Assignment #4
Due Tuesday, 2/10
Complete the following exercises from chapter 1.7:
� 12, 18, 32
Complete the following exercises from chapter 2.1:
5 4, 8, 10, 16, 20, 27, 28
p (For 20 in particular, justify your answer)
Chapter 1.7 Exercise 12: Prove or disprove that if
a
and
b
are rational numbers, then
a
b
is also
rational.
Disprove by counterexample: Let a = 2 and b = ½. Then 2
½
=
2
, which is irrational
from Example 10 in p. 80.
Chapter 1.7 Exercise 18: Prove that given a real number
x
there exist unique numbers
n
and
ε
such that
x
=
n
+
ε
,
n
is an integer, and 0
≤
ε
< 1.
Given
x
, let
n
be the greatest integer less than or equal to
x
, and let
ε
= x – n.
Clearly 0
≤
ε
≤
1, and
ε
is unique for this
n
.
Any other choice of
n
would cause the required
ε
to be
less than 0 or greater than or equal to 1, so
n
is unique as well.
Chapter 1.7 Exercise 32: Prove that
3
2
is irrational.
This is similar to Example 10 in Section 1.6 in p. 80. Let us assume that
3
2
is rational.
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 Spring '09

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