Theorem 7: The Square Root of 2 cannot be represented by the ratio of two integers
Proof:
(1) Assume that the squre root of 2 could be represented by a ratio of two integers with a and b the reduced form.
\frac{a}{b} = \sqrt{2}
(2) Squaring both sides:
a^2 = 2b^2
(3) Since a must be even, there exists c such that a=2c and:
a^2 = (2c)^2 = 4c^2 = 2b^2
(4) It follows that b must be even since:
2c^2 = b^2
(5) Then there exists d such that b=2d and:
\sqrt{2} = \frac{2c}{2d} = \frac{c}{d}
(6) But then we have a contradiction since a and b can be reduced to c and d.
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