My 254th book of science Overhauling Sequence theory, Polynomial theory, and including Pythagorean theorem.

The pain in writing this book, is to join it all together, for I am all over the map on this.

I will attempt to join Sequence theory with Polynomial theory, and throw on top of that Pythagorean Triples.

The case to be made on P-Triples is that they have integer solutions for quadratic equations and any higher exponent is broken into pieces of quadratic times the mx+b linear.

As I said before, every polynomial is rewritten as a Whole Counting number of its factors.

Now the question arises as to whether the Algebra Axiom that forces the rightside to be a singular positive Decimal Grid Number, does that also force the factors of the first few terms to be also positive and never negative. Such as in Stewart,Redlin,
Watson page 311, with their R(x) = -2x^5 +4 for which only a small portion of their function graph lies in the 1st Quadrant Only and the rest all lying in the negative quadrants.

In New Math we have only Reality Math, not Fictional Math. We do not have Loch Ness monster math nor Puff the Magic Dragon math nor Ghost Hobgoblin Witches on Flying Broomstick Math. All of math in New Math is Reality Math.

So in New Math, we can accept only that portion of -2x^5 +4 that is inside the 1st Quadrant Only. And maybe that is what is needed in order to cover all pathes as functions of the 1st Quadrant Only.

But now, a function such as S(x) = -2x^5 - 4 would not even cross the 1st Quadrant Only and thus it is not a function that exists, but imagination gone amok.

So it is my hope in this book to not only overhaul Sequence theory, Polynomial theory, but to join together both these theories and to include Pythagorean triples. Why Pythagorean Triples? Because Pythagorean Triples, like Polynomials if expressed as
fractions can always be turned into Whole Counting Numbers. And then, well sequences, since they can only be Decimal Grid Numbers, can be turned into Whole Counting Numbers.

So far, I am worried that not much new math will come of this. But there already, the New Math that forces Old Math polynomial theory into the garbage rubbish bin, is the solution solving. This alone is worthy of trashcanning Old Math's polynomial theory
and much of its algebra.

It is the idea that given any polynomial equation, we find the solutions not by some formula, but by eyeballing in where the polynomial is between two consecutive counting numbers. In Old Math, there was no quintic solution formula and there you had to
eyeball in two consecutive integers for a solution. Provided the equation is not factorable. Mind you we do not throw out everything of Old Math polynomial theory for we do use their zeroes in factors. So if we had a quintic factorable as (x-1)(x-2)(x-3)(
x-7)(x-9), then we have five positive Decimal Grid Number solutions.

But another important finding I want to achieve, if possible, is a broader view of all these disparate theories-- sequence, polynomial, pythagorean triples, tie all three together as one smooth bigger theory, all because the true numbers of mathematics
are Decimal Grid Numbers. Wish me luck. This is not easy.

AP
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Archimedes Plutonium
Jul 25, 2023, 1:04:33 AM (yesterday)
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The nasty habit in Old Math on polynomial graphing, make the y-axis uneven with x-axis.

I was looking to see if a polynomial approximates a quarter circle, and thought I had one in 5x^2, turns out for every 1 on x-axis the y-axis is in units of 10 per same length, distorting the entire graph. This distortion needs to stop. If you cannot
make the axes the same, do not bother in graphing the polynomial.

AP
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Archimedes Plutonium
Jul 25, 2023, 3:00:03 AM (23 hours ago)
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Alright, the 5x^2 looked so much like a length of the circle arc, that I had to investigate it further. We know that 3 noncollinear points in a plane determine a unique circle. So what I want to find out if at least 4 of the points of Y = 5x^2 start to
form a fraction of the circle, perhaps 5 points?

I am hoping for a quarter circle.

Y = 5x^2
x | y
0 | 0
0.25| 0.31
0.5| 1.25
0.75| 2.81
1 | 5

Now graphing those coordinate points keeping the x and y axis the same, not the ugly distortions of unequal lengths of Old Math.

Now getting out my compass to determine the center of a unique circle for the first three points and hoping that the 4th and 5th points are included.

I discover the first 4 coordinate points are included for a circle at center of (-3.2, 2.3)-- a case where the negative quadrants are of some use-- in practice.

And it appears that these 4 points of Y = 5x^2 are a 1/8 of a circle arc.

