Exercise 1571Algebra › Complex numbersOlympiadAdded 3 Nov 2026

A region in the complex plane

#region#inequality#geometric

Statement

Describe the region of the complex plane defined by $|z-1|<|z+i|$ .

Hint

Compare distances to $A(1,0)$ and $B(0,-1)$ .

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Interpretation. $MA<MB$ , the set of points closer to $A$ than to $B$ .

Boundary. The perpendicular bisector of $[AB]$ has equation $x+y=0$ (from a previous exercise).

Region. The half-plane containing $A(1,0)$ : $x+y>0$ .

Conclusion. The region is the open half-plane $\{z:\operatorname{Re}(z)+\operatorname{Im}(z)>0\}$ .