Third
Revisionist Version :

There
are No Irrational Numbers.

By this
i mean that In A Mathematical Reality;

They
Don’t make ‘Sense’.

Everything
in The Mathematical Reality has to Make Sense.

Everything
Numerical, Symbolic, Logical or Allegorical that Doesn’t make Sense falls into
The Mathematical Realm.

Jiggery
Pokery; The Faux Logic, Batman/Riddler Logic, Joke Logic, Schizophrenic
Narratives, Science Fiction, Theoretical Physics, Folk Wisdom, Children’s
Judiciousness, The Prudence of Neanderthals, A Sense of Folly, Political
Certainty, The Wonderland of Alice, or The Ring of Truth— All fall within The
Province of The Mathematical Realm.

The
Mathematical Realm can be Used to Prove any Desirable Thing.

The
Mathematical Reality can only Prove True Things.

- -

The
Tools of The Mathematical Reality however—

Includes
Fractional Propositional Values; Which Allow that Arguments may be Presented in
which Not All, or Any, Of your Axioms, Established Tautologies, Propositions,
Postulates, Premises, Suppositions or Conjectures; Have a Truth Value of Less
than Complete Certainty; Such that your Irrefutable Conclusion may only be
slightly better than a Louisiana Whore’s Intuition.

- -

Arguments
made within The Mathematical Realm are always Contributed with Undeniable
Certainty.

: - - -
- - - - - - - - - - - - : o

Some (
Not All ) of The Warning Signs of An Argument Created with The UnConfined
Machinations from The Mathematical Realm of Jiggery Pokery are :

The
Conclusion is Irrefutable.

Their
Argument Explains Everything. It’s been noted elsewhere that any theory that
takes into account all Observable Data is most certainly wrong, as some of The
Observational Data is Always Wrong.

Their
Argument/Theory includes Variables/Patches/Adjustable Parameters/WholeCloth
‘Constants’ or Other UnObserved or Untestable Conditions to Make it Viable.

Although
Science Insists that A Good Scientific Theory is DisProvable; Many such
Theories or Models Do Not Allow for Their Theories to Be Disproven. e.g.; No
Matter How Unlikely or Improbable a Biological Artifact of Irreducible
Complexity may be; It Must Necessarily have been The Result of Evolutionary
Mutations, As there is No Other Organizing Principle that is ‘Allowed’. e.g.;
No Matter How Many Additional Assumptions are Required to Make The Big Bang
Theory Viable; They Must be True ( Dark Matter, Dark Energy & Expansionist
Periods ) Because Orthodox Astronomers with Tenure have Decided that The Big
Bang has already been too heavily Invested in.

The
Proof is Convoluted; Consisting of Dozens or Hundreds of Steps Which Contain
Terms that are Vaguely Defined, or if A Clear Definition is Requested; It is
said to be Obvious, Childish or Too Complicated for you to Understand.

The
Argument is Entirely Symbolic.

The
‘Plain English’ Jiggery Pokery Argument will contain numerous Well Worn or
Imaginative ‘Fallacies’. This Encyclopedia of Fallacies is Incomplete, As
Jiggery Pokery is always adding New Misconstructions, Inconsistencies,
Wordplay, MisDirections or Regionally Defended Mythological Beliefs to This
Catalogue.

i would
like to offer a few ‘Rules of Thumb’ for Detecting The Validity of A Good
Mathematical Reality Proof;

But i
believe that All Logic is Bunk.

The
Entire Point of ‘Real Logic’ in which only True Things can be Proven to Be
’True’; & Thus Provide The Creator or Witness of Such an Argument with An
Undeniable Certainty of Some Truth— But Jiggery Pokery is Indistinguishable
from Real Logic ( ! )

Just as
The Best Liar nearly always tells The Truth, The Best, Most Convincing Jiggery
Pokery Arguments use Their Tools of Nonsense Sparingly.

Finding
The Nearly Invisible Subtle MisStep, Is The Onus of The Epistemologist.