Now I need to find a way to revolve that 5x^2 to revolve it around so it fills in a full circle. An interpolation such that the 5x^2 fills in a whole circle, not just 1/8 of a circle.

AP
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Archimedes Plutonium
Jul 25, 2023, 3:49:58 AM (22 hours ago)
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On Tuesday, July 25, 2023 at 3:00:03 AM UTC-5, Archimedes Plutonium wrote:

Alright, the 5x^2 looked so much like a length of the circle arc, that I had to investigate it further. We know that 3 noncollinear points in a plane determine a unique circle. So what I want to find out if at least 4 of the points of Y = 5x^2 start to

form a fraction of the circle, perhaps 5 points?

I am hoping for a quarter circle.

Y = 5x^2
x | y
0 | 0
0.25| 0.31
0.5| 1.25
0.75| 2.81
1 | 5

What I am looking for here is a way of pivoting around Y = 5x^2 from its center, so as to create a full circle based upon Y = 5x^2.

And possibly this is going to end up biting me in the behind. I mean, I spent so much time tossing out negative numbers, that it just may happen to be, that I need the other three quadrants with their negative numbers to create a full circle.

It maybe the case that the most elaborate 1/8 circle in all of Polynomial theory is Y = 5x^2 and where we need the other 3 quadrants to fill in the other 7/8 of the circle.

If so, if true, one has to ask why on Earth is 5x^2 so key? Perhaps it is because of Primitive Pythagorean Triples, that the smallest is 3,4,5. And that would directly link up Polynomial theory with Pythagorean Triples.

Ahh, this is getting more and more enticing and wonderful.

AP
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Archimedes Plutonium
Jul 25, 2023, 1:41:49 PM (12 hours ago)
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I was just estimating yesterday that my arc portion of Y = 5x^2 is 1/8 of a circle.

So double checking today. The diameter of circle for the first 4 points is radius 4 thus diameter 8. The arclength from (0,0) to (0.75, 2.81) is 3.2. The circumference is 3.14*8 = 25.12. And 25.12 divided by 3.2 is 7.85. That is within Sigma Error 8/7.85
= 1% Sigma Error.

It probably is the case that Y = 5x^2 comes closest of all of being a arc of a circle using 4 coordinate points and that 4x^2 or 6x^2 fall on accuracy.

Now there is a tiny portion of this circle that still is inside the 1st Quadrant Only, whose arclength is approximately 2, but the majority of this circle with center at (-3.2, 2.3), the majority of this circle 6/8 to 7/8 lies in the 2nd and 3rd
quadrants, the negative numbers region.

Of course I have a polynomial of Y=5x^2 for 3.2 of the 1st Quadrant Only but will need another polynomial for that portion of 2 still in 1st quadrant.

This is a beautiful coming together of geometry with algebra. That we have a 5x^2 polynomial that produces a circle. I had done the Lagrange Interpolation of the quarter circle, but nothing comes of that as simple as Y = 5x^2.

So why is, of all polynomials 5x^2 the one that resembles the circle best of all? Again, I believe it relates the smallest Primitive Pythagorean Triple of 3,4,5 with circular geometry. A direct connection of Polynomials with Pythagorean Theorem and of
Sequences.

AP
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Archimedes Plutonium
1:59 AM (12 hours ago)
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In this overhaul, the graphing of Polynomials of Old Math is all screwed up. For in the graphing there has to be 2 figures. All takes place in 1st Quadrant Only and one figure is a straight flat line such as in 2x^3 +3x^2 + 4x = 6. Where one figure is
the flat line Y = 6 and the other figure is the graphing of Y = 2x^3 + 3x^2 +4x. And where those two figures intercept is a solution/s to the polynomial.

AP
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Archimedes Plutonium
4:20 AM (9 hours ago)
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Alright, a problem I have been storaging for a long time, but no appropriate environment to present it. Here in Sequence and Polynomial theory is just the appropriate setting to pose the question and seek a solution.

The smallest of all Decimal Grid Number System is the 10 Grid which has 100 numbers in total, not counting zero. And is in all of mathematics, functions are not curves but a collection of straightline segments strung together. The reason we think we see
a smooth curve of a circle, is that it is in a large Grid where the numbers are packed close together and our eyes cannot see the points connected by such tiny straightline segments that we think it is a smooth curve.

But the 10 Grid never fools us for there is too much empty space from one point to the next point.

