ចំនួនកុំផ្លិច ៖ គឺជាចំនួនដែលមានទម្រង់
a
+
b
i
{\displaystyle a+bi\,}
a
{\displaystyle a\,}
b
{\displaystyle b\,}
i
{\displaystyle i\,}
ឯកតានិមិ្មត (
i
=
−
1
,
i
2
=
−
1
{\displaystyle i={\sqrt {\color {Red}-1}},\quad i^{2}=-1}
និយមន័យ ឯកតានិមិ្មត
i
=
−
1
,
i
2
=
−
1
{\displaystyle i={\sqrt {\color {Red}-1}},\quad i^{2}=-1}
Z
=
a
+
b
i
.
{\displaystyle Z=a+bi.\,}
a ជាផ្នែកពិត នៃចំនួនកុំផ្លិច Z (Real Part)
b ជាផ្នែកនិម្មិត នៃចំនួនកុំផ្លិច Z (Imaginary part) ប្រមាណវិធី
i
=
−
1
,
i
2
=
−
1
{\displaystyle i={\sqrt {\color {Red}-1}},\quad i^{2}=-1}
ផលបូក:
(
a
+
b
i
)
+
(
c
+
d
i
)
=
(
a
+
c
)
+
(
b
+
d
)
i
{\displaystyle \,(a+bi)+(c+di)=(a+c)+(b+d)i}
ផលដក:
(
a
+
b
i
)
−
(
c
+
d
i
)
=
(
a
−
c
)
+
(
b
−
d
)
i
{\displaystyle \,(a+bi)-(c+di)=(a-c)+(b-d)i}
ផលគុណ:
(
a
+
b
i
)
(
c
+
d
i
)
=
a
c
+
b
c
i
+
a
d
i
+
b
d
i
2
=
(
a
c
−
b
d
)
+
(
b
c
+
a
d
)
i
{\displaystyle \,(a+bi)(c+di)=ac+bci+adi+bdi^{2}=(ac-bd)+(bc+ad)i}
ផលចែក:
(
a
+
b
i
)
(
c
+
d
i
)
=
(
a
c
+
b
d
c
2
+
d
2
)
+
(
b
c
−
a
d
c
2
+
d
2
)
i
{\displaystyle \,{\frac {(a+bi)}{(c+di)}}=\left({ac+bd \over c^{2}+d^{2}}\right)+\left({bc-ad \over c^{2}+d^{2}}\right)i\,}
ប្លង់កុំផ្លិច លក្ខណៈធរណីមាត្រនៃ
z
{\displaystyle z}
z
¯
{\displaystyle {\bar {z}}}
តំលៃដាច់ខាតនៃចំនួនកុំផ្លិចឆ្លាស់
z
+
w
¯
=
z
¯
+
w
¯
{\displaystyle {\overline {z+w}}={\bar {z}}+{\bar {w}}}
z
⋅
w
¯
=
z
¯
⋅
w
¯
{\displaystyle {\overline {z\cdot w}}={\bar {z}}\cdot {\bar {w}}}
(
z
/
w
)
¯
=
z
¯
/
w
¯
{\displaystyle {\overline {(z/w)}}={\bar {z}}/{\bar {w}}}
z
¯
¯
=
z
{\displaystyle {\bar {\bar {z}}}=z}
z
¯
=
z
{\displaystyle {\bar {z}}=z}
z ជាចំនួនពិតសុទ្ធ
z
¯
=
−
z
{\displaystyle {\bar {z}}=-z}
z ជាចំនួននិម្មិតសុទ្ធ
|
z
|
=
|
z
¯
|
{\displaystyle |z|=|{\bar {z}}|}
|
z
|
2
=
z
⋅
z
¯
{\displaystyle |z|^{2}=z\cdot {\bar {z}}}
z
−
1
=
z
¯
⋅
|
z
|
−
2
{\displaystyle z^{-1}={\bar {z}}\cdot |z|^{-2}}
z ខុសពីសូន្យប្រភាគនៃចំនួនកុំផ្លិច
a
+
b
i
c
+
d
i
=
(
a
+
b
i
)
(
c
−
d
i
)
(
c
+
d
i
)
(
c
−
d
i
)
=
(
a
c
+
b
d
)
+
(
b
c
−
a
d
)
i
c
2
+
d
2
=
(
a
c
+
b
d
c
2
+
d
2
)
+
i
(
b
c
−
a
d
c
2
+
d
2
)
.
