The Art and Craft of Problem Solving

122 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS

4.2.4 Multip lication. All algebraic manipulations of complex numbers follow the

usual rules, with the additional proviso that j^2 = - 1. Hence, for example,

(2+ 3i)(4+5i) = 8+ l2i + lOi + l5P = -7 + 22i.

A straightforward use of trigonometric identities verifies that if z = rCisa and w =

s Cis f3, then

i.e.,

zw = (rCis a)(sCisf3) = rsCis (a + f3);

The length of zw is the product of th e lengths of z and w, and the angle of

zw is th e sum of the angles of z and w.

This trigonometric derivation is a good exercise, but is not really illuminating. It

doesn't really tell us why the mUltiplication of complex numbers has this satisfying

geometric property. Here is a different way to see it. We will do a specific case:

the geometric action of mUltiplying any complex number z by 3 + 4i. In polar form,

3 +4i = 5Cis e, where e = arctan(4/3), which is approximately 0.93 radians.

1. Since i(a + bi) = -b + ai, it follows (draw a picture !) that
   "Multip lication by i" means "rotate by n/2 counterclockwise:'


2. Likewise, if a is real,
   "Multiplication by a" means "expand by afactor of a."
   For example, multiplication of a complex vector z by 3 produces a new vector
   that has the same direction, but is three times as long. Multiplication of z by 115
   produces a vector with the same direction, but only 115 as long.

3. Multiplying z by 3 + 4i = 5 Cis e means that z is turned into (3 + 4i)z = 3z + 4iz.

This is the sum of two vectors, 3z and 4iz. The first vector is just z expanded by a

factor of 3. The second vector is z rotated by 90 degrees counterclockwise, then

expanded by a factor of 4. So the net result (draw a picture!!!) will be a vector

with length 5 1z1 and angle e + argz.