Now the question I have which goes back some decades is the question of how many functions are needed to cover every possible constructible function.

For instance Y = x in 10 Grid is in sequence form 0, 0.1, 0.2, 0.3, . . . 9.9, 10. And in coordinate point form is (0,0), (0.1,0.1), (0.2,0.2) ... (9.9, 9.9)(10,10).

The function Y = 5 is the flat line straight across and its sequence is 5,5,5,...5,5. In coordinate point form is (0,5), (0.1, 5) (0.2, 5)...(9.9,5)(10,5). So far I have two polynomials that represent two functions in 10 Grid.

But what if I asked for a function that is only connecting the first two points of Y = 5. Only connecting (0,5) and (0.1,5) and no other connection, perhaps all the other points are on the x-axis as (0.2, 0) etc. This is a function but its graph is a
line segment. So how does a polynomial represent that function? And to the bigger question. If we draw line segments connecting points in 10 Grid arbitrary, and all being functions, the question is how many polynomials do I need to cover all those
functions? The definition of function in the strict form of each x value has a unique Y value. Can all functions in 10 Grid be covered with just exponent 1 and exponent 2 polynomials?

AP
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Archimedes Plutonium
1:27 PM (9 minutes ago)
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Alright, looks like I will be able to sharpen up the definition of Function.

In the past I was thinking that a function need not entangle with every number on the x-axis, that it can stop along the way and still be a function. So like in 10 Grid, if we had the function Y = x^2 and because the y and x axes go only to 10 and stop
there, that the graph of x^2 stops at between 3.1 and 3.2 because the value is above 10. So the function abruptly ends in 10 Grid. And I need not proceed along the x axis any further for the 10 Grid has no y value beyond 3.1^2.

But then I saw Borrowing as a practical use, that we borrow from 100 Grid or if the function is x^3 we borrow from the 1000 Grid. So all problems solved in the Borrowing technique. And so, all functions can go out to the infinity borderline 10^604 or
even algebraic completeness 10^1208, never a worry of running out of numbers for y axis.

But now a different problem arises-- Step functions. Are they even functions at all in New Math? Or are they a violation of function definition??? I think they are a violation of function definition.

The smallest Grid System (not decimal) is the 2 by 2 Grid, next the 3 by 3 Grid (not decimal).

*...*
*...*


*...*...*
*...*...*
*...*...*

Now we are assigned the task of figuring out how many functions are possible to exist in the 2 by 2 and the 3 by 3. Now Probability theory Counting is not one of my strongest suits in math, for my strongest are logic and geometry.

Given the standard definition of function-- every point on x-axis is assigned a y-value that is a unique y value, and so that eliminates graphs with a perpendicular to x-axis, even eliminates graphs with 2 y values for a given x-value.

So how many functions if we connect one point to the next point is possible in 2by2?? I am guessing the Counting procedure is 2 possibilities in the first column and 2 in the second makes for 2*2 = 4. And for the 3by3 grid it is 3*3*3 = 27.

Let me see if that is true.

*---* one
*...*

*...* two
*---*

*\..*
*...* three

*...*
*/..* four

yes, that exhausts the possibilities for we cannot do a vertical.

Let me get started on the 3by3 to be assured of correctness

*...*...*
*...*...*
*---*---* one

*...*...*
*---*---* two
*...*...*

*---*---* three
*...*...*
*...*...*

*\..*...*
*...*\..*
*...*...* four

*...*...*
*...*/..*
*/..*...* five

*...*...*
*...*---*
*/..*...* six

*...*...*
*...*\..*
*/..*...* seven

to be continued....

Now, the reason Step Functions cannot exist in New Math is because the numbers are discrete and so you have no connection or link up of a next number in the next column. In Old Math, they could have step functions because of their assumed but spurious
notion of continuity that you can have a point next to the step function trailing point, but when the numbers are discrete, this no longer is true. And this exposes the Well Defined Function in New Math, and the ill defined function in Old Math. I cannot
show this in the 3by3 grid but I can show this in the 4 by 4 Grid.

