Draw a set of values in complex plane where the complex number $w=1-3i$ is pure imaginary number.

Question

How would you draw a set of values (in complex plane) where the complex number $w=1-3i$ is pure imaginary number?

Could this be the solution? If $Rew=0$.

Answer 1

If you speak about "set of values", I think to something that moves, like a function, not something stationary, as a number.

Hence you can substitute your number with the following function $$ w=w(t)=t-3i $$ that is an horizontal line, passing thru the point $(0,-3)$ (once you see $\mathbb C$ as the real plane $\mathbb R^2$).

Then the question "when does $w$ is imaginary?" makes sense.

By definition a complex number $a+ib$ is imaginary iff its real part is zero, i.e when $a=0$. So looking at $w(t)=t-3i$, the set of values where the complex number $w(t)$ is imaginary is where $t=0$, i.e. $w(0)=-3i$ that is the point you draw in your picture.