COMPLEX NUMBERS
JOS
´
E MALAG
´
ONL
´
OPEZ
What is a Number?
Numbers are “symbols” used for measuring and
counting purposes. Examples of numbers are:
•
Natural Numbers:
N
=
{
0
,
1
,
2
,
3
, . . .
}
;
•
Integers:
Z
=
{
0
,
±
1
,
±
2
,
±
3
, . . .
}
;
•
Rationals:
Q
=
{
a
b

a
and
b
are integers
}
.
All these numbers are part of a larger collection of such symbols, the
Real Numbers
, denoted by
R
.
Are There More Numbers?
From Calculus we know that there is
another collection of numbers, the irrationals.
We can say, in short,
that the irrationals are the real numbers that are not rationals. For a
more precise definition we need to consider limits, which we will avoid
since is not part of this course.
Examples of irrational numbers are:
√
2,
π
,
e
, etc.
In general, numbers have been “obtained” when we want to consider
the solution set of a given type of equations. For example, the integers
are obtained from equations of the form
x
+
a
= 0
,
a
∈
N
,
or the rationals, obtained from equations of the form
ax
=
b,
a, b
∈
Z
.
In this course we will consider two extensions of the real numbers, the
complex numbers and matrices.
Complex Numbers
The complex numbers
are
the solutions of equations of the form
x
2
=
a,
a
∈
R
.
1
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Fortunately, we only need to be concerned with the solution of
x
2
=
−
1.
Definition.
The symbol
ı
=
√
−
1
will be called the
imaginary unit
. Notice that
ı
2
=
−
1.
Definition.
The
complex numbers
, denoted by
C
, are defined as
the collection of ALL the expressions of the form
a
+
b ı,
a, b
∈
R
.
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 Fall '10
 JoseMalagonLopez
 Linear Algebra, Algebra, Integers, Natural Numbers, Complex Numbers, Counting, Complex number

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