Product of Complex Conjugates/Examples/3 Arguments

Theorem

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:

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

Proof 1

\(\ds \overline {z_1 z_2 z_3}\)

\(=\)

\(\ds \overline {\paren {z_1 z_2} \paren {z_3} }\)

\(\ds \)

\(=\)

\(\ds \paren {\overline {z_1 z_2} } \cdot \overline {z_3}\)

Product of Complex Conjugates

\(\ds \)

\(=\)

\(\ds \paren {\overline {z_1} \cdot \overline {z_2} } \cdot \overline {z_3}\)

Product of Complex Conjugates

\(\ds \)

\(=\)

\(\ds \overline {z_1} \cdot \overline {z_2} \cdot \overline {z_3}\)

$\blacksquare$

Proof 2

From Product of Complex Conjugates: General Result:

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

The result follows by setting $n = 3$.

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $55 \ \text {(b)}$