Let Z and W be complex numbers such that |Z|=|W|, and Z denotes the principal argument.

Question

Let $Z$ and $W$ be complex numbers such that $|Z|=|W|$, and $\arg Z$ denotes the principal argument.

Statement 1: If $\arg Z+\arg W=\pi$, then $Z=-W$.

Statement 2: $|Z|=|W|$ implies $\arg Z-\arg W=\pi$.

Accepted Answer

Let $Z=re^{i	heta},\ W=re^{i\varphi}$ with $	heta+\varphi=\pi$. Then $Z=re^{i(\pi-\varphi)}=-re^{-i\varphi}=-W$, so Statement 1 is true. Statement 2 clearly need not hold in general (only equality of moduli is given, not a fixed argument relation) --- false.

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