}{x_{2}^{2}+y_{2}^{2}}}+i{\frac {y_{1}x_{2}-x_{1}y_{2}}{x_{2}^{2}+y_{2}^{2}}},\quad {\textrm {za}}\quad z_{2}\neq 0}
∀
z
∈
C
)
(
z
≠
0
)
∃
z
′
∈
C
{\displaystyle \forall z\in \mathbb {C} )(z\neq 0)\exists z'\in \mathbb {C} }
Neka je
z
=
x
+
y
i
≠
0
{\displaystyle z=x+yi\neq 0}
x
2
+
y
2
≠
0
{\displaystyle x^{2}+y^{2}\neq 0}
z
′
=
x
x
2
+
y
2
+
−
y
x
2
+
y
2
i
{\displaystyle z'={\frac {x}{x^{2}+y^{2}}}+{\frac {-y}{x^{2}+y^{2}}}i}
1
z
=
z
¯
z
z
¯
=
z
¯
x
2
+
y
2
=
x
x
2
+
y
2
−
y
x
2
+
y
2
i
.
{\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.}
imamo
z
′
∗
z
=
z
∗
z
′
=
1
{\displaystyle z'*z=z*z'=1}
z
′
=
z
−
1
=
1
z
{\displaystyle z'=z^{-1}={\frac {1}{z}}}
Konjugovano kompleksni brojevi [ ] Kompleksan broj
z
¯
=
x
−
y
i
=
r
−
i
θ
{\displaystyle {\overline {z}}\ =x-yi=r^{-i\theta }}
z
=
x
+
y
i
=
r
i
θ
{\displaystyle z=x+yi=r^{i\theta }}
z
{\displaystyle z}
z
¯
{\displaystyle \overline{z}}
R
e
z
=
1
2
(
z
+
z
¯
{\displaystyle Rez={\frac {1}{2}}(z+{\overline {z}}}
I
m
z
=
1
2
i
(
z
−
z
¯
{\displaystyle Imz={\frac {1}{2i}}(z-{\overline {z}}}
Lako se provjerava da vrijedi
z
1
+
z
2
¯
=
z
1
¯
+
z
2
¯
{\displaystyle \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}}
z
1
−
z
2
¯
=
z
1
¯
−
z
2
¯
{\displaystyle {\overline {z_{1}-z_{2}}}={\overline {z_{1}}}-{\overline {z_{2}}}}
z
1
∗
z
2
¯
=
z
1
¯
∗
z
2
¯
{\displaystyle {\overline {z_{1}*z_{2}}}={\overline {z_{1}}}*{\overline {z_{2}}}}
(
z
1
z
2
)
¯
=
z
1
¯
z
1
¯
{\displaystyle {\overline {({\frac {z_{1}}{z_{2}}})}}={\frac {\overline {z_{1}}}{\overline {z_{1}}}}}
Moavrova formula [ ] Neka je
z
=
r
(
c
o
s
θ
+
i
s
i
n
θ
)
=
r
c
i
s
θ
{\displaystyle z=r(cos\theta +isin\theta )=r\ cis\theta }
z
2
=
z
∗
z
{\displaystyle z^{2}=z*z}
z
2
=
r
c
i
s
θ
∗
r
c
i
s
θ
=
r
2
c
i
s
(
θ
+
θ
)
=
r
2
c
i
s
2
θ
{\displaystyle z^{2}=r\ cis\theta *r\ cis\theta =r^{2}\ cis(\theta +\theta )=r^{2}\ cis2\theta }
z
3
=
r
2
c
i
s
2
θ
∗
r
c
i
s
θ
=
r
3
c
i
s
(
2
θ
+
θ
)
=
r
3
c
i
s
3
θ
{\displaystyle z^{3}=r^{2}\ cis2\theta *r\ cis\theta =r^{3}\ cis(2\theta +\theta )=r^{3}\ cis3\theta }
z
4
=
r
3
c
i
s
3
θ
∗
r
c
i
s
θ
=
r
4
c
i
s
(
3
θ
+
θ
)
=
r
4
c
i
s
4
θ
