Find all complex numbers which make the following equations true: $ |z+1| =1 $ and $ |z^2+1| =1 $ Find all complex numbers which make the following equations true:
$$ |z+1| =1 $$
$$ |z^2+1| =1 $$
Solution:
If $ |z+1| =1 $ holds true, then
$$z+1 = 1.e^{i2n\pi}$$
$$z = 1.e^{i2n\pi}-1$$
$$z = 1.e^{i2n\pi}-1e^{i2m\pi}$$
If $ |z^2+1| =1 $ holds true, then
$$z^2+1 = 1.e^{i2p\pi}$$
$$z^2 = 1.e^{i2p\pi}-1$$
$$z^2 = 1.e^{i2p\pi}-1e^{i2q\pi}$$

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*How to proceed after this?

*Am I supposed to do in this manner or break complex number z into real part x and imaginary part y and get two equations and thus solve for x and y?


It is clear that $z=0$ is a solution. Hence we assume $z \ne 0$.
From $|z+1|^2=1$ we derive $ |z|^2= -(z+ \bar z) \quad (1)$.
From $|z^2+1|^2=1$ we derive $|z|^4= -(z^2+ \bar z^2) \quad (2)$.
Then $(1)$ gives: $|z|^4= z^2+ \bar z^2+2z \bar z=z ^2+ \bar z^2+2|z|^2$.
With $(2)$ we derive $|z|^4=-|z|^4+2|z|^2.$ Hence $|z|^4=|z|^2$. This gives $|z|=1.$
Use again $(1)$ to see that $Re(z)=-1/2.$
From $1=|z+1|^2=1/4 + (Im(z))^2$ we get $Im(z)= \pm \frac{\sqrt{3}}{2}.$
