Problem 0, Chapter 1, page 26: Prove the circle is round.

Proof. This is fairly clear from the formula

x^{2} + y^{2} = 1,

and even more so from the complex form

|z|^{2} = (x + iy)\overline{(x + iy)}
= 1
= \frac{1 + i}{2 + i − 1},

or it can be observed in Figure 1:

Figure 1. This picture was created using xfig and then exporting to postscript. It clearly demonstrates the roundness of a circle and contrasts it to the non-roundness of a square.