Discussion
Which of the following is true for a complex number \(z = x + iy\) ?
Answer: C
The true relation is \(|x| + |y| \ge |z|\).
\[ \begin{aligned} (|x| + |y|)^{2} &= x^{2} + y^{2} + 2|x||y| \\ &\ge x^{2} + y^{2} = |z|^{2} \end{aligned} \]
Taking square roots gives \(|x| + |y| \ge |z|\), with equality only when \(x = 0\) or \(y = 0\). This is just the triangle inequality for \(z = x + iy\).
Equality (A) fails for \(z = 1 + i\), where \(2 \gt \sqrt{2}\). The \(\sqrt{2}\) version (B) is the upper bound \(|x| + |y| \le \sqrt{2}|z|\) turned into an equation, which holds only when \(|x| = |y|\), and D reverses the inequality.
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