对于复数
z
,
w
{\displaystyle z,w}
：
z
+
w
¯
=
z
¯
+
w
¯
z
−
w
¯
=
z
¯
−
w
¯
z
w
¯
=
z
¯
w
¯
(
z
w
)
¯
=
z
¯
w
¯
(
w
≠
0
)
z
¯
=
z
(
z
∈
R
)
z
n
¯
=
z
¯
n
(
n
∈
Z
)
|
z
¯
|
=
|
z
|
|
z
¯
|
2
=
z
z
¯
(
z
¯
)
¯
=
z
z
−
1
=
z
¯
|
z
|
2
(
z
≠
0
)
{\displaystyle {\begin{array}{l}{\overline {z+w}}={\overline {z}}+{\overline {w}}\\{\overline {z-w}}={\overline {z}}-{\overline {w}}\\{\overline {zw}}={\overline {z}}\,{\overline {w}}\\{\overline {\left({\dfrac {z}{w}}\right)}}={\dfrac {\overline {z}}{\overline {w}}}&(w\neq 0)\\{\overline {z}}=z&(z\in \mathbb {R} )\\{\overline {z^{n}}}={\overline {z}}^{n}&(n\in \mathbb {Z} )\\|{\overline {z}}|=|z|\\|{\overline {z}}|^{2}=z{\overline {z}}\\{\overline {({\overline {z}})}}=z\\z^{-1}={\dfrac {\overline {z}}{|z|^{2}}}&(z\neq 0)\end{array}}}
一般而言，如果复平面上的函数
ϕ
{\displaystyle \phi }
能表为实系数幂级数，则有：
ϕ
(
z
¯
)
=
ϕ
(
z
)
¯
{\displaystyle \phi ({\overline {z}})={\overline {\phi (z)}}}
最直接的例子是多项式，由此可推得实系数多项式之复根必共轭。此外也可用于复指数函数与复对数函数（取定一分支）：
exp
(
z
¯
)
=
exp
(
z
)
¯
log
(
z
¯
)
=
log
(
z
)
¯
(
z
≠
0
)
{\displaystyle {\begin{array}{l}\exp({\overline {z}})={\overline {\exp(z)}}\\\log({\overline {z}})={\overline {\log(z)}}&(z\neq 0)\end{array}}}
透过欧拉公式，在极坐标表法下，复数共轭可以写成
r
e
i
θ
¯
=
r
e
−
i
θ
{\displaystyle {\overline {re^{i\theta }}}=re^{-i\theta }}
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