135
Chapter 5
33. Proof:
Step 1
Assume that
Ï
2
w
is a rational number.
Step 2
If
Ï
2
w
is a rational number, it can be
written as
}
a
b
}
,where
a
and
b
are integers
with no common factors, and
b
Þ
0. If
Ï
2
w
5
}
a
b
}
,then 2
5
}
a
b
2
2
}
, and 2
b
2
5
a
2
.Thus
a
2
is an even number, as is
a
.Because
a
is
even it can be written as 2
n
.
2
b
2
5
¬
a
2
2
b
2
5
¬
(2
n
)
2
2
b
2
5
¬
4
n
2
b
2
5
¬
2
n
2
b
2
is an even number. So,
b
is also an even
number.
Step 3
Because
b
and
a
are both even numbers,
they have a common factor of 2. This
contradicts the definition of rational
numbers. Therefore,
Ï
2
w
is not rational.
34.
Sample answer: Indirect proof is sometimes used
in mystery novels. Answers should include the
following.
• Sherlock Holmes would disprove all possibilities
except the actual solution to a mystery.
• medical diagnosis, trials, scientific research
35.
D;
x
1
80
5
¬
140
x
5
¬
60
A, B, and C are true. D is not true.
36.
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 Fall '10
 Dr.Zhan
 Calculus, PreCalculus, Factors, Integers, Rational number

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