The Bad
Jiggery Pokery Arguments are easier to Spot.

But If
The Mathematicians or Physicists, Clergy or Politicians hold Such an Argument
or Proof close to their Bosoms, as they would a New Born Baby or Freshly Baked
Pie; They will be loath to Recognize how Ugly their own Spawn may be to An Unbiased
Railway Ticket Agent.

- -

An
Additional Hindrance to Living in A Truly Logical World,

Is that
Logic is Nearly Irrelevant.

We Live
in a World of Pragmatic Solutions which usually only have to stick to The Wall
until The End of The Day.

Physicists
& Mathematicians like to Claim that they are Responsible for The Grandiose
Bridges, iPods, SkyScrappers, Over priced Military Weapons Systems, Airliners,
Elegant Tupperware, The Wide Expanse of Shoes & Scuba Flippers,
Pharmaceuticals & Unappreciated Materials Sciences.

But all
of these things were actually tinkered together by Engineers.

Engineers
have a long history of making things that Fail Catastrophically; Upon Which The
Engineers or Next Generation of Engineers, try something a little Different,
Until it finally Works, Pretty Much.

Pharmaceuticals
are Notorious for Squandering The Lives of Millions of Experimental Lab Animals
as they Try Thousands of Found Compounds or Witches Potions on them; Looking
for one that ‘Seems’ to Cure a Given Disease or Recently Recognized Personality
Disorder.

- -

Reality
is Holistic & UnDelimited.

Mathematics
( at it’s Best & Purist ) Is Digital & Bounded.

It is
deeply laughable that it is such a well established folktale that Mathematics
somehow ‘Define’ The Parameters of The Physical Universe.

: - - - - - - - - - - - - - - - : o

What
got me really going on this idea that there are No Irrational Numbers, &
also; The Concept of Infinity is being horrifically misused in Mathematics—

Started
Quite A While ago;

But more
recently; i was reading a proof that Cantor created to prove that there were a
class of UnCountable Numbers, Notably, Irrational Numbers.

To do
this; He first requires that a List of All Irrational Numbers be Created (
Hypothetically ) & even more annoyingly;

That
is; More Annoying than Creating something that can’t even Hypothetically being
Created without addressing The First Elemental Issue that Such a List would be
impossible;

Is that
he Completely Blows Off The First Step of this List by Suggesting that it Be
Created using ‘Some Clever Method’.

If you
were going to create a list of All Irrational Numbers;

How
would you Begin ?

First
Tier : All Irrational Numbers that begin with .000…

that
continues for Infinity, but ends with a 1, then a 2, then a 3, & so on.

When
you get to 10, You’d Delete One Zero from The Prefix Infinite Series of Zeros,
So that according to This Crazy Rule, After an Infinity of Itinerations, You’d
have The Biggest Infinitely Large Number that would be Free of PreFixatory
Zeros.

Would
The Number right after The Decimal Point be a One or a Nine ?

So that
The Biggest Infinitely Large Number would be .999… …999

Does
that Make Sense ?

Can you
Delete An Intermediate Zero from A Series of Infinite Zeros ?

- - -

The
Problem with Creating a List of Irrational Numbers, Is that it can’t start out
with Non-Irrational Numbers ( see below ), From a Non-Irrational Base; It will
Never ( Not even after an Infinite Number of Itinerations ) Transit to An
Irrational Number, Numbers ( Intermittently Distributed Irrational Numbers ),
Or Reach The Beginning of a Series of Irrational Numbers, Never Returning to
Numbers of A Finite Length.

Such a
List would have to Start out with An Irrational Number & Then Proceed to
Include Nothing But Irrational Numbers.

What is
The First Irrational Number ?

.1000…
…000 ( ? )

.2000…
…000 ( ? )

.3000…
…000 ( ? )

.4000…
…000 ( ? )

…

i’m
thinking that there would be something here on The End to Define it as an
Irrational Number, But is that Possible in A Mathematical Reality ?

