Find all z for which $|z| = ze^{-i\frac \pi 2}$ How would one go about answering the 
following
Find all $z$ for which $$|z| = ze^{-i\frac \pi 2}$$
 A: A perfectly good solution could start by evaluating $e^{-i\pi/2}$ to $-i$.
But, depending on which tools you already have, you could also start by dividing through by $e^{-i\pi/2}$ as it is, producing
$$ z = |z|\cdot e^{\frac{\pi}2 i} $$
You will recognize the right-hand side of this as a complex number in polar form, so the equation states directly what the argument of $z$ needs to be.
A: Method 1
Expand in Cartesian form $x+i y$:
$$
\begin{align}
%
\left|z\right| &= z e^{-i\frac{\pi}{2}} \\\ 
%
\sqrt{x^2+y^2} &= -i (x + i y) = y - ix
%
\end{align}
$$
The left hand side is real, so the imaginary part of the right hand side must be $0$:
$$
\boxed{
x = 0
}
$$
Now the identity looks like
$$
\sqrt{y^{2} =  y}
$$
which restricts 
$$
\boxed{
y>0
}
$$

Method 2
Look for the roots of
$$
 f(z) = |z| - z e^{-i\frac{\pi}{2}}
$$
Solve for the zeros of the real and imaginary parts
$$
\text{Re } f = \sqrt{x^2+y^2}-y = 0 \qquad \Rightarrow \qquad \boxed{x = 0, y > 0}
$$
$$
\text{Im } f = x = 0 \qquad \Rightarrow \qquad x = 0
$$
Check your work by quick computation of $f(i)$ and $f(1)$.

Method 3
The method of @Henning Makholm: polar form
$$
z = re^{i\theta}
$$
$$
|z| = r, \qquad z e^{-i\frac{\theta}{2}} = re^{i\left(\theta-\frac{\theta}{2}\right)}
$$
$$
\begin{align}
  |z| &= z e^{-i\frac{\pi}{2}} \\
  r   &= r e^{i\left(\theta-\frac{\pi}{2}\right)}
\end{align}
$$
When does $e^{i\left(\theta-\frac{\pi}{2}\right)} = 0$? When $\theta = \frac{\pi}{2}$, when we are on the positive imaginary axis.

Conclusion
The only numbers which satisfy the identity are purely imaginary. That is, Re $z = 0$.
Furthermore, this process maps points on the positive imaginary axis to points on equal magnitude on the positive real axis.

A: Another approach is geometric consideration. We know $e^{-i\frac{\pi}{2}}$ is a rotation about origin with $-\dfrac{\pi}{2}$ radiant and $|z|$ is a real number, so we can ask for what number in complex plane whose rotation by $-\dfrac{\pi}{2}$ will be a real number!