{\displaystyle z^{4}=r^{3}\ cis3\theta *r\ cis\theta =r^{4}\ cis(3\theta +\theta )=r^{4}\ cis4\theta }
Na ovaj način dobijamo opšti oblik De Moavrova teorema koji ima važnu ulogu u elektrotehnici
z
n
=
r
n
c
i
s
n
θ
{\displaystyle z^{n}=r^{n}\ cisn\theta }
(
c
o
s
θ
+
i
s
i
n
θ
)
n
=
c
o
s
n
θ
+
i
s
i
n
n
θ
(
n
∈
Z
)
{\displaystyle (cos\theta +isin\theta )^{n}=cosn\theta +isinn\theta (n\in Z)}
Stepenovanje kompleksnog broja [ ]
z
n
=
r
n
(
c
o
s
n
θ
+
i
s
i
n
n
θ
)
=
r
n
e
i
n
θ
{\displaystyle z^{n}=r^{n}(cosn\theta +isinn\theta )=r^{n}e^{in\theta }}
n
∈
N
{\displaystyle n \in N}
z
m
z
n
=
z
m
+
n
{\displaystyle z^{m}z^{n}=z^{m+n}}
(
z
1
z
2
)
n
=
(
z
1
z
2
)
n
{\displaystyle (z_{1}z_{2})^{n}=(z_{1}z_{2})^{n}}
(
z
m
)
n
=
z
m
n
{\displaystyle (z^{m})^{n}=z^{mn}}
Korjenovanje kompleksnog broja [ ]
z
n
=
{
u
0
,
u
1
.
.
.
u
n
}
{\displaystyle {\sqrt[{n}]{z}}={\begin{Bmatrix}u_{0},u_{1}...u_{n}\end{Bmatrix}}}
n
∈
N
{\displaystyle n \in N}
gdje je
u
k
=
r
n
(
c
o
s
r
n
n
+
i
s
i
n
θ
+
2
k
π
n
)
{\displaystyle u_{k}={\sqrt[{n}]{r}}(cos{\frac {\sqrt[{n}]{r}}{n}}+isin{\frac {\theta +2k\pi }{n}})}
k
=
0
,
1
,
.
.
.
(
n
−
1
)
{\displaystyle k=0,1,...(n-1)}
u
k
=
r
n
e
i
(
θ
+
2
k
π
)
/
2
{\displaystyle u_{k}={\sqrt[{n}]{r}}e^{i(\theta +2k\pi )/2}}
k
=
0
,
1
,
.
.
.
(
n
−
1
)
{\displaystyle k=0,1,...(n-1)}
Kvadratni korjen imaginarnog broja [ ]
i
=
1
2
2
+
i
1
2
2
=
2
2
(
1
+
i
)
.
{\displaystyle {\sqrt {i}}={\frac {1}{2}}{\sqrt {2}}+i{\frac {1}{2}}{\sqrt {2}}={\frac {\sqrt {2}}{2}}(1+i).}
Ovaj rezultat možemo dobiti na sljedeći način
i
=
(
a
+
b
i
)
2
{\displaystyle i=(a+bi)^{2}\!}
i
=
a
2
+
2
a
b
i
−
b
2
.
{\displaystyle i=a^{2}+2abi-b^{2}.\!}
Dobijamo dvije jednačine
{
2
a
b
=
1
a
2
−
b
2
=
0
{\displaystyle {\begin{cases}2ab=1\!\\a^{2}-b^{2}=0\!\end{cases}}}
čija su rješenja
a
=
b
=
±
1
2
.
{\displaystyle a=b=\pm {\frac {1}{\sqrt {2}}}.}
Izbor glavnog korjena daje
a
=
b
=
1
2
.
{\displaystyle a=b={\frac {1}{\sqrt {2}}}.}
Rezultat možemo dobiti pomoću Moavrova formule
i
=
cos
(
π
2
)
+
i
sin
(
π
2
)
{\displaystyle i = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right)}
i
=
(
cos
(
π
2
)
+
i
sin
(
π
2
)
)
1
2
=
cos
(
π
4
)
+
i
sin
(
π
4
)
=
1
2
+
i
(
1
2
)
=
1
2
(
1
+
i
)
.
