The Art and Craft of Problem Solving

120 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS

4.1.24 If you place the digits 0, I, I, 0 clockwise on

a circle, it is possible to read any two-digit binary num­

ber from 00 to II by starting at a certain digit and then

reading clockwise. Is it possible to do this in general?

4.1.25 In a group of nine people, one person knows

two of the others, two people each know four others,

four each know five others, and the remaining two each

know six others. Show that there are three people who

all know each other.

4.1.26 Devise a graph-theoretic recasting of De

4.2 Complex Numbers

Bruijn's rectangle problem (Example 3.4.11 on

page 98).

4.1.27 (Bay Area Mathematical Olympiad 2005)

There are 1000 cities in the country of Euleria, and

some pairs of cities are linked by dirt roads. It is pos­

sible to get from any city to any other city by traveling

along these dirt roads. Prove that the government of

Euleria may pave some of these dirt roads so that ev­

ery city will have an odd number of paved roads lead­

ing out of it.

Long ago you learned how to manipulate the complex numbers C, the set of numbers

of the form a + bi, where a, b are real and i = R. What you might not have learned is

that complex numbers are the crossover artist's dream: like light, which exists simulta­

neously as wave and particle, complex numbers are both algebraic and geometric. You

will not realize their full power until you become comfortable with their geometric,

physical nature. This in tum will help you to become fluent in translating between the

algebraic and the geometric in a wide variety of problems.

We will develop the elementary properties of complex numbers below mostly as

a sequence of exercises and problems. This section is brief, meant only to open your

eyes to some interesting possibilities.^6

Basic Operations

4.2.1 Basic notation and representation of complex numbers. A useful way to depict

complex numbers is via the Gaussian or Argand plane. Take the usual Cartesian

plane, but replace the x-and y-axes with real and imaginary axes, respectively. We

can view each complex number z = a + bi as a point on this plane with coordinates

(a,b). We call a the real part of z and write a = Rez. Likewise, the imaginary

part Imz is equal to b. We can also think of Rez and Imz as the real and imaginary

components of the vector that starts at the origin and ends at (a, b). Hence the complex

number z = a + bi has a double meaning: it is both a point with coordinates (a,b)

and simultaneously the vector that starts at the origin and ends at (a,b). Do keep in

mind, though, that a vector can start anywhere, not just at the origin, and what defines

a vector uniquely is its magnitUde and direction. The magnitude of the complex

number z = a + bi is

(^6) For much more information, we strongly urge you to read at least the first few chapters of our chief inspi­
ration for this section, Tristan Needham's Visual Complex Analysis [ 2 9]. This trail-blazing book is fun to read,
beautifully illustrated, and contains dozens of geometric insights that you will find nowhere else.