*...*...*...*
*...*...*...*
*...*...*...*
*...*...*...* This is the 4by4 with no function

Now we try a step function in the 4by4

*---*...*...*
*...*---*...* not a function for it uses two y values
*...*...*...*
*...*...*...*

*---*...*...*
*...* *---* not a function for a function must be a continual linking and here is a gap and hole
*...*...*...*
*...*...*...*

In Old Math they could paper mache that blunder of thinking a step function exists by what they called point deletion from one step to the next step. In New Math, Step function never exist (poor Apostol who wrote a Calculus textbook pretty much basing it
on the existence of step functions).

But getting ahead of myself, the 10 Decimal Number Grid has 100 points along the x axis and 100 on the y-axis.
So if my Probability Counting has no mistakes in it, then the total number of possible functions in 10 Grid is 10^10, which is 1 with 10 zeroes after it 10,000,000,000.

And getting even further ahead of myself, I asked how many polynomials would cover all the functions in the 10 Grid? If we used simply the straight line polynomial Y = mx+b, how would we engineer those polynomials to cover every function in 10 Grid????

AP, King of Science

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Now the sequence as a function shows best why step functions are bogus, and phony baloney. Because a sequence would list one point after another and if a you had step functions, you have of the same points in a vertical as not allowed for a function.

And this whole idea of sequences having a derivative and integral is very exciting in New Math for it builds the function as we enter one cell and go to the next cell. We have to include the connection of one cell to the next in order to have a
derivative and integral in that cell (column).

So the derivative and integral force a function to connect up from one column (cell) to the next.

*---*...*...*
*...*...*---* not a function for a function must be a continual linking and here is a gap and hole
*...*...*...*
*...*...*...*
.......^ cell
You see, in one cell marked with the arrow has no connect up and the derivative and integral of Calculus demands and forces every function to be linked and connected in each column (each cell).

AP

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So, well, I added something spectacularly new to Calculus, the most important math subject there is.

I added the Cell concept to Calculus, that you have to do calculus in each cell of the function graph. And the polynomial of straightline Y = mx+b covers each cell. I dismiss the Lagrange Interpolation and made Calculus even far far easier.

I am using the term cell at the moment, but I can rename it if another better name comes up.

In the 10 Grid there are exactly 100 cells on the x-axis. In the 100 Grid there are exactly 100^2 cells, and each cell has a straightline derivative and the area for integral is the rectangle inside that cell built from the midpoint of the derivative
slope line.

AP

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Archimedes Plutonium<plutonium.archimedes@gmail.com>
3:01 PM (2 hours ago)
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The big take away lesson of today-- a function of mathematics has to engulf the entire x-axis. It cannot pick and chose parts of the x-axis and still be a function. This all ties in nicely with the calculus, for the calculus is such that the function is
a time Sequence on the x-axis, as each x-point delivers a unique y-point.

Step functions are phony and fakery, they have no link up of one step to another step, and that is the purpose and magic of the derivative-- it has to link up one column or one cell with the next cell. It is not that numbers are continuous, but in
calculus, the linkup is continuous. A function is with pen or pencil where you cannot lift up the pencil and continue somewhere else, for a function stays put from x=0 to the end of the Grid System.

This is not taught in Old Math education and students go through life assuming or falsely assuming what the answer is to a function linking up. As I myself had the notion in College that a function can stop and start in places. No, once you start a
function at x= 0, it stays a continual running along the x-axis, exhausting all the x-axis points. And the reason Old Math cannot come out and say this is because of their continuum b.s., their continuum allows them error filled ideas such as the step
function that a function can have breaks in it. A function is continuous but a number system must be discrete. And this is probably what confused those in Old Math, especially Cohen. Confused over what is continuous in mathematics. The numbers are not
continuous, but the run of a function, is continuous with no breaks. And the run of a function has to engulf every point of the x-axis, it cannot stop and call it quits, it must go out to infinity borderline 10^604, some even out to algebraic
completeness 10^1208.

So this idea is very new and very correcting of Old Math. Continuity exists in math, not in numbers packed close together, no, not in numbers. Continuity exists in the connection of one number to the Next Discrete Number, moving through empty space
between one number and the next number.

So, already a fantastic new find and clearing up of Old Math, in this project of sequence and polynomial theory.

AP, King of Science
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
4:16 PM (1 hour ago)
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Now I am looking for a method of covering every function in a Decimal Grid System with a polynomial. Not just any polynomial but a straightline polynomial of Y = mx + b.

In New Math, our equations are polynomial on leftside and a positive Decimal Grid number on rightside all alone.