{\displaystyle {\begin{aligned}{\sqrt {i}}&=\left(\cos \left({\frac {\pi }{2}}\right)+i\sin \left({\frac {\pi }{2}}\right)\right)^{\frac {1}{2}}\\&=\cos \left({\frac {\pi }{4}}\right)+i\sin \left({\frac {\pi }{4}}\right)\\&={\frac {1}{\sqrt {2}}}+i\left({\frac {1}{\sqrt {2}}}\right)={\frac {1}{\sqrt {2}}}(1+i).\\\end{aligned}}}
Apsolutna vrijednost argumenta [ ] Apsolutna vrijednost (ili modul ili veličine) kompleksnog broja
z
=
x
+
y
i
{\displaystyle z = x + yi}
r
=
|
z
|
=
x
2
+
y
2
.
{\displaystyle \textstyle r=|z|={\sqrt {x^{2}+y^{2}}}.\,}
Kvadrat apsolutne vrijednosti je
|
z
|
2
=
z
z
¯
=
x
2
+
y
2
.
{\displaystyle \textstyle |z|^{2}=z{\bar {z}}=x^{2}+y^{2}.\,}
φ
=
arg
(
z
)
=
{
arctan
(
y
x
)
if
x
>
0
arctan
(
y
x
)
+
π
if
x
<
0
and
y
≥
0
arctan
(
y
x
)
−
π
if
x
<
0
and
y
<
0
π
2
if
x
=
0
and
y
>
0
−
π
2
if
x
=
0
and
y
<
0
indeterminate
if
x
=
0
and
y
=
0.
{\displaystyle \varphi =\arg(z)={\begin{cases}\arctan({\frac {y}{x}})&{\mbox{if }}x>0\\\arctan({\frac {y}{x}})+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan({\frac {y}{x}})-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\{\mbox{indeterminate }}&{\mbox{if }}x=0{\mbox{ and }}y=0.\end{cases}}}
Množenje i dijeljenje u polarnom obliku [ ] Iz trigonometrijskih identiteta
cos
(
a
)
cos
(
b
)
−
sin
(
a
)
sin
(
b
)
=
cos
(
a
+
b
)
{\displaystyle \cos(a)\cos(b) - \sin(a)\sin(b) = \cos(a + b)}
cos
(
a
)
sin
(
b
)
+
sin
(
a
)
cos
(
b
)
=
sin
(
a
+
b
)
{\displaystyle \cos(a)\sin(b) + \sin(a)\cos(b) = \sin(a + b)}
imamo
z
1
z
2
=
r
1
r
2
(
cos
(
φ
1
+
φ
2
)
+
i
sin
(
φ
1
+
φ
2
)
)
.
{\displaystyle z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).\,}
Primjer
(
2
+
i
)
(
3
+
i
)
=
5
+
5
i
.
{\displaystyle (2+i)(3+i)=5+5i. \,}
π
4
=
arctan
1
2
+
arctan
1
3
{\displaystyle \frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{3} }
Dijeljenje
z
1
z
2
=
r
1
r
2
(
cos
(
φ
1
−
φ
2
)
+
i
sin
(
φ
1
−
φ
2
)
)
.
{\displaystyle \frac{z_1}{ z_2} = \frac{r_1}{ r_2} \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right).}
Izvori [ ] http://lav.fesb.hr/mat1/predavanja/node19.html http://www.elfak.ni.ac.rs/downloads/informacije/studenti/praktikumi/kompleksni-brojevi.pdf http://www.mim-sraga.com/formule/kompleksni-brojevi.htm https://element.hr/artikli/file/1333 http://www.dms.rs/DMS/data/nis/pripreme_2012/Primene%20kompleksnih%20brojeva%20u%20geometriji_Radoslav%20Dimitrijevic.pdf https://profesorka.wordpress.com/2014/02/21/moavrova-formula-i-n-ti-koren-kompleksnog-broja/ 
![{\displaystyle {\sqrt[{n}]{z}}={\begin{Bmatrix}u_{0},u_{1}...u_{n}\end{Bmatrix}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/708b85bedce3095d2d4d5ec8ea2798b3a2ae38a4)
![{\displaystyle u_{k}={\sqrt[{n}]{r}}(cos{\frac {\sqrt[{n}]{r}}{n}}+isin{\frac {\theta +2k\pi }{n}})}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/6032890e2cea54d7649673fe1964392266dc6400)
![{\displaystyle u_{k}={\sqrt[{n}]{r}}e^{i(\theta +2k\pi )/2}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/3d383ccf8d377b87f783737df34ac82aa6221d5f)
