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14 1. Fundamentals

The polar decomposition. Let r = 1z1,6’ = arg z, then z = r(cos B+isin 0).

z=xtyi

Through the introduction of complex numbers, we can find an expression
for the solution of any quadratic equation with real coefficients. Suppose
we try to solve quadratic equations with complex coefficients or polynomial
equations of higher degree. Would it be necessary to extend our number
system still further to accommodate the situation? For example, Leibniz
recognized that a root of t4 + a4 = 0 is given by a-, but apparently
did not realize that u&i could be expressed in the form a + bi. It is a
remarkable fact that no further extension of the number system is required
in order to solve any polynomial equation. In Exercises 10 and 12, this
will be shown insofar as quadratic equations with complex coefficients are
concerned; the more general result will be discussed in Chapter 4.

Exercises

1. Given that the square of every real number is nonnegative, show that
   a complex number can be written in exactly one way in the form
   x + yi with x and y real.


2. (a) Show that the transformation z - iz corresponds to a rotation
   of the complex plane counterclockwise through an angle of 7r/2.
   (b) Describe the result of applying the transformation in (a) twice.
   (c) Let w be an arbitrary fixed complex number. Give a geomet-
   ric description of the transformation z - wz on the complex
   plane.


3. Let z = x+yi = r(cosO+isinO), w = u+vi be two complex numbers.
   Show that

(a) z + w = (x + u) + (y + v)i

(b) zw = (2~ - yv) + (xv + yu)i