2
1. Algebra and Geometry of Complex Numbers
(based on §§17.1–17.3 of Zill)
Definition 1.1
A
complex number
has the form
z = (x, y),
where
x
and
y
are real
numbers.
x
is referred to as the
real part
of
z,
and
y
is referred to as the
imaginary part
of
z
. We write
Re
(z) = x
, Im
(z) = y.
Denote the set of complex numbers by C
I
. Think of the set of real numbers as a subset of
C
I
by writing the real number
x
as
(x, 0)
. The complex number
(0, 1)
is called
i.
Examples
3 = (3, 0), (0, 5), (-1, -
π
), i = (0, 1).
Geometric Representation of a Complex Number- in class.
Definition 1.2
Addition and multiplication of complex numbers, and also multiplication
by reals are given by:
(x, y) + (x
'
, y
'
) = ((x+x
'
), (y+y
'
))
(x, y)(x
'
, y
'
) = ((xx
'
-yy
'
), (xy
'
+x
'
y))
(x, y) = ( x, y).
Geometric Representation of Addition- in class.
(Multiplication later)
Examples 1.3
(a)
3+4 = (3, 0)+(4, 0) = (7, 0) = 7
(b)
3
4 = (3, 0)(4, 0) = (12-0, 0) = (12,
0) = 12
(c)
(0, y) = y(0, 1) = yi
(which we also write as
iy).
(d)
In general,
z = (x, y) = (x, 0) + (0, y) = x + iy.
z = x + iy
(e)
Also,
i
2
= (0, 1)(0, 1) = (-1, 0) = -1.
i
2
= -1
(g)
4 - 3i = (4, -3)
.
Note
In view of (d) above, from now on we shall write the complex number
(x, y)
as
x+iy
.
Definitions 1.4
The
complex conjugate
, z
, of the complex number
z = x+iy
given by
z = x - iy.
The
magnitude
,
|z|
of
z = x+iy
is given by
|z| =
x
2
+y
2
.
Examples and Geometric Representation of Conjugation and Magnitude - in class.
Notes
1.
z + z = (x+iy) + (x-iy) = 2x = 2
Re
(z)
. Therefore,
Re
(z) =
1
2
(z+z)