Complex Arithmetic/Examples/conj ((z 2 + z 3) (z 1 - z 3))

Example of Complex Arithmetic

Let $z_1 = 1 - i$, $z_2 = -2 + 4 i$ and $z_3 = \sqrt 3 - 2 i$.

Then:

$\overline {\paren {z_2 + z_3} \paren {z_1 - z_3} } = -7 + 3 \sqrt 3 + \sqrt 3 i$

$\ds \overline {\paren {z_2 + z_3} \paren {z_1 - z_3} }$

$=$

$\ds \overline {\paren {\paren {-2 + 4 i} + \paren {\sqrt 3 - 2 i} } \paren {\paren {1 - i} - \paren {\sqrt 3 - 2 i} } }$

$\ds $

$=$

$\ds \overline {\paren {-2 + \sqrt 3 + 2 i} \paren {1 - \sqrt 3 + i} }$

$\ds $

$=$

$\ds \overline {\paren {-2 + \sqrt 3} \paren {1 - \sqrt 3} + \paren {-2 + \sqrt 3} i + \paren {1 - \sqrt 3} 2 i + 2 i^2}$

$\ds $

$=$

$\ds \overline {\paren {-2 + \sqrt 3 + 2 \sqrt 3 - 3} - \sqrt 3 i - 2}$

$\ds $

$=$

$\ds \overline {-7 + 3 \sqrt 3 - \sqrt 3 i}$

$\ds $

$=$

$\ds -7 + 3 \sqrt 3 + \sqrt 3 i$

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(g)}$