The [[Complex Conjugate|Conjugate]] of a [[Complex Numbers|Complex Number]] $z$, denoted by $\bar{z}$ or $z^*$ is defined as the [[Function|reflection]] of $z$ over the [[Real Numbers|Real]] axis.
$\huge
\begin{align}
\pa{a+bi}^* &= a-bi \\
(re^{\theta i})^* &= re^{-\theta i}
\end{align}
$
![[../../00 Asset Bank/Pasted image 20250406220532.png|invert_Sepia]]
>[!tldr] [[Real Numbers]] [[Identity]]
>The [[Complex Conjugate|Conjugate]] of a [[Real Numbers|Real Number]] $r$ is equal to $r$.
>$\large \forall r \in \R : r^* = r $
The [[Complex Conjugate]] is [[Distributive]] over addition, subtraction, multiplication, and division.
$\large
\begin{align}
(z+w)^* &= z^* + w^*\\
(z-w)^* &= z^* - w^*\\
(zw)^* &= z^* w^*\\
\left( \frac{z}{w} \right)^* &= \frac{z^*}{w^*} \text{if } w\neq 0\\
\end{align}
$
The product of any [[Complex Numbers|Complex Number]] $z$ with its [[Complex Conjugate|Conjugate]] is equal to the [[Absolute Value]] of $z$ squared.
$\huge zz^* = \lvert z \rvert ^2 $
[[Complex Conjugate|Conjugation]] is [[Commutative]] under [[Exponential Functions]] and [[Logorithmic Function]].
$\large
\begin{align}
(z^n)^* &= (z^*)^n \\
\exp(z^*) &= \exp(z)^* \\
\ln(z^*) &= \ln(z)^*
\end{align}
$