# Product of Complex Conjugates

## Theorem

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\overline z$ denote the complex conjugate of the complex number $z$.

Then:

$\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$

### General Result

Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers.

Let $\overline z$ be the complex conjugate of the complex number $z$.

Then:

$\displaystyle \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$

## Proof

Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.

Then:

 $\ds \overline {z_1 z_2}$ $=$ $\ds \overline {\paren {x_1 x_2 - y_1 y_2} + i \paren {x_2 y_1 + x_1 y_2} }$ Definition of Complex Multiplication $\ds$ $=$ $\ds \paren {x_1 x_2 - y_1 y_2} - i \paren {x_2 y_1 + x_1 y_2}$ Definition of Complex Conjugate $\ds$ $=$ $\ds \paren {x_1 x_2 - \paren {-y_1} \paren {-y_2} } + i \paren {x_2 \paren {-y_1} + x_1 \paren {-y_2} }$ $\ds$ $=$ $\ds \paren {x_1 - i y_1} \paren {x_2 - i y_2}$ Definition of Complex Multiplication $\ds$ $=$ $\ds \overline {z_1} \cdot \overline {z_2}$ Definition of Complex Conjugate

$\blacksquare$

## Examples

### $3$ Arguments

Let $z_1, z_2, z_3 \in \C$ be complex numbers.

Let $\overline z$ denote the complex conjugate of the complex number $z$.

Then:

$\overline {z_1 z_2 z_3} = \overline {z_1} \cdot \overline {z_2} \cdot \overline {z_3}$