Difference 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:

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

Proof

Let $w = -z_2$.

Then:

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

\(=\)

\(\ds \overline {z_1 + \paren {-z_2} }\)

Definition of Complex Subtraction

\(\ds \)

\(=\)

\(\ds \overline {z_1 + w}\)

Definition of $w$

\(\ds \)

\(=\)

\(\ds \overline {z_1} + \overline w\)

Sum of Complex Conjugates

\(\ds \)

\(=\)

\(\ds \overline {z_1} + \overline {-z_2}\)

Definition of $w$

\(\ds \)

\(=\)

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

Definition of Complex Subtraction

$\blacksquare$

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugate (of a complex number)