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 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

|z| =   _____
a2+b2
 
,
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| = (4+9) = 13.

Polar Form

For z = a + b i, let

a
=
r cos(θ)
b
=
r sin(θ)
from which we can also obtain
r
=
  _____
a2+b2
 
= |z|
 
tanθ
=
b
a
.
Let:
φ = tan-1 y
x
,
θ = φ if x > 0,
θ = φ+π if x < 0.
Then
z = r(cosθ+ i sinθ)

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

z = r eiθ.

Here, r is the magnitude of z and θ is called the argument of z (arg z). The argument is not unique; we can add multiples of 2π to θ without changing z. We define Arg z, the principal value of the argument, to be in (-π,π]. 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 π to -π. This jump is called a branch cut.

Examples

  • eiπ = cosπ+ isinπ = -1

  • 3eiπ/2 = 3(cos[π / 2] + isin[π / 2]) = 3i

  • 2eiπ/6 = 2(cos[π / 6] + isin[π / 6]) = 3 + i

Multiplication and division of complex numbers is amazingly simple in polar form! If z1 = r1eiθ1 and z2 = r2eiθ2, then
z1z2
=
r1r2ei(θ1 + θ2)
z1
z2
=
r1
r2
ei(θ1-θ2)
If z = reiθ, then _
z
= re-iθ (Do you see why?) and so z _
z
= (reiθ)(re-iθ) = r2.

Example

To calculate (1+i)8, we can first rewrite 1+i as 2eiπ/4. Then

(2eiπ/4)8
=
(2)8ei8π/4
=
16e2πi
=
16.
  _____
12+12
 
=
2
tan-1



1
1




=
π
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 = eiθ. Then

(eiθ)n
=
1
einθ
=
ei(2πk)
nθ
=
2πk
θ
=
2 πk
n
(eiθ)n = einθ , together with Euler's Equation, gives us deMoivre's Formula:
(cosθ+ isinθ)n = cos nθ+ i sin nθ
so the nth roots of unity are of the form
z = ei[(2 πk)/ n].
1 = e0i = e2πki
for k = 0,±1,±2,…
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[(2π)/ 3]
=
- 1
2
+ i 3
2
e-i[(2π)/ 3]
=
- 1
2
- i 3
2
You can verify that (-1/2+ i[(3)/ 2])3 = 1 and (-1/2 - i[(3)/ 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 [index]

Standard Form Polar Form
z
=
a +bi
a
=
Re z
b
=
Im z
|z|
=
______
a2+b2
 
_
z
=
a-bi
a
=
r cosθ
b
=
rsinθ
r
=
______
a2 + b2
 
tanθ
=
b/a
z
=
reiθ
r
=
|z|
θ
=
arg z
_
z
=
re-iθ
Euler's Equation,

eiθ = cosθ+ isinθ,
provides the connection between these two representations of complex numbers.