Can an
Irrational Number have a Last ( Right Most ) Digit ?

i don’t
think so.

So— How
would you Keep The Irrational Window ‘Open’ ?

And
Even so; If you Proceeded with this Series; They would never be Truly
Irrational Numbers; they Would always be Finite Number with a ad hoc tail
attached to them. Numbers like π would never appear on this list.

What if
you Started with π, Removed The Whole Number 3…

That’s
it.

i got
it.

Start
off with an Irrational Fractional Number like π-3.

.14159265358979323846264338327950288419716939

Then
remove The First Digit from each New Number.

.4159265358979323846264338327950...

.1592653589793238462643383279502...

.5926535897932384626433832795028...

.9265358979323846264338327950288...

.2653589793238462643383279502884...

.6535897932384626433832795028841…

Wouldn’t
that Include all Irrational Numbers ( ? )

But
Wouldn’t Cantor’s Proof that This Wouldn’t Include all Irrational Number Still
Hold ?

Clearly;
There is Something ( ? ) Wrong with Cantor’s Proof,

Or that
you can Make a List ( By any Means ) of a All Irrational Numbers without
applying Cantor’s Proof,

Or—
There is something Wrong with Irrational Numbers Themselves.

- - -

What is
worse; Is that by Introducing a Wondrously Elegant Jiggery Pokery Argument; He
then goes on to Insist that This Hypothetical List of All Irrational Numbers (
Surprise Surprise ) Doesn’t Contain all Irrational Numbers.

**Cantor’s Proof**

The
Objection that Cantor Proposed to Disallow this Series from Representing all of
The Irrational Fractional Values in ( 0,1 ) was that if you first allow that
The numbers on this Grid Represented Only those with An Infinite Series of
Digits—

The
Indices on The Left Represent some Arbitrary Run of Such Fractional Numbers
somewhere along this Series. Also; Since Cantor’s Matrix is Arranged by Some
Esoteric Rule, They do Not Appear to Be Sequential, As The Above Matrix is
Arranged ( ! )

Cantor’s
Proof uses The Argument of Contradiction;

Assuming
that all such Infinite Strings for Any Given Value Does Exist,

So that
all he has to do; Is Create Such a Number According to some Given Rule, Then
Find that Number in The Series Matrix.

The
Rule that he used to Create this New Arbitrary Number is to Pick some Arbitrary
Spot along The Series; Such as ( τ+x1 ) & Then Mark off Each Digit Sliding off at a Diagonal
from That First Position.

This
Rule Generates The New Number : .806673692238…

Then he
Simply Adds 1 to Each Digit, Creating : .917784703349…

Keep in
mind that this Process goes off into Infinity ( To The Right ).

Now
Then :

This
Arrangement Assumes that for Every Point on ( 0,1 ),

It has
a Position Indicated by The τ+xn Indices.

So that
somewhere along this Continuum,

There
Exists The Number .917784703349…

This
seems extremely Obvious Doesn’t it ?

But
Cantor Argues that This Number is Not on this Continuum ( ! )

The
Notation of this Proof Calls each of The Indexed Numbers τ

&
All of The Numbers Highlighted in The Diagonal as §.

- -

Keep in
Mind also that this Rule has affected Every Number in this Matrix;

So that
if we Assume that The Sequence of Digits .917784703349… must Exit somewhere
along this Matrix Continuum, All we Have to do is Find that τ Indices.

Simple
Enough ?

So we
look through The List & Find one that begins with .917784703349.

We then
go to The § along The Diagonal for this Discovered Number is 3, But in that
Same Position of our Number, We’ve Changed it to a 4.

This
Crazy, Completely Contrived Rule applies to Each Number we Find that Almost
Matches our .917784703349… Number. In Each Case; The § Position Number will be
off by 1 ( ! ).

So that
our .917784703349… Doesn’t Exist Anywhere in This Matrix Continuum ( !! )

How
could This Be ?

