#1 TEACHING TRUE MATHEMATICS, 2018, by Archimedes Plutonium

[Archimedes Plutonium]

#1 TEACHING TRUE MATHEMATICS, 2018, by Archimedes Plutonium

Preface:

Alright I devote the entire year 2018 to doing this textbook on math. A textbook like none other for it takes the student from age 8 until age 26, throughout the subject of mathematics. The one book for all of mathematics.

It is historic in the sense that it cleans out all the errors of Old Math, and believe you me, the errors in Old Math were vast. So vast, that the best thing is to rewrite all the math textbooks, throw out the old.

And historic in the sense the book is used from ages 8 to 26 as a single math text, where teachers supplement blank pages to tailor fit their class. I write this textbook with the Common Core Curriculum in the USA, in mind. So I know what Grade level students are taught specific topics.

I write this textbook in a style never seen before in science textbooks, for I am talking most often to the student, but also, talking very much to the teacher. So both the student and teacher are addressed in this textbook.

This book is unique, in that you need only this text of math for all school needs. None of this buying a new book for each course. And it requires work by the teacher to fill in blanks. I cover only the very basic. But the beauty of that, is many students bored with the lessons on hand will keep on looking further ahead in the pages, at more advanced math-- for it is available and open to their eyes to wander on into what their future math lessons are going to be. I am a pragmatist teacher myself, and know that if math is available, only more advanced, that the students will often exceed expectations and learn what was meant for them years into the future. I see it the case that some very bright High School students will have mastered the College level math from this textbook, just because it is in plain sight.

Now in writing this text I need commentary along with lessons. So the structure of the book is lessons and then commentary. Arranged in chapters, and arranged for age level of students. The 8 year old starts at the beginning. The High School student starts near the middle, the college student starts near the end. But all ages can wander through this textbook learning more and more about math.

And this textbook is unique in another way, it is written Logically, a logical minded author, am I. Logic means clarity and order. So few, so very very few books of science are written by authors who possess logic.

But, I was a teacher myself, and know the key number one element of being a teacher, a good teacher, is that a good teacher teaches students at a level of age, that the student can readily comprehend and understand the material. If a student cannot understand what is being taught, is a bad teacher. It is not the student's fault, but the teachers fault. The teacher is not at the level where the student can learn. The teacher is "over the heads of the students". And that is the worse kind of teacher. Over the heads of the student and makes the student fear math, and forever turned off by math. Math, never needs to be feared.

But another sign of a excellent teacher, is he/she comes to class every day, prepared, not ad lib an hour, but prepared the night before.

So, teacher, to young students, tell them this book is used by them throughout their life, but for the classroom at a certain age, only this part of the book are they going to learn. Tell them the rest of the book they will learn in later classes in later years, not your class. Your class only is responsible for this part of the textbook. And of course, well, the brightest students will be bored with what they are responsible for, and will peek ahead into future lessons. And, let them do that.

But, in all my lessons of this textbook, I strive for CLARITY, strive for Understanding, and strive for Logic and Order. A text on math written by a logical author.

Bon voyage

Lesson 1, Chapter 1, Counting, and "What are Numbers" Re: _Textbook :: TEACHING TRUE MATHEMATICS, 2018

Lesson 1, Chapter 1, Counting, and "What are Numbers"

Sad to say, the most important use of mathematics by every person alive is to be able to know how much money, count money and know numbers to count correctly. But, then again, that is why everyone needs to learn math-- to be able to handle money. Very difficult to live in modern society and not be able to handle money because you have no math.

We start this book for 8 year old students and to Count and then from Counting learn "What is a number".

Have the students write out all the numbers from 0 to 100. They are Counting

Something like this::

90, 91, 92, 93, 94, 95 96, 97, 98, 99, 100
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
00, 01, 02, 03, 04, 05, 06, 07, 08, 09,

Now, teacher, warn the students that writing 1 as 01 or 2 as 02, etc, does not affect the number. Teach them a 0 in front of a number does not affect the value, but a 0 after does affect the value. Such as 1 is 01 but 10 is ten, and 02 is 2 but 20 is twenty. So, teacher, spend a good deal of time teaching when 0 affects the value of a number. Tell the students that the reason they are writing a 0 in front of 1 or in front of 0 itself is so there are two digits, yes two digits for all these numbers, except for 100.

