This preview shows pages 1–3. Sign up to view the full content.
2.35
SECTION 2.4: COMPLEX NUMBERS
Let
a
,
b
,
c
, and
d
represent real numbers.
PART A: COMPLEX NUMBERS
i
, the Imaginary Unit
We define:
i
=
−
1
.
i
2
=
−
1
If
c
is a positive real number
c
∈
R
+
( )
, then
−
c
=
i c
.
Note: We often prefer writing
i c
, as opposed to
c
i
, because we don’t want to
be confused about what is included in the radicand.
Examples
−
15
=
i
15
−
16
=
i
16
=
4
i
−
18
=
i
18
=
i
9
⋅
2
=
3
i
2
Here, we used the fact that 9 is the largest perfect square that divides
18 evenly. The 9 “comes out” of the square root radical as
9
, or 3.
Standard Form for a Complex Number
a
+
bi
, where
a
and
b
are real numbers
a
,
b
∈
R
( )
.
a
is the real part;
b
or
bi
is the imaginary part.
C
is the set of all complex numbers, which includes all real numbers.
In other words,
R
⊆
C
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Examples
2, 3
i
, and
2
+
3
i
are all complex numbers.
2 is also a real number.
3
i
is called a pure imaginary number, because
a
=
0
and
b
≠
0
here.
2
+
3
i
is called an imaginary number, because it is a nonreal complex
number.
PART B: THE COMPLEX PLANE
The real number line (below) exhibits a linear ordering of the real numbers.
In other words, if
c
and
d
are real numbers, then exactly one of the following must be
true:
c
<
d
,
c
>
d
, or
c
=
d
.
The complex plane (below) exhibits no such linear ordering of the complex numbers.
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '11
 staff
 Calculus, Real Numbers, Complex Numbers

Click to edit the document details