{\displaystyle {\begin{aligned}{a+bi \over c+di}&={(a+bi)(c-di) \over (c+di)(c-di)}={(ac+bd)+(bc-ad)i \over c^{2}+d^{2}}\\&=\left({ac+bd \over c^{2}+d^{2}}\right)+i\left({bc-ad \over c^{2}+d^{2}}\right).\,\end{aligned}}}
ទម្រង់ប៉ូលែរ កូអរដោនេប៉ូលែក្នុងតម្រុយដេកាត
x
=
r
cos
φ ふぁい
{\displaystyle x=r\cos \varphi }
y
=
r
sin
φ ふぁい
{\displaystyle y=r\sin \varphi }
ផ្ទុយមកវិញ
r
=
x
2
+
y
2
{\displaystyle r={\sqrt {x^{2}+y^{2}}}}
φ ふぁい =
arg
(
z
)
=
a
r
c
t
a
n
y
x
{\displaystyle \varphi =\arg(z)=arctan{\frac {y}{x}}}
x
+
i
y
=
r
e
i
φ ふぁい
{\displaystyle x+iy=re^{i\varphi }\!}
ទម្រង់ត្រីកោណមាត្រ និង ម៉ូឌុលនៃចំនួនកុំផ្លិច
a
+
b
i
=
r
(
c
o
s
α あるふぁ +
i
s
i
n
α あるふぁ )
{\displaystyle a+bi=r(cos\alpha +isin\alpha )\!}
r
{\displaystyle r\!}
a
+
b
i
{\displaystyle a+bi\!}
r
=
|
z
|
=
a
2
+
b
2
{\displaystyle r=|z|={\sqrt {a^{2}+b^{2}}}\!}
c
o
s
α あるふぁ =
a
r
;
s
i
n
α あるふぁ =
b
r
{\displaystyle cos\alpha ={\frac {a}{r}};sin\alpha ={\frac {b}{r}}\!}
ទ្រឹស្តីបទ៖
បើគេមានទម្រង់ត្រីកោណមាត្រនៃចំនួនកំផ្លិច
z
1
{\displaystyle z_{1}\!}
z
2
{\displaystyle z_{2}\!}
z
1
=
r
1
(
c
o
s
α あるふぁ
1
+
i
s
i
n
α あるふぁ
1
)
{\displaystyle z_{1}=r_{1}(cos\alpha _{1}+isin\alpha _{1})\!}
z
2
=
r
2
(
c
o
s
α あるふぁ
2
+
i
s
i
n
α あるふぁ
2
)
{\displaystyle z_{2}=r_{2}(cos\alpha _{2}+isin\alpha _{2})\!}
ក)
z
1
z
2
=
r
1
r
2
[
c
o
s
(
α あるふぁ
1
+
α あるふぁ
2
)
+
i
s
i
n
(
α あるふぁ
1
+
α あるふぁ
2
)
]
{\displaystyle z_{1}z_{2}=r_{1}r_{2}[cos(\alpha _{1}+\alpha _{2})+isin(\alpha _{1}+\alpha _{2})]\!}
z
1
z
2
=
r
1
r
2
[
c
o
s
(
α あるふぁ
1
−
α あるふぁ
2
)
+
i
s
i
n
(
α あるふぁ
1
−
α あるふぁ
2
)
]
{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}[cos(\alpha _{1}-\alpha _{2})+isin(\alpha _{1}-\alpha _{2})]\!}
ទ្រឹស្តីបទ៖
បើ
z
{\displaystyle z\!}
|
z
|
2
=
z
⋅
z
¯
{\displaystyle |z|^{2}=z\cdot {\bar {z}}\!}
លក្ខណៈ
គេឲ្យ
w
{\displaystyle w\!}
z
{\displaystyle z\!}
ក)
|
w
z
|
=
|
w
|
⋅
|
z
|
{\displaystyle |wz|=|w|\cdot |z|\!}
|
w
z
|
=
|
w
|
|
z
|
;
z
≠
0
{\displaystyle |{\frac {w}{z}}|={\frac {|w|}{|z|}};z\neq 0\!}
|
w
+
z
|
≤
|
w
|
+
|
z
|
{\displaystyle |w+z|\leq |w|+|z|\!}
ស្វ័យគុណទី
n
{\displaystyle n\!}
គេមាន
Z
=
r
(
c
o
s
α あるふぁ +
i
s
i
n
α あるふぁ )
{\displaystyle Z=r(cos\alpha +isin\alpha )\!}
Z
1
Z
2
=
r
1
r
2
[
c
o
s
(
α あるふぁ
1
+
α あるふぁ
2
)
+
i
s
i
n
(
α あるふぁ
1
+
α あるふぁ
2
)
]
{\displaystyle Z_{1}Z_{2}=r_{1}r_{2}[cos(\alpha _{1}+\alpha _{2})+isin(\alpha _{1}+\alpha _{2})]\!}
Z
Z
=
r
r
[
c
o
s
(
α あるふぁ +
α あるふぁ )
+
i
s
i
n
(
α あるふぁ +
