My lecturer only explained how to plot complex numbers on the complex plane, but he didn't explain how to plot a set of complex numbers.
I did some research online but I didn't find any clear explanation or method. I have an exercise to practice but I don't know how to even start!.
Any help would be really appreciated!
Best Answer
HINT
Let's tackle the first one. A similar approach is required for the other two.
For $z=x+iy$, $x,y\in \mathbb{R}$ we get $$|z-1+i|=1\Rightarrow\\|x-1+i(y+1))|=1\Rightarrow\\\sqrt{(x-1)^2+(y+1)^2}=1\Rightarrow\\(x-1)^2+(y+1)^2=1$$
Does this-hopefully it does-remind you a more general equation of a circle?
Let's also look at the third one as well.
We have $$\operatorname{Re}\Big(\frac{z+1}{z-1}\Big)=\Re\Big(\frac{z+1}{z-1}\Big)=\Re\Big[\frac{(z+1)(\bar{z}-1)}{(z-1)(\bar{z}-1)}\Big]=\Re\Big[\frac{(z+1)(\bar{z}-1)}{|z-1|^2}\Big]\gt1$$ Now for $z=x+iy$, $x,y\in \mathbb{R}$ you can substitute on the last relationship and obtain an equation for $x,y$-an inequality to be precise.