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
$$z^2+1 = 1.e^{i2p\pi}$$
$$z^2 = 1.e^{i2p\pi}-1$$
$$z^2 = 1.e^{i2p\pi}-1e^{i2q\pi}$$

*

*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?


If you write $z$ as $a+bi$, then you get the system$$\left\{\begin{array}{l}(a+1)^2+b^2=1\\(a^2-b^2+1)^2+(-2ab)^2=1,\end{array}\right.$$which is equivalent to$$\left\{\begin{array}{l}a^2+b^2+2a=0\\2 a^2 b^2+a^4+2 a^2+b^4-2 b^2=0.\end{array}\right.$$From the first equation, you get that $b^2=-2a-a^2$, and if you replace $b^2$ by $-2a-a^2$ in the second equation, you get $8a^2+4a=0$. So, $a=0$ or $a=-\frac12$, and therefore the solutions of your system are $0$ and $-\frac12\pm\frac{\sqrt3}2i$.