Cantor
apparently believed, & Argued from this Contrivance that These Created
Numbers Did Exist, As they Must ( Duh ! ) But they were ‘UnCountable’. ( ? )

i think
that if any reasonable traffic cone were to look at this argument; Their
Correct Conclusion is that this is a curious artifice of Jiggery Pokery, &
is merely Gibberish.

Like
All Arguments created with Jiggery Pokery; The Argument seems both Simple &
Irrefutable. It is a little Convoluted; But Deep Thinkers are Used to that.

: - - -
- - - - - - - - - - - - : o

You
would think that any reasonably rational person would step back from that &
say; ‘Hmmm, maybe your initial suggestion that it would be possible to create a
list of all Irrational Numbers was seriously flawed, or maybe The Idea of
Irrational Numbers itself is Flawed.

But no.

Instead;
Mathematicians have universally accepted this proof & adopted The Idea that
Some Infinities are Bigger than other Infinities.

i
suspect that Mathematicians have so readily accepted this; Is because they are
in complete denial or are just unaware that there is This ‘Alternative’ School
of Logic that i Call Jiggery Pokery Logic.

The
Kind of Logic that i call Fractional Propositional Logic can only be used to
Prove True Things.

While
Jiggery Pokery Logic can prove anything.

The
Wondrous thing about Jiggery Pokery Logic is that it is Indistinguishable from
Fractional Propositional Logic.

Just by
Acknowledging that there is a Kind of Logic called Jiggery Pokery Logic &
allowing that many Jiggery Pokery Logical Arguments are being passed off as
Fractional Propositional Logical Arguments—

Proves
that Logic is Bunk.

The
Entire Purpose of Fractional Propositional Logic is to Prove things with
Absolute Certainty; But this Simple & Direct Proof for Jiggery Pokery Logic
Takes that away from Fractional Propositional Logic. You can Never be sure if
your Argument is Really Fractional Propositional Logic or Jiggery Pokery Logic.
You may believe that it’s Really Fractional Propositional Logic, But
Mathematicians & Philosophers have been fooled many times in The Past.

: - - -
- - - - - - - - - - - - : o

Fractional
Propositional Logic ( btw )

Is just
like Classical Propositional Logic, Except that it’s Not Restricted by
Arbitrarily Asserting that some Observed Conditional is 100% True or 100%
False. There’s a lot of times when you’re Not sure or are unable to determine
The precise Truth Value of your Antecedent or Succedent.

The
Operators for Fractional Proposition work like this :

: - - -
- - - - - - - - - - - - : o

Returning
to Irrational Numbers for a Moment :

i would
suggest that it would be possible to Create a List of all Fractional Numbers
using this series :

.0

.1

.2

.3

…

.9

.01

.11

.21

…

.99

.001

.101

.201

…

&
So on.

This
would Include All Fractional Values between Zero & One

Excluding
One itself.

But
they would all be Finite in Length.

It
would Continue to Infinity, & Provide a Seamless Continuum of All
Additional Points between any Two Specified Points.

It
would Exclude All Irrational Numbers.

They’re
Not needed.

The
Problem then is that The ‘Idea’ of Irrational Numbers is Sooooo Easy to
‘Imagine’. You’re initial response is; Of course they Exist, How could they Not
Exist ?

Like π
for instance; It has been proven that π is Irrational !

Really.

If you
Take A Circle on a Sheet of Paper & Try to Measure The Diameter ( which is
Defined as a Given Length ) & then try to Measure The Circumference, You’re
going to find that it’s very Difficult to Get a Good Measurement.

So
then; You think; Let’s just imagine that A Circle is a Polygon with Infinite
Sides! We can Easily Calculate The Circumference of a Polygon to any Desired
Degree of Accuracy—

If we
hold that A Circle is a Polygon with Infinite Sides, Which it is Not ( ! )

On The
Simplest Level; You have to define The Radius of The Polygon from The Center to
The Vertice or Middle of A Side; Both of Which are Wrong.

