Graphics Reference
In-Depth Information
Identify (a,b) Œ
R
2
with the formal expression a +
i
b. (0 +
i
b is abbreviated to
i
b and
i
·1, to
i
.) Using this notation define arithmetic operations + and · on
R
2
as
follows:
(
)
++
(
)
=+
(
)
++
(
)
ab cd ac bd
a
+
i
i
i
(
)
◊
(
)
=-
(
)
++
(
)
+
i
b
c
+
i
d
ac
bd
i
ad
bc
B.9.1. Theorem.
(
R
2
,+,·) is a field denoted by
C
which contains
R
as a subfield under
the identification of a with (a,0).
Proof.
Straightforward.
Thought of as field elements, the elements of
R
2
are called
complex numbers
. It is
easy to check that we have the well-known identity
i
2
=- ,
1
in other words, -1 has a square root in
C
.
Definition.
If z = a +
i
b Œ
C
, then a is called the
real part
of z and b is called the
imaginary part
of z. Define functions Re(z) and Im(z) by
Re
()
=
z
a
and
Im
()
=
z
b
.
The
complex conjugate
of z, , and the
modulus
of z, |z|, are defined by
z
2
2
.
za b
=-
i
and
z
=
ab
+
Here are two simple facts that are easily checked:
(1) The modulus function is multiplicative, that is, |z
1
z
2
| = |z
1
||z
2
| .
(2) If z = a +
i
b, then we have the identity
1
a
ab
b
ab
z
1
=
-
i
=
z
z
2
2
2
2
+
+
B.10
Vector Spaces
Vector spaces are discussed in more detail in Appendix C. We only define them here
and list those basic properties that are needed in the section on field extensions.
Definition.
A
vector space over a field k
is a triple (
V
,+,·) consisting of a set
V
of
objects called
vectors
together with two operations + and · called
vector addition
and
