The Art and Craft of Problem Solving

4.2 COMPLEX NUMBERS 121

which is, of course, the length of the vector from the origin to (a, b). Other terms
for magnitude are modulus and absolute value, as well as the more informal term
"length," which we will often use. The direction of this vector is conventionally de­
fined to be the angle that it makes with the horizontal (real) axis, measured counter­
clockwise. This is called the argument of z, denoted argz. Informally, we also call

this the "angle" of z. If (^0) = argz, and r = Izl, we have
z = r(cos 0 + isin 0).
This is called the polar fo rm of z. A handy abbreviation for cos 0 + i sin 0 is Cis 0;
thus we write
z = rCis 0.
For example (all angles are in radians),
(^57) = 57 CisO, - 12 i = 12Cis
3
;, 1 + i = VlCiS�.
bi
a + bi
a
4.2.2 Conjugation. If z = a + bi, we define the conjugate of z to be
z=a- bi.
Geometrically, z is just the reflection of z about the real axis.
4.2.3 Addition and Subtraction. Complex numbers add "componentwise," i.e.,
(a+ bi) + (c + di) = (a +c) + (b +d)i.
Geometrically, complex number addition obeys the "parallelogram rule" of vector ad­
dition: If z and w are complex numbers viewed as vectors, their sum z + w is the
diagonal of the parallelogram with sides z and w with one endpoint at the origin. Like­
wise, the difference z - w is a vector that has the same magnitude and direction as the
vector with starting point at wand ending point at z. Consequently, if Zl , Z 2 , •.• Zn are
complex numbers that sum to zero, when drawn as vectors and placed end-to-end they
form a closed polygon.