Algebra of complex numbers - Consider complex numbers z = 1 + j, w = -1 + j, v = -1 - j and u = 1 - j. You may use MATLAB compass to plot vectors corresponding to complex numbers to verify your analytic results.

(a) In the complex plane, indicate the point (x; y) that corresponds to z and then show a vector z̅ that joins the point (x; y) to the origin. What is the magnitude and the angle corresponding to z or z̅?

(b) Do the same for the complex numbers w, v and u. Plot the four complex numbers and find their sum z + w + v + u analytically and graphically.

(c) Find the ratios z/w, w/v, and u/z. Determine the real and imaginary parts of each, as well as their magnitudes and phases. Using the ratios find u/w.

(d) The phase of a complex number is only significant when the magnitude of the complex number is significant. Consider z and y = 10^-16z, compare their magnitudes and phases. What would you say about the phase of y?