May 2014
Intermediate to advanced
350 pages
9h 55m
English
Content preview from Computing
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20 ◾ Computing
Hippasus thrown overboard and drowned! An unfortunate footnote to
this is that
2
is oen called Pythagoras’s constant.
Let us present a simple (non-constructive [6]) proof that there can exist
no rational number x whose square is 2. Suppose such a number existed,
i.e.,x=m/n, where n is not 0, m/n is simplied to the lowest terms (i.e.,m/n
isan irreducible fraction), and x
2
= 2. It follows that (mm)/(nn) = 2. erefore,
both m and n cannot be even numbers—at least one of them must be odd
a
a
a
a
b
b
b
b
c
c
c
c
A
B
D
C
c
cc
c
a
a
a
a
b
b
b
b
FIGURE 3.1 ree dierent proofs of the Pythagorean theorem.
Rational and Irrational Numbers ◾ 21
since ...
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