Question

Prove that for any complex numbers \(z\),
\(\large |1+z| \ge \frac{ 1 }{ \sqrt{2} } \) or \(\large |1+z^2| \ge 1\).

Answer 1

the first statement can be rewritten

Answer 2

$$\large |z- (-1)| \ge \frac{ 1 }{ \sqrt{2} }$$

this means the distance betwen complex number z and z = -1 is greater than sqrt(2)/2

Answer 3

the complex number z = -1 is equal to -1 + 0i , you can plot that