Now, ask the students to Count those numbers as if they were whole dollars (in other countries, they may not or may have a currency- a money in decimal, if they do not, well, just use USA dollar with cents).

So, zero dollars, 1 dollar is 01, 2 dollars is 02, etc, etc. Have the entire class count up to 100 dollars.

90, 91, 92, 93, 94, 95 96, 97, 98, 99, 100
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
00, 01, 02, 03, 04, 05, 06, 07, 08, 09,


Now I am a firm believer in repetition by students, for they memorize and it boosts their confidence-- that they can do this, do math. They understand it and do it easily.

Now, we take our chart and place a decimal point "." between digits. We just took our above table of numbers 00 to 100 and added a decimal point. Teacher, it is not far away where we teach Decimals, but let the students do mostly counting for now, and the decimal explanation will come soon enough.

9.0, 9.1, 9.2, 9.3, 9.4, 9.5 9.6, 9.7, 9.8, 9.9, 10.0
8.0, 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9,
7.0, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, 7.9,
6.0, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9,
5.0, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9,
4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9,
3.0, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9,
2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9,
1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9,
0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9,

Now, have the students count these numbers.
Zero point zero, zero point 1, zero point 2, zero point 3, zero point 4, etc etc. Count all the way to nine point nine, ten point zero.

Now, teacher, have them count that same table as if they were dollars and cents. Zero cents, ten cents is .1, twenty cents is .2, thirty cents is point .3, forty cents is .4, on up to 9 dollars and ninety cents is 9.9, ten dollars is 10.0.

AP

[Archimedes Plutonium]

Lesson 2, Chapter 1, Counting & Decimals Re: _Textbook :: TEACHING TRUE MATHEMATICS, 2018

Picking up where we left off the last time. Only this time we move the decimal point over one more time.

.90, .91, .92, .93, .94, .95, .96, .97, .98, .99, 1.0
.80, .81, .82, .83, .84, .85, .86, .87, .88, .89,
.70, .71, .72, .73, .74, .75, .76, .77, .78, .79,
.60, .61, .62, .63, .64, .65, .66, .67, .68, .69,
.50, .51, .52, .53, .54, .55, .56, .57, .58, .59,
.40, .41, .42, .43, .44, .45, .46, .47, .48, .49,
.30, .31, .32, .33, .34, .35, .36, .37, .38, .39,
.20, .21, .22, .23, .24, .25, .26, .27, .28, .29,
.10, .11, .12, .13, .14, .15, .16, .17, .18, .19,
.00, .01, .02, .03, .04, .05, .06, .07, .08, .09,


Now teacher, prepare for class the night before, for students deserve to have a teacher that prepares, and does not walk into the class just ad-libing, just winging it. Come prepared for each lesson. The difference between a excellent teacher and those beneath, is preparedness. Although Grade schoolers may not recognize a prepared teacher from not prepared, by High School, the students do recognize.

That is why I like to blend in one lesson with the next, so that the blend shows the preparedness.

Now, have the students make the above table where we moved that decimal point over once more. Now count these numbers.

Point zero zero is 0 cents. Point zero 1 is 1 cent, point zero 2 is 2 cents, point zero 3 is 3 cents. Count all the way to point nine nine is 99 cents, one point zero is 1 dollar.

Now, teacher, if the class is not fluent in these tables, repeat them until they are fluent and do other exercises, until they are very good at this. In fact do exercises focused on decimals.

