The minimum value of the expression $|3z - 3| + |2z - 4|$ is equal to (where, $z$ is a complex number)
Step-by-Step Solution
Key Concept: Recognize that $z^2 + z + 1 = 0$ has solutions as non-real cube roots of unity, which have modulus 1.
From $z^2 + z + 1 = 0$, we have $z = \omega$ (a cube root of unity, where $\omega^3 = 1$). Using $z^{2003} = z^{3 \cdot 667 + 2} = z^2$ and noting $z^{2003} + z^{2004} = z^2 + z^3 = z^2 + 1 = -z$ (from $z^2 + z + 1 = 0$). Then $z^{2003} = z^2 = \omega^2$ with $|\omega^2| = |\omega|^2 = 1^2 = 1$, giving $|z| = 1$.
Correct Answer: 1