MATH 300 REVIEW
1.
The Arithmetic of Complex Numbers
1.1.
The Algebra of Complex Numbers.
(1) 1
,
2
,
3
,
many
(2) 1
,
2
,
3
, . . .
(3) Operations on positive integers; addition and multiplication:
a
+
b
and
ab
(4) The Greeks thought of numbers in terms of lengths or areas; the number line; inter
preting numbers in terms of Euclidean geometry.
(5) Solving the linear equation
x
+
b
=
a
: solution
x
=
b

a.
(6) The number 0 and negative numbers.
(7) Solving the linear equation
bx
=
a
: solution
x
=
a/b.
(8) The rational numbers
Q
.
(9) Arithmetic properties of
Q
: commutativity of addition and multiplication; existence
of 0 and 1; inverses for addition and multiplication; associativity of addition and mul
tiplication; the distributive law.
(10) Are there more numbers?
(11) The diagonal of the unit square; irrationality of
√
2; the number field
Q
(
√
2)
.
(12) The real numbers
R
; interpret in terms of the number line.
(13) Are there more numbers? Consider quadratic equations
ax
2
+
bx
+
c
= 0
,
where
a
= 0
.
The solution is
x
=

b
±
√
b
2

4
ac
2
a
.
If
b
2

4
ac <
0 this makes no sense, unless of
course we introduce some new numbers.
In particular
ı
will stand for a particular
square root of

1
.
(14) Adjoin
ı
to
R
to get the complex number field
C
;
a
+
bı,
where
a
and
b
are real numbers.
(15) Arithmetic properties of the complex number field.
(16) Algebraic numbers, transcendental numbers.
1.2.
Point Representation of Complex Numbers.
(1) The Argand diagram.
(2) Real and imaginary axes; real and imaginary parts of a complex number
z
=
x
+
ıy
are
Re
(
z
) =
x, Im
(
z
) =
y.
(3) The modulus of
z
=
x
+
ıy
is

z

:=
x
2
+
y
2
; the conjugate of a complex number,
z
=
x
+
ıy
=
x

ıy
; algebraic properties of conjugation;
z
=
z
⇐⇒
z
is real
.
(4)

z

z
0

=
r
represents a circle.
1.3.
Vectors and Polar Forms.
(1) The complex number
z
=
x
+
ıy
can be identified with the vector (
x, y
)
.
(2) The triangle inequality

z
1
+
z
2
 ≤ 
z
1

+

z
2

.
1
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2
(3) Exercise

z
2
  
z
1
 ≤ 
z
2

z
1

.
(4) The arguments of a complex number; the principal argument (

π < θ
≤
π.
)
(5)
z
=
x
+
ıy
=
r
(cos
θ
+
ı
sin
θ
)
,
where
r
=

z

.
(6)
cisθ
= cos
θ
+
ı
sin
θ
;
z
1
z
2
=
r
1
cisθ
1
r
2
cisθ
2
=
r
1
r
2
cis
(
θ
1
+
θ
2
)
.
(7) Examples 1 +
√
3 = 2
cisπ/
3;
1 +
ı
√
3

ı
=
1
√
2
cis
5
π/
12
.
1.4.
The Complex Exponential.
(1)
exp
(
x
) =
e
x
= 1 +
x
+
1
2!
x
2
+
1
3!
x
3
+
· · ·
converges for all
x.
(2) How do we define
e
z
=
e
x
+
ıy
for a complex number
z
? Want the usual laws of exponents
to hold for the complex exponential function, for example
e
z
1
e
z
2
=
e
z
1
+
z
2
.
The correct
definition is
exp
(
z
) =
e
x
= 1 +
z
+
1
2!
z
2
+
1
3!
z
3
+
· · ·
This converges for all
z.
(3)
e
x
+
ıy
=
e
x
(cos
y
+
ı
sin
y
);

e
z

=
e
x
.
(4) Polar form of a complex number
e
z
=
re
ıθ
.
(5)
z
1
z
2
=
r
1
r
2
e
ı
(
θ
1
+
θ
2
)
.
(6) De Moivre’s formula (cos
θ
+
ı
sin
θ
)
n
= cos
nθ
+
ı
sin
nθ.
1.5.
Powers and Roots.
(1) De Moivre’s formula allows us to find simple formulas for the
n
th
roots of a complex
number
z
=
re
ıθ
.
There are
n
roots, they are
ω
k
=
r
1
/n
e
(
θ
+2
kπ
)
/n
, k
= 0
,
1
, . . . , n

1
.
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