Product of Complex Conjugates

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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:

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


Sources