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!

My world is one where shapes collide — I act as a square that
seeks solace, but it’s forced to abide, through life's tight rings, it
strains to pass. Yearning for freedom, a lonely chance to amass.
A longing to think beyond the lines it knows, for a simple shift
—the means to a spirit, that it actually grows.  

As the nights call me softly, while days linger long, in the midst
of their familiar chaos, I must muster my song. While the burden
of now presses heavy and tight; slumber escapes me, lost away
in the night.

I wade through the shadows, each moment isn’t always a gem,
in this fragile ballet, still I cherish them – boxed in my heart; in
this life of a square.

As we walked through the old church, once more,
We saw little Andoni was there, sitting scared,
Asking us: have you forgotten our prayer?
He was angry and very square.

In the corner,
Shrouded by smoke,
Odilon Redon was there.
He watched on with an exalted air.

So we carried little Andoni to the aqueduct
And we sat in the aqueduct, square.
And we sat in the aqueduct until midnight,
Where we had first conceived of our prayer.