Question

Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].

Answer 1

Let \[z = x + iy\].

Then,

\[\left| z \right| = \sqrt{x^2 + y^2}\]

\[\therefore z^2 + \left| z \right|^2 = 0\]

\[ \Rightarrow \left( x + iy \right)^2 + \left( \sqrt{x^2 + y^2} \right)^2 = 0\]

\[ \Rightarrow x^2 + i^2 y^2 + 2ixy + x^2 + y^2 = 0\]

\[ \Rightarrow x^2 - y^2 + 2ixy + x^2 + y^2 = 0\]

\[ \Rightarrow 2 x^2 + 2ixy = 0\]

\[ \Rightarrow 2x(x + iy) = 0\]

\[ \Rightarrow x = 0 \text { or } x + iy = 0\]

\[ \Rightarrow x = 0 \text { or } z = 0\]

For

\[x = 0, z = 0 + iy\]

​Thus, there are infinitely many solutions of the form 

\[z = 0 + iy, y \in R\]