α あるふぁ )
]
{\displaystyle ZZ=rr[cos(\alpha +\alpha )+isin(\alpha +\alpha )]\!}
Z
2
=
r
2
(
c
o
s
2
α あるふぁ +
i
s
i
n
2
α あるふぁ )
{\displaystyle Z^{2}=r^{2}(cos2\alpha +isin2\alpha )\!}
Z
3
=
Z
2
⋅
Z
=
(
r
2
⋅
r
)
[
c
o
s
(
2
α あるふぁ +
α あるふぁ )
+
i
s
i
n
(
2
α あるふぁ +
α あるふぁ )
]
=
r
3
(
c
o
s
3
α あるふぁ +
i
s
i
n
3
α あるふぁ )
{\displaystyle Z^{3}=Z^{2}\cdot Z=(r^{2}\cdot r)[cos(2\alpha +\alpha )+isin(2\alpha +\alpha )]=r^{3}(cos3\alpha +isin3\alpha )\!}
Z
n
=
Z
n
−
1
⋅
Z
=
r
n
(
c
o
s
n
α あるふぁ +
i
s
i
n
n
α あるふぁ )
{\displaystyle Z^{n}=Z^{n-1}\cdot Z=r^{n}(cosn\alpha +isinn\alpha )\!}
ជាទូទៅ៖
Z
n
=
[
r
(
c
o
s
α あるふぁ +
i
s
i
n
α あるふぁ )
]
n
=
r
n
(
c
o
s
n
α あるふぁ +
i
s
i
n
n
α あるふぁ )
{\displaystyle Z^{n}=[r(cos\alpha +isin\alpha )]^{n}=r^{n}(cosn\alpha +isinn\alpha )\!}
n
∈
Z
{\displaystyle n\in \mathbb {Z} \!}
(
c
o
s
α あるふぁ +
i
s
i
n
α あるふぁ
)
n
=
(
c
o
s
n
α あるふぁ +
i
s
i
n
n
α あるふぁ )
{\displaystyle (cos\alpha +isin\alpha )^{n}=(cosn\alpha +isinn\alpha )\!}
ឧទាហរណ៍: គណនា
(
1
+
i
)
50
{\displaystyle (1+i)^{50}\!}
z
=
1
+
i
{\displaystyle z=1+i\!}
z
=
2
(
c
o
s
π ぱい 4
+
i
s
i
n
π ぱい 4
)
{\displaystyle z={\sqrt {2}}(cos{\frac {\pi }{4}}+isin{\frac {\pi }{4}})\!}
តាមទ្រឹស្តីបទដឺម័រ
(
i
+
i
)
50
=
2
50
[
c
o
s
(
50
⋅
π ぱい 4
)
+
i
s
i
n
(
50
⋅
π ぱい 4
)
]
=
2
25
(
c
o
s
25
π ぱい
2
+
i
s
i
n
25
π ぱい
2
)
=
2
25
[
c
o
s
(
12
π ぱい +
π ぱい 2
)
+
i
s
i
n
(
12
π ぱい +
π ぱい 2
)
]
=
2
25
(
c
o
s
π ぱい 2
+
i
s
i
n
π ぱい 2
)
{\displaystyle (i+i)^{50}={\sqrt {2}}^{50}[cos(50\cdot {\frac {\pi }{4}})+isin(50\cdot {\frac {\pi }{4}})]=2^{25}(cos{\frac {25\pi }{2}}+isin{\frac {25\pi }{2}})=2^{25}[cos(12\pi +{\frac {\pi }{2}})+isin(12\pi +{\frac {\pi }{2}})]=2^{25}(cos{\frac {\pi }{2}}+isin{\frac {\pi }{2}})\!}
(
1
+
i
)
50
=
2
25
i
=
33554432
i
{\displaystyle (1+i)^{50}=2^{25}i=33554432i\!}
ឫសទី
n
{\displaystyle n\!}
បើចំនួនកុំផ្លិចមិនសូន្យ Z មានឫសទី n គឺ W គេបាន
W
n
=
Z
{\displaystyle W^{n}=Z\!}
Z
=
r
(
c
o
s
θ しーた +
i
s
i
n
θ しーた )
{\displaystyle Z=r(cos\theta +isin\theta )\!}
W
=
s
(
c
o
s
α あるふぁ +
i
s
i
n
α あるふぁ )
{\displaystyle W=s(cos\alpha +isin\alpha )\!}
W
n
=
s
n
(
c
o
s
n
α あるふぁ +
i
s
i
n
n
α あるふぁ )
{\displaystyle W^{n}=s^{n}(cosn\alpha +isinn\alpha )\!}
W
n
=
Z
{\displaystyle W^{n}=Z\!}
s
n
(
c
o
s
n
α あるふぁ +
i
s
i
n
n
α あるふぁ )
=
r
(
c
o
s
θ しーた +
i
s
i
n
θ しーた )
{\displaystyle s^{n}(cosn\alpha +isinn\alpha )=r(cos\theta +isin\theta )\!}
ដូចនេះ
s
n
=
r
{\displaystyle s^{n}=r\!}
s
>
0
