If z is a complex number satisfying |Re(z)|+|Im(z)|=4, then |z| cannot be:

Question

If $z$ is a complex number satisfying $|\operatorname{Re}(z)|+|\operatorname{Im}(z)|=4$, then $|z|$ cannot be:

Accepted Answer

With $|x|+|y|=4$, $|z|=\sqrt{x^2+y^2}$ ranges over $[\sqrt8,4]=[2\sqrt2,4]$, i.e. $|z|\in[\sqrt8,\sqrt{16}]$. Since $\sqrt7<\sqrt8$, $|z|$ cannot equal $\sqrt7$.

Solutions · 1