03/25/2008, 08:28 AM
(This post was last modified: 03/26/2008, 12:57 AM by James Knight.)

Ok you guys are getting closer and closer to what I have defined as zeration.... Anyway I have scribbled a paper that I was going to formally type up but I want to share!!! Anyway, you don't have to believe anything that I put but please have an open mind! BTW I have been researching since September...

Preamble

I want to distinguish operation types (so we're on the same page)

1st type : Regular Operations (zeration, addition, multiplication,tetration etc.)

2nd Type : Left Inverse Operations (Root Type)

3rd type : Right Inverse Operations (Log Type)

First I want to define a special function N(n). This function will give the neutral value of a particular level/rank 'n'. This will be useful to determine values of the Zeration Operation.

A Neutral Value (I think it has been called the "right unit element u" or something like an element incidence, anyway) is the value which is "between" the Right Inverse Operation and the Regular Operation.

Values

Rank n N (n)

1 0

2 1

3 1

Ok so you want proof.... here!

+1 +1 +1

-1 0 1 2

Notice the differences are 1... Watch when they change to 2

+2 +2 +2 +2

-2 0 2 4 6

Notice that as we go LEFT to Right >>>> (some people can be dislexic!!)

We are ADDING "2"

Notice as we move from RIGHT to Left <<<<<<<<<

We are subtracting "2"

Notice that 0 is the "MIDDLE" number between the POSTIVE and the NEGATIVE quantities in both scenarios.

EXAMPLE TWO

x2 x2 x2 x2

1/4 1/2 1 2 4

Again we can see that the Neutral Value is situated in the "MIDDLE" between the NICE whole numbers and the UGLY fractions.

EXAMPLE THREE

2^ 2^ 2^ 2^

0 1 2 4 16

Notice going left to right >>> we are applying a base of two (ie. if we have 'n' then we are finding "2^n")

and right to left <<< we are taking log base two

1 really isn't in the middle but we can see that because we are iterating an operation we are finding the value of the operation of rank one higher.

Examples

eg3) 2#-1 2#0 2#1 2#2 2#3

0 1 2 4 16

eg2) 2^(-2) 2^-1 2^0 2^1 2^2

1/4 1/2 1 2 4

eg1 2(-1) 2(0) 2(1) 2(2)

-2 0 2 4

If you have noticed the pattern it is...

N(n) = x[n+1]0

where x is allowed to be any value

Ok Also I would like to lay down the LAW (sorry about these poor jokes... I am tired...)

The Minus One Law (and the Plus One Law)

-------------------

So if we want to find N[n] value then we need an equation.

It is known that going Left to Right the Sequence for any Level Becomes

x[n]P x[n]P (remember the +2 x2 2^ etc?)

x[n+1](-1) x[n+1]( 0) x[n+1] (1) etc...

where P is the Previous Term

Left to Right is the the Pluse One LAw

x[n](x[n+1](b)) = x[n+1](b+1)

VOILA!

OK the Minus one Law is Reversed <<<<< Right to Left

First I must define a notation

Let x [®n] (y) be the Right Inverse of the Regular Operation Rank n.

(BTW I have developed my own simple notation.. but computer eghh...)

MINUS ONE LAW

(x[n+1]b) [®n] (x) = x [n+1] (b-1)

Also (n = n-1)

(x[n]b) [®n-1] (x) = x [n] (b-1)

OK enough LAWS

Zeration

Rearanging the Minus One Law we get

x [n-1] (x[n](b-1)) = x [n] (b)

Substituting n = 1 and y = x+b -1

Definition 1

x [0] y = y + 1

This is where controversy and contradiction begins... Remember keep an open mind.

From this definition we can prove that zeration is NOT Commutative!!!

