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The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
(Continued )
Proof: Let us assume that 2
p
is a rational number, that is
2
p
=
n
q
with n 0 (since 2
p
0), q 0 (because division by 0 is not defined), n and q integers, with the
fraction written in reduced form (see Some Facts and Properties of Numbers at the front of the book).
So n and q have no common factors. Since this equality is our only starting point, we can try to perform
some algebra steps, to see what will happen. (Sometimes the logic steps in a proof are guided by
curiosity, or lack of alternatives.) Therefore,
2 =
n
2
q
2
:
Thus
n
2
= 2q
2
:
Since n
2
is a multiple of 2, it is even. Then n must be also even (see Exercise 45 in this chapter).
Therefore ...
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Publisher Resources

ISBN: 9780123822178