Additional Examples of Proof by Contradiction
Paul Skoufranis
October 4, 2011
The purpose of this document is to present two common proofs by contradiction that are not related to
linear algebra. The ﬁrst is a very common proof where the second was developed by Euclid.
Example)
Prove
√
2
is irrational.
Proof
: Notice that either
√
2 is rational or
√
2 is irrational. Suppose to the contrary that
√
2 is ratio
nal. Then there exists integers
a,b
∈
Z
such that
b
6
= 0,
a
and
b
have no common factors (meaning that no
natural number larger than 1 divides both), and
√
2 =
a
b
.
By rearranging the equation and squaring both sides, we obtain that
2
b
2
=
a
2
and thus
a
2
must be an even integer. Since the square of an odd integer is odd and
a
2
is even,
a
must be an
even integer. Therefore there exists a
c
∈
Z
such that
a
= 2
c
. By substituting 2
c
for
a
in the above equation,
we obtain that
2
b
2
= (2
c
)
2
= 4
c
2
so
b
2
= 2
c
2
.
Thus, by the same logic as before,
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 Fall '08
 Kong,L
 Math, Linear Algebra, Algebra, Addition, Prime Numbers, Natural number, Prime number, Euclid

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