Tuesday, April 08, 2014

Thrid Version / There are No Irrational Numbers

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.

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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.

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

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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.

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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.

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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.

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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 ?

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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.

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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 §.

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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.

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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.

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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 :

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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.

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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 ?

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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 ?

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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.

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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 = a2 / b2             : 2 = 72 / 52 = 1.96 ( Pretty Close ( ? ) )

a2 = 2 · b2             : 72 = 2 · 52 : 49 = 50

Given then that b2 is 52 = 25

And 25 x 2 is 50; An Even Number;

Forced to be Even by Multiplication by 2

Then a2 is an Even Number

Our Approximations make a2 = 49

But if a2 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 a2 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.; 32 = 9

So there’s something of a Problem with ±7.0710 being Even.

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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 !

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Returning to :

2 = a2 / b2             : 2 = 72 / 52 = 1.96

2 = (2k)2 / b2        : 2 = (2 · 3.5)2 / 52 = 1.96

(2k)2 = 4 · k2

2 = 4 · k2 / b2       : 2 = 4 · 3.52 / 52 = 1.96

2 · b2 = 4 · k2        : 2 · 52 = 4 · 3.52 : 50 = 49

b2 = 2 · k2             : 52 = 2 · 3.52 : 25 = 24.5

Which Means that b2  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 a2 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.

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

√36 = 6

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√9 = 3

√9 = a/b               : a = 3  b = 1 : a/b = 3

9 = a9 / b9             : 9 = 39 / 19 = 19683

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What if it was supposed to be :

9 = a2 / b2             : 9 = 32 / 12 = 9 / 1 = 9

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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

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

It should still work ( ? )

√9 = a/b

9 = a2 / b2             : 9 = 62 / 22 = 36 / 4 = 9

It Still Works !

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

√16 = a/b = 4

a = 12 : b = 3

16 = a2 / b2           : 16 = 122 / 32 = 144 / 9 = 16

Which Still Works.

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

Lettuce Continue :

a2 = 16 · b2           : 122 = 16 · 32 : 144 = 16 · 9 : 144 = 144

Then a2 is an Even Number

144 is an Even Number.

a = 12

Which is also an Even Number.

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

k would then be 6 : 2 · 6 = 12 : 122 = 144

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Returning to :

16 = a2 / b2           : 16 = 122 / 32 = 144 / 9 = 16

16 = (2k)2 / b2      : 16 = (2 · 6)2 / 32 = 16

16 · b2 = (2k)2      : 16 · 32 = (2 · 6)2 : 144 = 144

16 · b2 = 4 · k2      : 16 · 32 = 4 · 36 : 144 = 144

Divide Both Sides by 16 to Free up b2

b2 = .25 · k2          : 32 = .25 · 36 : 9 = 9

But here; b2 is supposed to be proven to be Even;

But .25 · k2 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 12 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 = a2 / b2           : 16 = 42 / 12 = 16 / 1 = 16

a2 = 16 · b2           : 42 = 16 · 12 : 15 = 16 · 1 : 16 = 16

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

k would then be 2 : 2 · 2 = 4 : 42 = 16

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Returning to :

16 = a2 / b2           : 16 = 42 / 12 = 16 / 1 = 16

16 = (2k)2 / b2      : 16 = (2 · 2)2 / 12 = 16

16 · b2 = (2k)2      : 16 · 12 = (2 · 2)2 : 16 = 16

16 · b2 = 4 · k2      : 16 · 12 = 4 · 4 : 16 = 16

Divide Both Sides by 16 to free up b2

b2 = .25 · k2          : 12 = .25 · 4 : 1 = 1

So here again; b2 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 = (x2 = 2.

The Confusion is; Where did The 2 in (x2 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 ( ! )