The Art and Craft of Problem Solving

126 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS

In other words, if you start with z and then reciprocate, negate, translate by

1 and then reciprocate again, you will get J (z ).

Next, use this "decomposition" of J(z) plus the geometry lemma that you

proved above to show that every point z on the imaginary axis is mapped to

a circle with diameter 1 with center at 1/2. Draw a good diagram!

(c) The "converse" of (b) is true: not only does J(z) transform the Cartesian grid to

circles, it transforms certain circles to Cartesian grid lines! Verify this explicitly

for the unit circle (circle with radius 1 centered at the origin). Show that J(z)

turns the unit circle into the vertical line consisting of all points with real part

equal to 1/2. In fact, show that

( i8 )

1 1. (J

J e = 2" -2"1 cot "2'

As in (b), do this in two different ways-algebraically and geometrically. Make

sure that you show exactly how the unit circle gets mapped into this line. For

example, it is obvious that - 1 is mapped to 1/2. What happens as you move

counterclockwise, starting from -1 along the unit circle?

Roots of Unity

The zeros of the equation y;!l = 1 are called the nth roots of unity. These numbers have

many beautiful properties that interconnect algebra, geometry and number theory. One

reason for the ubiquity of roots of unity in mathematics is symmetry: roots of unity, in

some sense, epitomize symmetry, as you will see below. (We will be assuming some

knowledge about polynomials and summation. If you are unsure about this material,

consult Chapter 5.)

,� •.• , .•.. --------___ ,=cos600+ isin 600