So our leftside is going to look like a function with variable x in every term.

Such as for example 3x^3 +2x^2 + x = 6. We can disregard the rightside. And focus only on the leftside and turn that leftside into nothing but Y = mx+ b.

We divide by x and have x(3x^2 +2x +1) that equals x((3x^2 +2x)+1) that equals x(x(3x +2) +1).

Now we have x(x(3x +2) +1) as our polynomial is all broken down into Y = mx+b as factors, for we have 3x+2 and x and x and 1. We have Y= 3x+2, and Y = x and Y=x and Y=1, all straightlines of Y = mx+b.

This can be done to every polynomial no matter what is the starting exponent. Can be done because the lowest term is x on the leftside. It cannot be done in Old Math because the lowest term is a constant.

In Old Math, the generalized Polynomial function was written as a_n(x^n) + a_n-1(x^n-1) + . . . + a_1x + a_0 = 0

In New Math, the generalized Polynomial function is written as a_n(x^n) + a_n-1(x^n-1) + . . . + a_1x = a_0 which is a positive Decimal Grid Number.

No having a zero and negative number or imaginary number on the rightside of equation makes all the difference in the world.

So let me graph x(x(3x +2) +1) using just counting numbers 0 to 10.

x....|...Y
________

0.........0
1.........6
2.........34
3.........102
etc etc

The curve in this polynomial will be a zig zag

Offering me some idea as to what polynomial functions cover the 10 Decimal Grid can expect.

Let me try another one which stays more inbound to the 100 Grid with x^3 -2x^2 +x. Which turns into x(x(x -2) +1) and its graph is.


x....|...Y
________

0.........0
1.........0
2.........2
3.........12
4.........36
5..........80
etc etc

Here I am trying to get a feel of a zig zag polynomial to cover all the points of a grid system under a function.

I am especially keen on finding sawtooth functions such as in 10 Grid

x....|...Y
________

0.........0
1.........10
2.........0
3.........10
4.........0
5..........10
etc etc

AP

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Archimedes Plutonium<plutonium.archimedes@gmail.com>
4:44 PM (1 hour ago)
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For a sawtooth polynomial function, I am going to need something like a remake of the sine or cosine.


Let me try this.


-1.44x^2 + .44x + 1 -> Y


x....|...Y
________

0.........1
1.........0 excellent so far, the up and down of sawtooth
2.........-3.88 and fizzles out.

So I am looking for a polynomial that at least gets me this much.

x....|...Y
________

0.........1
1.........0 excellent so far, the up and down of sawtooth
2.........1
3..........0

That gets me four x points to be sawtooth and dissect that equation as being a chain of Y= mx+b.

AP
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
5:10 PM (4 minutes ago)
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Alright, I found the solution. I seldom like being in a state of vexation.

The solution involves Series added to Sequence theory. Is it understandable, that you cannot have a Sequence theory without a Series theory, and that that two go hand in hand.

So to solve the Sawtooth function being a pure polynomial, I have to treat each Cell of a Decimal Grid System. In the 10 Grid there are 100 cells on the x-axis, and each cell has a polynomial function designed for that cell.

And what this does is throw out the Lagrange Interpolation, my pride and joy for the past decade, of turning all functions not polynomial, turning them into polynomials over a interval.

Here now, I throw out even the Lagrange Interpolation and ask the person doing the math to graph his/her function in the interval under purview. Graph it, then assign a polynomial Y = mx+b for those cells and write it as a Series of polynomials.

So for example my beloved Sawtooth function in 10 Grid that alternates from 0 to 10 and 10 to 0, up and down with 0 and 10.

x....|...Y
________

0.........0
1.........10
2.........0
3..........10
4..........0
5..........10

And the cell functions are Y=10x or Y= -10x.

Written as a Series for the above function:

P(x) = (Y=0)_0 + (Y=10x)_1 + (Y=-10x)_2 + (Y=10x)_3 etc etc

And well, this then says the total number of straightline functions for the 10 Grid, is exactly 10^10 needed to cover all the functions possible in that 10 Grid. The choices are exactly 10*10*10*10*10*10*10*10*10*10.

The upshot of this is that Sequence theory cannot be developed in isolation of Series theory, the two must be combined as one.

And well, I delight in simplicity and to throw out the Lagrange Interpolation except when it is useful is also a delight.

AP

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