Neither
can you Calculate Both Polygons, One for The Inside Limit & one for The
Outside Limit & Average them out; Because both are going to continue to
Infinity, By Passing The Correct Value.

: - - -
- - - - - - - - - - - - : o

This
Brings up The Problem of Series’ that purport to Create an Irrational Number
that Corresponds to A Labeled or Otherwise Defined Value. The Series is
=Designed= by Definition to continue on forever & ever. It is Not Created
to Suddenly Stop when it finds The Correct Measurement.

Even
when you have a Sigma Series that Converges on a Fixed Value; The Series will
never Reach it. It will get closer & closer & closer & closer,
without end.

There
are also very hinky Series’ that seem to converge on a fixed value, but if you
wait long enough, allowing The calculation to repeat it’s itineration cycle
enough times, it will slip past it’s previously assumed fixed value.

One of
The Principle Warning Signs of Jiggery Pokery, is Excessively &/or
Convoluted Steps to Prove (x.

The
Wikipedia Page that Professes to Reveal The Proof for The Irrationality of π is
page after page of Dense Calculus Gibberish.

It
would have been very nice if The Author were to Express The Proof in plain
English, A Step by Step Recounting for An Attentive 5th Grader to Follow. But i
very much Suspect that if you were to ask someone that ostensibly understood
this Explanation that was provided; They Would claim that such a Plain English
Translation would be Impossible.

But
what seems most obvious to me; What Justification is given that Relates these
proofs to π ? You’ve got your π, you’ve got The Proofs, why do you think one
corresponds to The other ?

: - - -
- - - - - - - - - - - - : o

Another
thing that occurred to me; If you going to allow that it is possible to
‘Define’ Irrational Numbers—

Would
it be possible to Define an Irrational Number like this;

.000…
…001

That
is; We have an Number that goes on to Infinity, But it’s Last Digit is 1. Of
Course; A Number like this that goes on forever, doesn’t have a last digit. But
if A Definition like this is Impossible, Then would this one also be
Impossible;

.000…
…010… …000…

You’ve
got a number that starts out as Zeros; Goes on for Infinity, Then somewhere in
The Middle, has a 1, then goes on for infinity with more zeros.

Can you
do that ?

Would
that Number have a Unique & Different Value than:

.000…
…001… …000…

?

or even

.000…
…020… …000…

Would
that be an entirely Different Number ?

Would
it be Greater or Lesser than The Number with a 1 in The Middle?

The
Problem would be that it’s Not ‘Really’ in The Middle.

You
can’t have a Middle to Infinity.

Which
brings up The Problem of A Universe that has Existed for Infinity. Sure; We’re
in A Universe that ‘Started’ only a few Billion Years ago, But if We’re in a
Larger ‘SuperVerse’, Then this Infinite Universe Hypothesis holds.

In
which case; The Universe should have achieved Conscious Perfection after an
Infinity of Existence, But We’re here now after an Infinity of Existence. Where
is The Perfection ?

Is this
The Best, Most Perfect of all Realities?

Is it
more Perfect than yesterday ?

: - - -
- - - - - - - - - - - - : o

The
Fallacy of The Elegant Solution

i
suspect that The Reason that Mathematicians believe that π is Irrational is
that it doesn’t seem to have a simple elegant solution. Wouldn’t it be nice if
gawd had provided this π ration with a nice simple fractional value ?

But
it’s Not.

What if
Numerator was a few billion digits long ?

There
is No Way that you could ‘Measure’ that by any means.

You
couldn’t even measure it if it were only a few dozen digits long.

So
without an Elegant Solution Forthcoming; The Mathematicians created a New Kind
of Elegant Solution, & Created all The Necessary Crazy Proofs to Prove it.

