Appendix B
Complex Numbers
P. Danziger
1 Some Useful Sets
1.1 The Empty Set
Deﬁnition 1
The empty set is the set with no elements, denoted by
φ
.
1.2 Number Sets
•
N
=
{
0
,
1
,
2
,
3
,...
}
 The natural numbers.
•
Z
=
{
...,

3
,

2
,

1
,
0
,
1
,
2
,
3
,...
}
 The integers.
•
Q
=
{
x
y

x
∈
Z
∧
y
∈
N
+
}
 The rationals.
•
R
= (
∞
,
∞
)  The Real numbers.
•
I
=
R

Q
(all real numbers which are not rational)  The irrational numbers.
•
C
=
{
x
+
yi

x,y
∈
R
}
 The Complex numbers.
Note:
There are many real numbers which are not rational, e.g.
π
,
√
2 etc.
2 Complex Numbers
2.1 Introduction
We can’t solve the equation
x
2
+ 1 = 0 over the real numbers, so we invent a new number
i
which
is the solution to this equation, i.e.
i
2
=

1.
Complex numbers are numbers of the form
z
=
x
+
iy,
where
x,y
∈
R
.
The set of complex numbers is represented by
C
. Generally we represent Complex numbers by
z
and
w
, and real numbers by
x,y,u,v
, so
z
=
x
+
iy, w
=
u
+
iv, z,w