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Saturday, April 10, 2021

The Square Root of 2 is irrational

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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