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S.S.N:
MAS 3300 – FINAL EXAM
Fall 1992
In giving proofs make sure your reasoning is clear.
The questions have equal value.
Write on ONE side of your paper.
1.
PROVE
If
a
∈
R
then 0
a
= 0.
2.
Use mathematical induction to show that for any positive integer
n
1 + 2 +
···
+
n
=
n
(
n
+ 1)
2
.
3.
Prove
√
20 is irrational. You may NOT assume Theorem 4.3(c).
4.
Show that if
a
∈
Z
n
and
a
and
n
are relatively prime then
a
has a multiplicative inverse in
Z
n
.
5
PROVE that no order relation can be placed on
C
so that
C
becomes an ordered ﬁeld.
6.
Use the
Rational Root Theorem 7.14
to prove that
3
√
5 is irrational.
7.
PROVE that there are inﬁnitely many primes.
8.
PROVE that
√
5
6∈
Q
[
√
3] :=
{
a
+
b
√
3 :
a,b
∈
Q
}
.
9.
PROVE
If
z
1
,
z
2
∈
C
then
z
1
+
z
2
=
z
1
+
z
2
.
10.
FIND all solutions in
C
to
x
4
+ 16 = 0
.
WRITE your solutions in
rectangular
form and
simplify
.
EXHIBIT your solutions geometrically.
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This note was uploaded on 06/03/2011 for the course MAS 3300 taught by Professor Staff during the Spring '08 term at University of Florida.
 Spring '08
 Staff

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