Strangely, I don't find easily on the internet sources about inequalities with complex numbers. In this moment, I am interested to absolute value inequalities with complex numbers but would be good having a wider perspective with all sorts of inequalities with complex numbers.
I have here an exercise which asks: Describe the set $C=\{z: |z−2i|≤ 2\} $ as a subset of the complex plane. Draw a picture.
It doesn't look hard but I have some questions. Usually, am I wrong if I say that we operate this way in R:
$-2 <= |z−2i|≤ 2$ then, we should isolate the variable within the absolute value – let's say that "-2" within the absolute value is a real constant – and we would end up with: $0 <= |x| <= 4$ which would be the interval solution for our inequality. This is because the absolute value theory in R states that we calculate the distance from the origin of the number line.
Anyhow, we are in C now. As I have recently learnt, we consider the magnitude of the complex number seen as a vector – which is quite intuitive and thus easy to learn. However, I don't see in this case the real part of the complex number – am I wrong if I say that we are dealing only with the imaginary part? As z ∈ C, so I am still not sure whether there is a real part or not…
Since this is the first time that I try to solve an inequality with a complex number and I don't find enough material on the internet, I ask here. This is my guess, however, based on logical thinking about inequalities: do I have consider as interval solution of the inequality all the values which don't make the magnitude of the vector zero or negative?
Best Answer
To Henry's comment here is the drawing: