Imaginary Part of Complex Product

Theorem

Let $z_1$ and $z_2$ be complex numbers.

Then:

$\map \Im {z_1 z_2} = \map \Re {z_1} \, \map \Im {z_2} + \map \Im {z_1} \, \map \Re {z_2}$

Proof

Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$.

By definition of complex multiplication:

$z_1 z_2 = x_1 x_2 - y_1 y_2 + i \paren {x_1 y_2 + x_2 y_1}$

Then

\(\ds \map \Im {z_1 z_2}\)

\(=\)

\(\ds x_1 y_2 + x_2 y_1\)

Definition of Imaginary Part

\(\ds \)

\(=\)

\(\ds \map \Re {z_1} \, \map \Im {z_2} + \map \Im {z_1} \, \map \Re {z_2}\)

Definition of Imaginary Part

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $166 \ \text{(b)}$