1) A mathemetician is willing to sell you something valued at $i^i. Would you pay him 20 cents for it? 2) Let z=(z1/z2) where z1 = a+ib and z2 = c+id. Show the angle of z is the difference between angle z1 and z2. 3) Show that multiplying any vector by e^ix doesn't alter its length. My attempts: 1) Tried using cosx + isinx to no avail. Honestly have no idea where to begin. 2) First I found the z by multiplying it out. let x be the angle. so i did tanx = bc-ad/ac+bd. Individually, you get tanx1 = b/a and tanx2=d/c which is not the same. 3) in the length formula i end up with cos^2-sin^2 which does alter length. ruined my thanks giving, so i give up onthese.