Inversion in complex variables.

Question

Hey guys I am given this question on my past midterm, and I cant come about the solution, i know its a mapping of all complex numbers minus the 0 and it maps to itself. So i tried to graph the points (1), (1,1),(1,-1),(1,2). Looks like a horizantal line to me but I dont know how to do this question. Please help out

The inversion is a map $\mathbb{C} \backslash \{0\} \rightarrow \mathbb{C} \backslash \{0\}$ given by $z \rightarrow \frac{1}{z}$ The points $1, 1+i, 1-i, 1+2i$ are on a line, find their images under the inversion. What shape do the image points lie on?

Thank you so much

Answer 1

Compute the inverses of $1,1+i,1-i,1+2i$, then plot those, and observe the shape.

The inverse of $1$ is $1$. (i.e. $(1,0)$)

The inverse of $1+i$ is $\frac{1}{1+i} = \frac{1-i}{1-i}\frac{1}{1+i} = \frac{1-i}{1 + 1} = \frac{1}{2} - \frac{1}{2}i$. (i.e. $(1/2,-1/2)$)

I'll leave the rest to you.