{\displaystyle s>0\!}
r
>
0
{\displaystyle r>0\!}
s
=
r
n
{\displaystyle s={\sqrt[{n}]{r}}\!}
c
o
s
n
α あるふぁ +
i
s
i
n
n
α あるふぁ =
c
o
s
θ しーた +
i
s
i
n
θ しーた
{\displaystyle cosn\alpha +isinn\alpha =cos\theta +isin\theta \!}
c
o
s
n
α あるふぁ =
c
o
s
θ しーた
{\displaystyle cosn\alpha =cos\theta \!}
n
α あるふぁ =
θ しーた +
2
k
π ぱい
;
α あるふぁ =
θ しーた +
2
k
π ぱい
n
;
k
∈
Z
{\displaystyle n\alpha =\theta +2k\pi \ ;\ \alpha ={\frac {\theta +2k\pi }{n}}\ ;\ k\in \mathbb {Z} \!}
α あるふぁ =
θ しーた +
2
k
π ぱい
n
{\displaystyle \alpha ={\frac {\theta +2k\pi }{n}}\!}
s
=
r
n
{\displaystyle s={\sqrt[{n}]{r}}\!}
W
{\displaystyle W}
w
=
r
n
[
c
o
s
(
θ しーた +
2
k
π ぱい
n
)
+
i
s
i
n
(
θ しーた +
2
k
π ぱい
n
)
]
{\displaystyle w={\sqrt[{n}]{r}}[cos({\frac {\theta +2k\pi }{n}})+isin({\frac {\theta +2k\pi }{n}})]\!}
k
=
0
;
1
;
2
;
.
.
.
;
n
−
1
{\displaystyle k=0;1;2;...;n-1}
ទ្រឹស្តីបទ៖
បើ
Z
=
r
(
c
o
s
θ しーた +
i
s
i
n
θ しーた )
{\displaystyle Z=r(cos\theta +isin\theta )\!}
w
k
=
r
n
[
c
o
s
(
θ しーた +
2
k
π ぱい
n
)
+
i
s
i
n
(
θ しーた +
2
k
π ぱい
n
)
]
{\displaystyle w_{k}={\sqrt[{n}]{r}}[cos({\frac {\theta +2k\pi }{n}})+isin({\frac {\theta +2k\pi }{n}})]\!}
w
0
;
w
1
;
w
2
;
.
.
.
;
w
n
−
1
{\displaystyle w_{0};w_{1};w_{2};...;w_{n-1}\!}
តាង Z = -1 + 0i គេបាន
r
=
1
=
1
{\displaystyle r={\sqrt {1}}=1}
c
o
s
θ しーた =
a
r
=
−
1
{\displaystyle cos\theta ={\frac {a}{r}}=-1}
s
i
n
θ しーた =
b
r
=
0
{\displaystyle sin\theta ={\frac {b}{r}}=0}
θ しーた =
π ぱい
{\displaystyle \theta =\pi }
Z
=
−
1
+
0
i
=
(
c
o
s
π ぱい +
i
s
i
n
π ぱい )
{\displaystyle Z=-1+0i=(cos\pi +isin\pi )\!}
w
k
=
c
o
s
(
π ぱい +
2
k
π ぱい
6
)
+
i
s
i
n
(
π ぱい +
2
k
π ぱい
6
)
{\displaystyle w_{k}=cos({\frac {\pi +2k\pi }{6}})+isin({\frac {\pi +2k\pi }{6}})\!}
w
k
=
c
o
s
(
π ぱい 6
+
k
π ぱい
3
)
+
i
s
i
n
(
π ぱい 6
+
k
π ぱい
3
)
{\displaystyle w_{k}=cos({\frac {\pi }{6}}+{\frac {k\pi }{3}})+isin({\frac {\pi }{6}}+{\frac {k\pi }{3}})\!}
k=0 នាំឲ្យ
w
0
=
c
o
s
π ぱい 6
+
i
s
i
n
π ぱい 6
=
3
2
+
1
2
i
{\displaystyle w_{0}=cos{\frac {\pi }{6}}+isin{\frac {\pi }{6}}={\frac {\sqrt {3}}{2}}+{\frac {1}{2}}i\!}
w
1
=
c
o
s
π ぱい 2
+
i
s
i
n
π ぱい 2
=
i
{\displaystyle w_{1}=cos{\frac {\pi }{2}}+isin{\frac {\pi }{2}}=i}
w
2
=
c
o
s
5
π ぱい
6
+
i
s
i
n
5
π ぱい
6
=
−
3
2
+
1
2
i
{\displaystyle w_{2}=cos{\frac {5\pi }{6}}+isin{\frac {5\pi }{6}}=-{\frac {\sqrt {3}}{2}}+{\frac {1}{2}}i}
w
3
=
c
o
s
7
π ぱい
6
+
i
s
i
n
7
π ぱい
6
=
−
3
2
−
1
2
i