(In my mind, it didn't make sense that zeration was commutatave and not associative)

PROOf

x [0] y = y+1

y [0] x = x + 1

y+`1 = x+1 only when y=x therefore NOT commutative

Again it's NOT Associative

a [0](b[0]c) = c+2

(a[0]b)[0]c = c+1

c+1 = c+2 (only at NON FINTE VALUES)THerefore NOT ASSOCIATIVE!

OK this is the good stuff now!

Because zeration is not commutative it has two inverses: a left and a right.

The left inverse of zeration is defined as deltation. Basically they are hyperreal infinite or infinitesimal .

A vertical line x=a can be made by the equation y = x Δ (a-1)

A "Black Graph" can be made by the equation y = x Δ (x-1)

A "White Graph" can be made by the equation y = x Δ x + c

where c is everything but -1

Ok that leaves the exciting right inverse!!!

Since last fall, I had my doubts over the commutativity of zeration as well as the discontinuity. I have spent numerous hours redoing laws and being frustrated. Ok now I would like to present to you

Knightation or Nitation (struggling on what to call it...)

Knightation is the Right Inverse of Zeration.

The Operator J is used to refer to Knightation (it's supposed to look like an ear lobe idea)

if x o y = z then

y = z J x

Definition 2

x J y = x - 1

Knightation and Zeration are two Parallel Lines basically

Also I looked into ranks lower than 0.

It's fairly easy you just have to use the rearranged minus one formula.

Basically Zeration is what you get but!!!

The Neutral Value changes to 1. This however doen'st affect anything...

Ok now I want to look back at the previous definitions of zeration

x o x o x o ... x = x + n

n

This doesn't work when n = 1

LS = x RS = x + 1

There has to be consistency otherwise nothing means anything.

x o x = x + 2 is one of the previous fundamental definitions.

However, if I use Knightation on both sides I get

x = x+1 which is not good.

I think the Neutral Values are to blame for this mishap.

notice that the neutral values are one one less than what they usually are. Also notice '2' is after '1'.

Well I hope you have gained something from this or have been entertained by my random jokes. I hope to actually post the paper I am working on currently.

Also I am a computer programmer and I am going to soon start a program that will compute and graph hyperoperations. Anyway, I am sooooo happy right now because I got accepted to the University of Waterloo!! Soo tired!

Talk to you later,

James

Preamble

I want to distinguish operation types (so we're on the same page)

1st type : Regular Operations (zeration, addition, multiplication,tetration etc.)

2nd Type : Left Inverse Operations (Root Type)

3rd type : Right Inverse Operations (Log Type)

First I want to define a special function N(n). This function will give the neutral value of a particular level/rank 'n'. This will be useful to determine values of the Zeration Operation.

A Neutral Value (I think it has been called the "right unit element u" or something like an element incidence, anyway) is the value which is "between" the Right Inverse Operation and the Regular Operation.

Values

Rank n N (n)

1 0

2 1

3 1

Ok so you want proof.... here!

+1 +1 +1

-1 0 1 2

Notice the differences are 1... Watch when they change to 2

+2 +2 +2 +2

-2 0 2 4 6

Notice that as we go LEFT to Right >>>> (some people can be dislexic!!)

We are ADDING "2"

Notice as we move from RIGHT to Left <<<<<<<<<

We are subtracting "2"

Notice that 0 is the "MIDDLE" number between the POSTIVE and the NEGATIVE quantities in both scenarios.

EXAMPLE TWO

x2 x2 x2 x2

1/4 1/2 1 2 4

Again we can see that the Neutral Value is situated in the "MIDDLE" between the NICE whole numbers and the UGLY fractions.

EXAMPLE THREE

2^ 2^ 2^ 2^

0 1 2 4 16

Notice going left to right >>> we are applying a base of two (ie. if we have 'n' then we are finding "2^n")

and right to left <<< we are taking log base two

1 really isn't in the middle but we can see that because we are iterating an operation we are finding the value of the operation of rank one higher.

