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:

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

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:

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

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

$\blacksquare$

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:

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

1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 2$. Conjugate Complex Numbers
1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: $(1.7)$
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $55 \ \text {(a)}$
1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.3$ Complex conjugation: $(4)$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugate (of a complex number)
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): conjugate (of a complex number)