{\displaystyle w_{3}=cos{\frac {7\pi }{6}}+isin{\frac {7\pi }{6}}=-{\frac {\sqrt {3}}{2}}-{\frac {1}{2}}i}
w
4
=
c
o
s
3
π ぱい
2
+
i
s
i
n
3
π ぱい
2
=
−
i
{\displaystyle w_{4}=cos{\frac {3\pi }{2}}+isin{\frac {3\pi }{2}}=-i}
w
5
=
c
o
s
11
π ぱい
6
+
i
s
i
n
11
π ぱい
6
=
3
2
−
1
2
i
{\displaystyle w_{5}=cos{\frac {11\pi }{6}}+isin{\frac {11\pi }{6}}={\frac {\sqrt {3}}{2}}-{\frac {1}{2}}i}
សូមមើលផងដែរ 
![{\displaystyle z_{1}z_{2}=r_{1}r_{2}[cos(\alpha _{1}+\alpha _{2})+isin(\alpha _{1}+\alpha _{2})]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f012127e31ef3da231009171afed189cb344bb15)
![{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}[cos(\alpha _{1}-\alpha _{2})+isin(\alpha _{1}-\alpha _{2})]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45457d95c24be129e0db2e20a798bbc967bdfa06)
![{\displaystyle Z_{1}Z_{2}=r_{1}r_{2}[cos(\alpha _{1}+\alpha _{2})+isin(\alpha _{1}+\alpha _{2})]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d34dd8049b3cd53bafd2273976550091add19192)
![{\displaystyle ZZ=rr[cos(\alpha +\alpha )+isin(\alpha +\alpha )]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7377783c44f95202c9d393eb4eb6a4eca31075)
![{\displaystyle Z^{3}=Z^{2}\cdot Z=(r^{2}\cdot r)[cos(2\alpha +\alpha )+isin(2\alpha +\alpha )]=r^{3}(cos3\alpha +isin3\alpha )\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22a36ac682b1ddc905ac13f0eb17db55c279aea7)
![{\displaystyle Z^{n}=[r(cos\alpha +isin\alpha )]^{n}=r^{n}(cosn\alpha +isinn\alpha )\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf3933dedb0bd12eb8495d2753f67ee77b3de05)
![{\displaystyle (i+i)^{50}={\sqrt {2}}^{50}[cos(50\cdot {\frac {\pi }{4}})+isin(50\cdot {\frac {\pi }{4}})]=2^{25}(cos{\frac {25\pi }{2}}+isin{\frac {25\pi }{2}})=2^{25}[cos(12\pi +{\frac {\pi }{2}})+isin(12\pi +{\frac {\pi }{2}})]=2^{25}(cos{\frac {\pi }{2}}+isin{\frac {\pi }{2}})\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa6f2c78b2ecea18a472dfaa616865227747f0ed)
![{\displaystyle s={\sqrt[{n}]{r}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e7e32c3357120dd301ebb34a2f2dfe4f1c8556)
![{\displaystyle w={\sqrt[{n}]{r}}[cos({\frac {\theta +2k\pi }{n}})+isin({\frac {\theta +2k\pi }{n}})]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31a4f64039287ae204471f250736a29a0c0cbb85)
![{\displaystyle w_{k}={\sqrt[{n}]{r}}[cos({\frac {\theta +2k\pi }{n}})+isin({\frac {\theta +2k\pi }{n}})]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/959fe371556d6bc965d538101e9386329d587435)
