Complex Conjugate/Examples/conj z1 - conj z2

Example of Complex Conjugates

Let $z_1 = 4 - 3 i$ and $z_2 = -1 + 2 i$.

Then:

$\overline {z_1} - \overline {z_2} = 5 + 5 i$

Proof 1

An illustration of the modulus of the difference of the complex conjugates of the complex numbers:

$z_1 = 4 - 3 i$

$z_2 = -1 + 2 i$

is given below:

$\blacksquare$

Proof 2

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

\(=\)

\(\ds \paren {\overline {4 - 3 i} } - \paren {\overline {-1 + 2 i} }\)

Definition of $z_1$ and $z_2$

\(\ds \)

\(=\)

\(\ds \paren {4 + 3 i} - \paren {-1 - 2 i}\)

Definition of Complex Conjugate

\(\ds \)

\(=\)

\(\ds \paren {4 - \paren {-1} } + \paren {3 - \paren {-2} } i\)

Definition of Complex Subtraction

\(\ds \)

\(=\)

\(\ds 5 + 5 i\)

$\blacksquare$