Now the next exercise once the students are fluent in the above tables-- homework, quizzes, tests, and make them easy quiz and test, build confidence. Is the teaching of Decimal Numbers themselves. From a look at the Common Core Curriculum, either the students of 8 years age have seen the decimals before, or they have not. So the teacher here has to determine if the students have been already taught the decimal numbers or not, and with that determination the teacher must supplement my teaching with a good chapter of common core on decimals. I do not know how the Common Core teaches decimals. And my own personal memory can no longer pull that experience up and what age I was. I vaguely recall doing fractions, multiply, add, subtract even divide fractions. And here is where I argue with Common Core. That more time needs be spent on Decimals for decimals are the True Numbers in mathematics, whereas Fractions are a bit of exotica, for fractions are just Long Division without doing division. So, what I propose is that Fractions be cut deeply from the curriculum and with that extra time, devote it to Decimals.

Now this is just my gut instinct, but I think, Decimals as a concept are far more difficult to learn than is fractions, because of the definition of Decimals, with its place values. Place values for 8 year olds seems too difficult. So, what I propose is teaching Decimals every year, just gradually a bit more and more, so that by High School, they surely will have comprehended the Decimals, and its tough definition. For me, I vaguely just vaguely remember that the definition I would gloss over, sort of like surf the definition-- only the word "surf" was never around when I was a Grade School in 1958 an 8 year old. And the first time I used the word surf was from some songs out of California in the 1960s. But now we have surfing the web and internet. So, why not surf the definition of Decimal. By surfing Decimal definition at age 8 and for the next 7 years reinforce that understanding of the Decimal number.

I must emphasize that Decimals is a key crucial part of mathematics, a essential part. So the teacher must have Decimals taught long and hard.

So here is a juncture point for the teacher and me as author. I am not going to write long lessons on teaching decimals or long division. But at this moment of this textbook, what the teacher has to accomplish and the student has to learn well and hard. Is Decimals and Long Division.


1. Number & Operations in Base Ten - Common Core State Standards ...www.corestandards.org/Math/Content/5/NBT/‎
Read, write, and compare decimals to thousandths. CCSS.Math.Content.5.NBT.A .3.a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 ...‎CCSS.Math.Content.5.NBT.B.7 - ‎CCSS.Math.Content.5.NBT.B.5

And I would be content if they only learned to the thousandths place value in decimals. The above number 347.392 is good for 8 year olds.

Definition of Decimal:: a number such as 347.392 = 3 × 100 + 4 × 10 + 7 × 1 . + 3 × (1/10) + 9 × (1/100) + 2 x (1/1000)

So, teacher, give the students many numbers and let them write that number according to the definition.

Exercises::

1.01

231.94

444.555

980.002 written out is 9x(100) + 8x(10) + 0x(1) . + 0x(1/10) + 0x(1/100) + 2x(1/1000)

So, do many exercises of writing out decimals, and point out the Place Values. The "ones" place value, the tens place value, the hundreds place value. Point out the decimal point then the 1/10 place value, etc etc.

Now out of curiosity I had to see what age is Long Division taught, and apparently in 5th grade, ages 10-11. So here the teacher has to decide whether 8 year olds can handle Long Division. If not, well, wait for 10 years of age. So here I need guidance by teachers who teach 8, 9, 10, 11, 12 year olds. When does Long division start and how is it best taught? So let the teacher decide on this.

But in the meantime. I wonder if we can add, subtract, multiply decimals.

So here I ask the teacher to determine how much the students know of decimals and know how to add, subtract, multiply. If weak, then months go by with the teacher using Common Core materials to ground the students well in Decimals. Use other books for the teacher to do the lessons. That means the teacher has to well prepare these month long lessons.

DECIMALS:

So here, let us focus on Decimals. And according to Core Curriculum, the 8 to 14 year old students learned what Decimals were, and how to add, subtract, multiply and divide with them.

It would be nice, very nice to repeat lessons on Decimals in all the age groups from 8 to 14, giving them refreshment exercises each year. Unless they learn decimals well, they cannot go into science, or engineering.

So, first, let me repeat the meaning of a decimal with an example using the number 321.987.

