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!