: - - -
- - - - - - - - - - - - : o

Proof

Lettuce
assume that a/b = √2

For
this a/b to be The Simplest Terms

Both a
or b may Not be Even or Divisible by a Common Factor

√2 =
1.4142…

√2 =
a/b : a = 7 b = 5 : a/b = 1.4

*a & b are Wildly Approximated*

*So that we can see how The Algebraics are Working ( ? )*

2 = a

^{2}/ b^{2}: 2 = 7^{2}/ 5^{2}= 1.96*( Pretty Close ( ? ) )*
a

^{2}= 2 · b^{2}: 7^{2}= 2 · 5^{2}: 49 = 50
Given
then that b

^{2 }is 5^{2}= 25
And 25
x 2 is 50; An Even Number;

Forced
to be Even by Multiplication by 2

Then a

^{2 }is an Even Number
Our
Approximations make a

^{2}= 49
But if
a

^{2}were 50 : a = 7.0710
But
we’re also asserting that a is axiomatically a Whole Number

So that
if This were to work out so that a

^{2}were to be a Whole Number
&
(a would also be a Whole Number,

Then (a
would be Necessarily Even,

Since
any Odd Number Squared is Odd.

e.g.; 3

^{2}= 9
So
there’s something of a Problem with ±7.0710 being Even.

- - -

But
Never Mind that —

Let us
then Arbitrarily Replace (a with 2·k ( or 2k )

k would
then be 3.5

But for
this to ‘Work Out’; k would have to be a Whole Number

But by
working with Symbolic Algebraics;

These
Fractions are Swept under The Carpeting !

- -

Returning
to :

2 = a

^{2}/ b^{2}: 2 = 7^{2}/ 5^{2}= 1.96
2 =
(2k)

^{2}/ b^{2}: 2 = (2 · 3.5)^{2}/ 5^{2}= 1.96
(2k)

^{2}= 4 · k^{2}
2 = 4 ·
k

^{2}/ b^{2}: 2 = 4 · 3.5^{2}/ 5^{2}= 1.96
2 · b

^{2 }= 4 · k^{2}: 2 · 5^{2 }= 4 · 3.5^{2}: 50 = 49
b

^{2 }= 2 · k^{2}: 5^{2 }= 2 · 3.5^{2}: 25 = 24.5
Which
Means that b

^{2}must be Even by The Same Logic Expressed above ( 2 · x ) must be an Even Number.
Now
both a & b are Even according to this Juggling Act,

But
this all Assumes that a & b Start out as Whole Numbers;

And all
of The Permutations that they Endure allow their SubDivisions to Remain Whole
Numbers too.

Which
They don’t.

Even if
you were to somehow allow that these Conditionals were Met;

The
Jiggery Pokery here is Only Asserting that (a or (b are Not Odd or Even. It is
somehow insisting that (a or (b are Outside The Realm of Whole Numbers.

It
seems far more Reasonable to assume that this ‘Argument’ is A Paradox of The
Zeno Type; And that while it seems Reasonable; It tacitly asserts things that
it shouldn’t.

e.g.:
That If a

^{2 }is a Whole Even Number; (a must be a Whole Number as Well. It was assumed that (a was a Whole Number at The Beginning of The Argument; But then Craziness set in.
- - -

On a
More Obvious Level; Doesn’t this Argument Structure assert that all Square
Roots are Irrational; Which is Clearly Wrong.

√36 = 6

- - -

√9 = 3

√9 =
a/b : a = 3 b = 1 : a/b = 3

9 = a

^{9}/ b^{9}: 9 = 3^{9}/ 1^{9}= 19683
- - -

What if
it was supposed to be :

9 = a

^{2}/ b^{2}: 9 = 3^{2}/ 1^{2}= 9 / 1 = 9
- - -

Oh! So
2 works for Everything.

So that
The 2 didn’t come from The 2 in √2

It came
from The Square of (x : √x

- - -

So if
we try a = 6 & b = 2 : 6/2 = 3

It
should still work ( ? )

√9 =
a/b

9 = a

^{2}/ b^{2}: 9 = 6^{2}/ 2^{2}= 36 / 4 = 9
It
Still Works !

