B U Department of Mathematics
Math 232 Introduction To Complex Analysis
Spring 2008 Exercise Sheet 3
1
1.
A complex number
z
=
x
+
iy
may also be visualised as a 2
×
2 matrix
±
x
y

y x
²
Verify that addition and multiplication of complex numbers deﬁned via matrix operations are con
sistent with the usual addition and multiplication rules. What is the matrix representation corre
sponding to
z

1
?
2.
A complex number of the form
α
+
iβ
, where
α,β
∈
Z
\ {
0
}
is called a
Gaussian integer
. A
Gaussian integer
a
is said to be composite if it can be factored into the form
a
=
bc
, where
b
and
c
are both Gaussian integers (excluding
±
1,
±
i
); otherwise it is said to be prime. Show that, as
Gaussian integers, 2 is composite but 3 is prime.
3.
Consider the circline given by the equation
³
³
³
³
z

α
z

β
³
³
³
³
=
λ
, (
λ >
0). We know, if
λ
= 1 we have a
line, say
L
, and the points
α
and
β
are reﬂections of each other in
L
. Now, let
λ
6
= 1. Prove that
there are exactly two points
z
1
,
z
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 Spring '11
 gurel
 Addition, Multiplication, Complex Numbers, Complex number, Gaussian integers, U Department of Mathematics

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