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

Example of Complex Arithmetic

Let $z^3 = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$.

Then:

$\paren {\overline {z_3} }^4 = -\dfrac 1 2 - \dfrac {\sqrt 3} 2 i$

Proof

\(\ds \paren {\overline {z_3} }^4\)

\(=\)

\(\ds \paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2 i}^4\)

Definition of Complex Conjugate

\(\ds \)

\(=\)

\(\ds \paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2 i}^3 \times \paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2 i}\)

But from Cube Roots of Unity:

$\paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2 i}^3 = 1$

Hence the result.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $2 \ \text{(d)}$