Problem of the Month
Hint for Problem 3: Multiplication in two dimensions

1. 1. Expand out the left-hand side, and equate it to the right-hand side of the given equation.

   2. If it’s true for all \((c,d)\) it must be true when \((c,d) = (1,1)\) (or any other specific point).

2. Compute the first six powers and plot them on the Cartesian plane. Question 1(b) may come in handy here.

3. For both (a) and (b), it may help to plot \((0,2), (1,1)\), and \((0,2)*(1,1)\) on the Cartesian plane. For (b), try to write the coordinates of \(D_1\) in terms of \(r_1\) and \(\phi_1\). To do this, begin by assuming \(D_1\) is in the first quadrant of the Cartesian plane and draw out a diagram relating \(r_1\), \(\theta_1\), and the coordinates of \(D_1\).

4. Suppose \(C\) is a solution to \(x^4 = (-119,-120)\). Using \(3\)(b), see if you can figure anything out about \(|C|\) and \(\theta(C)\).

5. Interpret the conditions given in the question in terms of \(|F|\), \(|Y|\), \(\theta(F)\), and \(\theta(Y)\). Then apply 3(b).