Examples

eg3) 2#-1 2#0 2#1 2#2 2#3

0 1 2 4 16

eg2) 2^(-2) 2^-1 2^0 2^1 2^2

1/4 1/2 1 2 4

eg1 2(-1) 2(0) 2(1) 2(2)

-2 0 2 4

If you have noticed the pattern it is...

N(n) = x[n+1]0

where x is allowed to be any value

Ok Also I would like to lay down the LAW (sorry about these poor jokes... I am tired...)

The Minus One Law (and the Plus One Law)

-------------------

So if we want to find N[n] value then we need an equation.

It is known that going Left to Right the Sequence for any Level Becomes

x[n]P x[n]P (remember the +2 x2 2^ etc?)

x[n+1](-1) x[n+1]( 0) x[n+1] (1) etc...

where P is the Previous Term

Left to Right is the the Pluse One LAw

x[n](x[n+1](b)) = x[n+1](b+1)

VOILA!

OK the Minus one Law is Reversed <<<<< Right to Left

First I must define a notation

Let x [®n] (y) be the Right Inverse of the Regular Operation Rank n.

(BTW I have developed my own simple notation.. but computer eghh...)

MINUS ONE LAW

(x[n+1]b) [®n] (x) = x [n+1] (b-1)

Also (n = n-1)

(x[n]b) [®n-1] (x) = x [n] (b-1)

OK enough LAWS

Zeration

Rearanging the Minus One Law we get

x [n-1] (x[n](b-1)) = x [n] (b)

Substituting n = 1 and y = x+b -1

Definition 1

x [0] y = y + 1

This is where controversy and contradiction begins... Remember keep an open mind.

From this definition we can prove that zeration is NOT Commutative!!!

(In my mind, it didn't make sense that zeration was commutatave and not associative)

PROOf

x [0] y = y+1

y [0] x = x + 1

y+`1 = x+1 only when y=x therefore NOT commutative

Again it's NOT Associative

a [0](b[0]c) = c+2

(a[0]b)[0]c = c+1

c+1 = c+2 (only at NON FINTE VALUES)THerefore NOT ASSOCIATIVE!

OK this is the good stuff now!

Because zeration is not commutative it has two inverses: a left and a right.

The left inverse of zeration is defined as deltation. Basically they are hyperreal infinite or infinitesimal .

A vertical line x=a can be made by the equation y = x Δ (a-1)

A "Black Graph" can be made by the equation y = x Δ (x-1)

A "White Graph" can be made by the equation y = x Δ x + c

where c is everything but -1

Ok that leaves the exciting right inverse!!!

Since last fall, I had my doubts over the commutativity of zeration as well as the discontinuity. I have spent numerous hours redoing laws and being frustrated. Ok now I would like to present to you

Knightation or Nitation (struggling on what to call it...)

Knightation is the Right Inverse of Zeration.

The Operator J is used to refer to Knightation (it's supposed to look like an ear lobe idea)

if x o y = z then

y = z J x

Definition 2

x J y = x - 1

Knightation and Zeration are two Parallel Lines basically

Also I looked into ranks lower than 0.

It's fairly easy you just have to use the rearranged minus one formula.

Basically Zeration is what you get but!!!

The Neutral Value changes to 1. This however doen'st affect anything...

Ok now I want to look back at the previous definitions of zeration

x o x o x o ... x = x + n

n

This doesn't work when n = 1

LS = x RS = x + 1

There has to be consistency otherwise nothing means anything.

x o x = x + 2 is one of the previous fundamental definitions.

However, if I use Knightation on both sides I get

x = x+1 which is not good.

I think the Neutral Values are to blame for this mishap.

notice that the neutral values are one one less than what they usually are. Also notice '2' is after '1'.

Well I hope you have gained something from this or have been entertained by my random jokes. I hope to actually post the paper I am working on currently.

Also I am a computer programmer and I am going to soon start a program that will compute and graph hyperoperations. Anyway, I am sooooo happy right now because I got accepted to the University of Waterloo!! Soo tired!

Talk to you later,

James