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Complex Numbers

The complex numbers are an extension of the real numbers containing all roots of quadratic equations. If we define i to be a solution of the equation x2 = -1, then the set C of complex numbers is represented in standard form as

{ a+bi | a,b Î R}.
We often use the variable z = a+bi to represent a complex number. The number a is called the real part of z (Re z) while b is called the imaginary part of z (Im z). Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

We represent complex numbers graphically by associating z = a+bi with the point (a,b) on the complex plane.

Basic Operations

The basic operations on complex numbers are defined as follows:

(a+bi) + (c+di)
=
(a+c) + (b+d)i
(a+bi) - (c+di)
=
(a-c) + (b-d)i
(a+bi)(c+di)
=
(ac-bd) + (bc+ad)i
a+bi
c+di
 = a+bi
c+di
 · c-di
c-di
 = ac+bd
c2+d2
 + bc-ad
c2+d2
 i

In dividing a+bi by c+di, we rationalized the denominator using the fact that (c+di)(c-di) = c2 -cdi +cdi -d2i2 = c2 + d2. The complex numbers c+di and c-di are called complex conjugates.
If z = c+di, we use _
z		 to denote c-di.

Viewed as a vector in the complex plane, z=a+bi has magnitude

,
Notice that z _
z		 =|z|2.
which we call the modulus or absolute value of z.

Examples

  • (2+3i)(2-3i) = 4-6i+6i-9i2 = 4+9 = 13.

  • |2+3i| = |2-3i| = Ö(4+9) = Ö13.

Polar Form

For z = a+bi, let

a
=
rcosq
b
=
rsinq
from which we can also obtain
r
=
  _____
Öa2+b2
 
 = |z|
 
tanq
=
b
a
 .
If you write
q = tan-1 y
x		 ,
be careful to
choose the value
for q in the
correct quadrant.
Then
z = rcosq+ irsinq

Euler's Equation:
eiq = cos q + i sin q
and so, by Euler's Equation, we obtain the polar form

z = reiq.

Here, r is the magnitude of z and q is called the argument of z (arg z). The argument is not unique; we can add multiples of 2p to q without changing z. We define Arg z, the principal value of the argument, to be in (-p,p]. The principal value is unique for each z but creates unavoidable (yet interesting!) complications due to its discontinuity across the negative real axis where it jumps from p to -p. This jump is called a branch cut.

Examples

  • eip = cosp+ isinp = -1

  • 3eip/2 = 3(cos[p / 2] + isin[p / 2]) = 3i

  • 2eip/6 = 2(cos[p / 6] + isin[p / 6]) = Ö3 + i

Multiplication and division of complex numbers is amazingly simple in polar form! If z1 = r1eiq1 and z2 = r2eiq2, then
z1z2
=
r1r2ei(q1 + q2)
z1
z2
=
r1
r2
 ei(q1-q2)
If z = reiq, then _
z		 = re-iq (Do you see why?) and so z _
z		 = (reiq)(re-iq) = r2.

Example

To calculate (1+i)8, we can first rewrite 1+i as Ö2eip/4. Then

(Ö2eip/4)8
=
(Ö2)8ei8p/4
=
16e2pi
=
16.
  _____
Ö12+12
 
=
Ö2
tan-1 æ
ç
è 		1
1
 ö
÷
ø
=
p
4

Roots of Unity

The equation

zn = 1
has n complex-valued solutions, called the nth roots of unity. Since we know each root has magnitude 1, let z = eiq. Then

(eiq)n
=
1
einq
=
ei(2pk)
nq
=
2pk
q
=
2 pk
n
(eiq)n = einq , together with Euler's Equation, gives us deMoivre's Formula:
(cosq+ isinq)n = cos nq+ i sin nq
so the nth roots of unity are of the form
z = ei[(2 pk)/ n].
1 = e0i = e2pki
for k = 0,±1,±2,¼
There are n distinct roots, after which we start duplicating roots already found.
These are evenly spaced around the unit circle.

Example

The 3rd roots of unity are

1
ei[(2p)/ 3]
=
- 1
2
 + i Ö3
2
e-i[(2p)/ 3]
=
- 1
2
 - i Ö3
2
You can verify that (-1/2+ i[(Ö3)/ 2])3 = 1 and (-1/2 - i[(Ö3)/ 2])3 = 1.

This tutorial has reviewed the basics of complex arithmetic. The methods of complex analysis, which build on this background, are both intriguing and powerful!

Key Concepts

Standard Form Polar Form
z
=
a +bi
a
=
Re z
b
=
Im z
|z|
=
Ö______
a2+b2
 
_
z
=
a-bi
a
=
rcosq
b
=
rsinq
r
=
Ö______
a2 + b2
 
tanq
=
b/a
z
=
reiq
r
=
|z|
q
=
arg z
_
z
=
re-iq
Euler's Equation,

eiq = cosq+ isinq,
provides the connection between these two representations of complex numbers.