THEOREM: √2 is not a rational number

ASSUME: √2 is a rational number
i.e. √2 = X/Y where X, Y are integers with no common factors    (1)
      (cos by definition that what it means to be a rational number)

==> (√2)^2 = (X/Y)^2    (2)  (by squaring (^) each side)
==> 2 = X^2/Y^2    (3)  (by multiplying everything out)
==> 2Y^2 = X^2    (4)  (by multiplying both sides by Y^2)
==> X^2 = 2Y^2    (5)  (by reversing the equation)
==> X^2 is even    (6)  (cos 2 times anything is even)
==> X is even    (7)  (cos if X it were odd, X^2 would be odd)
==> X = 2P where P is some other positive integer     (8)
       (cos by definition that’s what it means for X to be even)
(4) and (8) ==> 2Y^2 = (2P)^2    (9)  (by substituting for X)
==> 2Y^2 = 4P^2    (10)  (by multiplying everything out)
==> Y^2 = 2P^2    (11)  (by dividing both sides by 2)
==> Y^2 is even    (12)  (cos 2 times anything is even)
==> Y is even    (13)  (cos if Y it were odd, Y^2 would be odd)
==> Y = 2Q where Q is some other positive integer     (14)
       (cos by definition that’s what it means for Y to be even)
(1), (8) and (14) ==> X/Y = 2P/2Q  (17)  (by substituting for X and Y)
==> X and Y have a common factor of 2  (18)  (cos we can see the 2’s)
==> The assumption at (1) is false  (19)
       (cos it says: “X, Y are integers with no common factors”)
==> √2 is NOT rational 
Q.E.D


POETRY!