Complex Arithmetic/Examples/Real (2 z 1^3 + 3 z 2^2 - 5 z 3^2)

Example of Complex Arithmetic

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

Then:

$\map \Re {2 {z_1}^3 + 3 {z_2}^2 - 5 {z_3}^2} = -35$

$\ds \map \Re {2 {z_1}^3 + 3 {z_2}^2 - 5 {z_3}^2}$

$=$

$\ds \map \Re {2 \paren {1 - i}^3 + 3 \paren {-2 + 4 i}^2 - 5 \paren {\sqrt 3 - 2 i}^2}$

$\ds $

$=$

$\ds \map \Re {2 \paren {1 - i} \paren {-2 i} + 3 \paren {4 - 16 i + 16 i^2} - 5 \paren {3 - 4 \sqrt 3 i + 4 i^2} }$

$\ds $

$=$

$\ds \map \Re {2 \paren {-2 - 2 i} + 3 \paren {-12 - 16 i} - 5 \paren {-1 - 4 \sqrt 3 i} }$

$\ds $

$=$

$\ds \map \Re {\paren {-4 - 4 i} + \paren {-36 - 48 i} + \paren {5 + 20 \sqrt 3 i} }$

$\ds $

$=$

$\ds \map \Re {-35 + \paren {-52 + 20 \sqrt 3} i}$

$\ds $

$=$

$\ds -35$

$\blacksquare$

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