92.305 Homework 1
Solutions
September 18, 2009
Exercise 1.2.1.
(a) Prove that
√
3 is irrational.
Proof:
Suppose that
√
3 is rational. Then we can write 3 = (
p/q
)
2
with
p,q
∈
Z
and
q
negationslash
= 0.
Without loss of generality, we can arrange this so that
p,q
share no common factors. We
can rewrite the above equation as
p
2
/q
2
= 3, or
p
2
= 3
q
2
. Thus
p
3
is divisible by 3, and so
p
itself must be divisible by 3. Writing
p
= 3
t
with
t
∈
Z
, we have 9
t
2
= (3
t
)
2
= 3
q
2
, or
3
t
2
=
q
2
. This implies that 3 divides
q
2
and hence 3 divides
q
as well. We see that 3 must
divide both
p
and
q
, contradicting their lack of common factors. Therefore, we cannot write
3 as the square of a rational number.
squaresolid
(b) Where does the proof of Theorem 1.1.1 break down if we try to use it to prove
√
4 is
irrational?
Answer:
Let us try to use the method of Theorem 1.1.1 to prove that
√
4 is irrational. We
would suppose
√
4 is rational, so 4 = (
p/q
)
2
with
p,q
∈
Z
and
q
negationslash
= 0, and
p
2
= 4
q
2
. We
deduce that
q
2
and hence
q
is even, so that we can write
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 Spring '10
 MetCalfe
 Logic, Set Theory, Inductive Reasoning, Prime number, Mathematical logic

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