1.The inversion mapping takesztow= 1/z. Find the fixed points, meaning points inzthat get mapped to themselves.

2.Each line through the origin in thezplane maps inwworld to a line also through the origin. Try to show this in both rectangular and polar coordinates.

3.A line that does not include the origin inzhas a traditional equation of the formax+by=cwherec0since the origin is not on the line. Find out what this looks like inwworld using algebra in rectangular coordinates.

4.(Strategic ways to think about mathematics - simplify, understand that part well, then go back) Consider the stereographic projection connecting the usual real number line, viewed at thexaxis in thexyplane, with the usual unit circleX²+Y²= 1minus the 2-D North pole(0,1).Write out the algebra that links(x,0)to(X, Y).

5.Staying in this reduced dimensional case, use your answer above to see where the point(1/x,0)goes in relation to where(x,0)went. This presumesx0. Can you extend the mapping on the circle to include the excluded(0,0)and its partner, the point at infinity?

6.We can view the previous problem as thinking about a point on the circle (except the North Pole), mapping it down to the real line (call thatSfor stereographic, finding the inverse point (with exception noted above), (call thatRfor reciprocal), and mapping back to the circle (S^-1). Try to express this symbolically and then see how the symbols suggest that the circle mapping, like the reciprocal, is an involution - do it twice and you end up in your original position!

7.Now lift the last two problems up a dimension to discuss the complex plane and the corresponding sphere in 3-D. Does the symbolic version look different? Is that good or bad in your opinion?