The Square Root of 2

by Ari Rosenfield

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08:30

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One proof of the number's irrationality:

Assume that √2 is a rational number, meaning that there exists an integer a and an integer b in general such that a / b = √2.
Then √2 can be written as an irreducible fraction a / b such that a and b are coprime integers and (a / b)^2 = 2.
It follows that a^2 / b^2 = 2 and a^2 = 2b^2.
Therefore a^2 is even because it is equal to 2b^2. (2b^2 is necessarily even because it is 2 times another whole number and even numbers are multiples of 2.)
It follows that a must be even (as squares of odd integers are never even).
Because a is even, there exists an integer k that fulfills: a = 2k.
Substituting 2k from step 6 for a in the second equation of step 3: 2b^2 = (2k)^2 is equivalent to 2b^2 = 4k^2, which is equivalent to b^2 = 2k^2.
Because 2k^2 is divisible by two and therefore even, and because 2k^2 = b^2, it follows that b^2 is also even which means that b is even.
By steps 5 and 8 a and b are both even, which contradicts that a / b is irreducible as stated in step 2.

Q.E.D.

credits

released January 1, 2013

Thank you to pat, who recorded several of these. I have no idea how to get in touch with you, nor do I even remember your last name. Also, thanks to Indiana at Secret Pennies Records for releasing the cassettes!

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Ari Rosenfield Springfield, Oregon

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Track Name: Infinite Regress
little one, are you all that's left of those days before they put you to work,
before all men were the slaves of death?
it's empty promises now: the yoke and the harness.
we all work for our bread, but not in infinite regress;
and as the sparrow is my prey, so i'll be your witness.

little one, are you all that's left of those giants who fed on the fruits of the dust,
the first who were pulled from the salt of the depths?
it's a smaller world now, for man and his infinite axe.
we all work for our bread, and the wheat is the sweat of my back.
i'm only a well-spoken ape, it's more than your wingspan i lack.

where skin meets the sky and the ground still frosts over,
you cannot touch it, cannot hold it, cannot know it,
it will elude you forever.
Track Name: Go Where You Will
once constructed a simple plan,
and once with yours it was overlapping
i have no longer the heels to find you,
no longer the desert sky to cry and bind you

oh, you go where you will, oh, little darlin, you go where you will
due to some arbitrary motion, i'll be here still
but oh, little darlin, you go where you will

the great synthesizers of our lifetime must lie pale and fallow
perhaps they mourn for some unborn whose time will come tomorrow
and the further you go, the faster they come
we'll watch you speed across the distance like some lonesome arrow

your breath comes like a constellation, like flipping pages
i have no practical idea what will become of me
hope i'll fall into the sunset like a thousand starlings
leave you to quiet contemplation of these empty ages
Track Name: Never Get to Heaven When You Die
well i fall on my knees, and beggin you please
won't you come and stand awhile by me
you told me more lies than stars in the sky
and you'll never get to heaven when you die

i wish to the lord i'd never been born
or died when i was young
i never would have kissed those rosy red cheeks
or heard your lying tongue

well it's look up and look down that lonesome old road
and it's hang down your little head and cry
you told me more lies than stars in the sky
and you'll never get to heaven when you die
Track Name: The Binding Rope
diamonds in your mouth and pearls in your hair
the binding rope dwells not upon the false, nor the fair
this was to be our story, it's got no end
and where the road may take us:
i beg, the arms of my steadfast friend

for all your children and all your cells,
for all my hope and all my will
when we meet again, i'll be more broken still
and it's then i'll need you most to nurse me back to health

the bow's in my hand, the flail's in my fist
these travelers care not that they'll be sorely missed
but golden fruit from that little tree still hangs
and for it to last forever
so delicious it'd be worth the pain

so i wish it now like i wished it then
for that is half the battle, to be unbound again
wait long enough, perhaps you'll take me off the shelf
it's then i'll need you most to nurse me back to health

it's hot and it's forenoon, the curtains drawn
my twin stands shirtless in the fluorescent dawn
so pay heed my sisters, for it won't be long
and to look into those eyes
all your reason might be scattered and gone
Track Name: The Callers Call, the Weepers Weep
oh, those were the nights:
i had to sing to escape/defeat them,
still louder, in fever

i crossed upon your hearth, i lean upon your shoulder
a liar, and never the bolder
well you are the fool now. you have always been the fool.
lay aside your rice-paddy dreams

oh, liar, you will swallow your tongue,
your eyes will cease to open
you will live in my shadow all your days,
all your days, i will haunt you

oh, youth, they will offer you their backs
they will offer you their money
they would offer their pride, they would offer their very eyes
if only they could have your youth
Track Name: Big Sciota
lived by the river all my life
never knew nothin but sorrow and strife
can't go home again, love
well the river is deep, the river is wide
satan stands on every side
can't go home again, love

east and west the highway roam,
lord have mercy on my poor soul
wish my blue-eyed baby home
pretty little blue-eyed baby-o

pretty little gal i loved so well
she go to heaven, i go to hell
can't go home again, love
well the river is deep, the river is wide
satan stands on every side
can't go home again, love