:*’``’*:-.,_,.-:*’``’*:-.,_,.-:*’``’*:-.,_,.-:*’``’*:-.,_,.-:*’``’*

√16 =
a/b = 4

a = 12
: b = 3

16 = a

^{2}/ b^{2}: 16 = 12^{2}/ 3^{2}= 144 / 9 = 16
Which
Still Works.

- -

So
shouldn’t this mean that √16 is Irrational ?

Lettuce
Continue :

a

^{2}= 16 · b^{2}: 12^{2}= 16 · 3^{2}: 144 = 16 · 9 : 144 = 144
Then a

^{2 }is an Even Number
144 is
an Even Number.

a = 12

Which
is also an Even Number.

- - -

Let us
then Arbitrarily Replace (a with 2·k ( or 2k )

k would
then be 6 : 2 · 6 = 12 : 12

^{2}= 144
- -

Returning
to :

16 = a

^{2}/ b^{2}: 16 = 12^{2}/ 3^{2}= 144 / 9 = 16
16 =
(2k)

^{2}/ b^{2}: 16 = (2 · 6)^{2}/ 3^{2}= 16
16 · b

^{2 }= (2k)^{2}: 16 · 3^{2 }= (2 · 6)^{2 }: 144 = 144
16 · b

^{2 }= 4 · k^{2}: 16 · 3^{2 }= 4 · 36 : 144 = 144
Divide
Both Sides by 16 to Free up b

^{2}
b

^{2 }= .25 · k^{2}: 3^{2 }= .25 · 36 : 9 = 9
But
here; b

^{2 }is supposed to be proven to be Even;
But .25
· k

^{2 }doesn’t prove that.
What
Happened ?

In The
Original Proof for √2

a &
b were supposed to be Whole Numbers that Weren’t Both Even,

But in
our ReProof with √16—

Instead
of Using The Simplest Fraction for 4, Which would have been 4/1; We Used 12/3,
Because if we Used 1, when you use 1

^{2 }in an Expression; Instead of getting a ‘responsible’ Answer; You’ll get 1.
If we
Rework this with a = 4 & b = 1

We Get
:

√16 =
a/b = 4

a = 4 :
b = 1

16 = a

^{2}/ b^{2}: 16 = 4^{2}/ 1^{2}= 16 / 1 = 16
a

^{2}= 16 · b^{2}: 4^{2}= 16 · 1^{2}: 15 = 16 · 1 : 16 = 16
Let us
then Arbitrarily Replace (a with 2·k ( or 2k )

k would
then be 2 : 2 · 2 = 4 : 4

^{2}= 16
- -

Returning
to :

16 = a

^{2}/ b^{2}: 16 = 4^{2}/ 1^{2}= 16 / 1 = 16
16 =
(2k)

^{2}/ b^{2}: 16 = (2 · 2)^{2}/ 1^{2}= 16
16 · b

^{2 }= (2k)^{2}: 16 · 1^{2 }= (2 · 2)^{2 }: 16 = 16
16 · b

^{2 }= 4 · k^{2}: 16 · 1^{2 }= 4 · 4 : 16 = 16
Divide
Both Sides by 16 to free up b

^{2 }
b

^{2 }= .25 · k^{2}: 1^{2 }= .25 · 4 : 1 = 1
So here
again; b

^{2}is supposed to be proven to be Even; And It’s Not.
The
Original Proof is A Jiggery Pokery Argument that is Founded on The Confusion of
The √2 which means √2 = (x

Which
means that (x · (x = (x

^{2}= 2.
The
Confusion is; Where did The 2 in (x

^{2}come from ?
Also;
By Using Symbolic a’s & b’s; We’re never able to see if a or b is actually
Even or Odd. The Algebraics tell us to believe if a or b is Even or Odd, When
they are Not ( ! )

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