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34 The Nuts and Bolts of Proof, Third Edition
where ny^O,q^O, and n and q are integers, with the fraction written in
reduced form. (See front material on rational numbers.) Therefore,
n^
Thus,
2 2
n' = pq\
Because n^ is a multiple of p, which is a prime number, then n must be
a multiple of p. (See Exercise 27.) Therefore, we can write n = pk for
some positive integer k. This implies:
or
ph^ = pq^
pk' = q\
Because c^ is a multiple of p, which is a prime number, then ^ must be
a multiple of p. (See Exercise 27 and explain how to use it in this case.)
Therefore, we can write q = pm for some positive integer m and
n pk k
q pm m'
This contradicts the fact that the fraction n/q is already in reduced
form, and proves that ^ is an irrational number. •