Decimals are place values and that number represents (3x100) + (2x10) + (1 x 1) . + (9x 1/10) + (8 x 1/100) + (7x1/1000)

Teacher, show the students the progression leftward of the decimal point starting with 1 then 10 then 100 according to the place value. Then show them the progression of place value going rightward.

I think the Common Core even shows diagrams and pictures of place value.

So let me get a list of practice exercises, and the teacher preparing each day for months on end with practice on decimals, and add, subtract, multiply with decimals.

With add and subtract, it is pretty much the same as if we dealt with numbers without a decimal point, just make sure you ___line up the decimal point___ before adding or subtracting.

16.45
45.39
78.02
-----

187233.32
-67894.56
----------

87 x .01

1.2 x 2.8

Now teach the students a key idea of multiplication, one item that gave me trouble when I was a 8-14 year old. The idea in multiplication that if you multiply two small numbers, and small numbers are less than 1, that the product is smaller than either of the two numbers you started with. But, any multiplication of two large numbers, numbers bigger than 1, the product is always larger than either one of its multipliers.

For example, two large numbers 1.1 x 1.9, that the product 2.09 is bigger than either 1.1 or 1.9

Example, .1 x .93 that the product .093 is smaller than either .1 and .93.

It takes time for young minds to understand when you multiply little things, numbers between 0 and 1, you get smaller but when you multiply big things (numbers bigger than 1) you get larger.

History Note:: So in covering Decimals, the key thing that students are going to slowly learn over the next seven years of teaching decimals every year, is that Numbers are things that you can count. You cannot count fractions, because fractions are not orderly, they pop up everywhere, but you can count decimals. That is a key idea in math and science-- numbers that exist, like Decimals cover all numbers that exist because you can count them. If it is a number, it is a decimal because you can count decimals. And, in the history of mathematics, math was poorly sputtering along until about 1202 when an Italian mathematician named Fibonacci wrote a book called Liber Abaci (book of calculations) using Decimals, it was only after 1202 that mathematics exploded in growth because finally the True Numbers of mathematics were discovered.

Decimals are to math what the microscope is to the biologist.

Decimals are to math, what the telescope is to the astronomer.

Decimals are to math, what electricity and magnetism is to the physicist.

Decimals are to math, what the table of atomic elements is to chemistry.

AP

[Archimedes Plutonium]

_Lesson3 Decimals Re: _Textbook :: TEACHING TRUE MATHEMATICS, 2018


Alright, it is important that students of age 8 onwards get the correct view and understanding of "What are Numbers" This book starts with age 8, because it is about here, where fake and corrupt Old Math sets hold. I do not want education where in one year you learn something and a few years later, be told, forget what you learned and this is the true math.

In the past history of math education, the student was mislead and then when they reach College, they have such a error filled view of numbers, and geometry. For this reason, I needed to start this book for age 8, to catch the student before corruption of math can set in.

Now in the Core Curriculum too much time is spent on Fractions. So, what I want to see here, is cut that time in half, and with that extra time, spend it on teaching Decimals better.

I want the 8 year old to be able to do these decimals without much trouble.

534.232
+234.119
-----------


109.008
+111.32
----------


324.567
-177.122
----------

444.11
-9.08
-------

12.89
x7.38
------

772.953
x265.5
--------
     _________
6.3| 829.200

       ___________
137| 1.000000

Teacher, plan out your own quizzes and tests to see if the student has mastered Decimals at 8 years of age, and I include long division, but if the 8 year olds are not ready for long division, then skip it for another year. And now we go on to the next topic of Logic connectors. Make it simple, for keep in mind, these are 8 year olds.

So spend less time on Fractions and more time on Decimals.

Teacher's Story Note:: Now one story about Decimals that is telling. Is that many people who never had Decimals in school, come to believe that 1/2 is not that of 0.5, that

  ______
2| 1.00

is not .5. Why? Is it because they were not formally taught in classroom? Perhaps they cannot understand how 1/2 has a 2 involved and they see no "2" in .5. So the same kind of a "mental block" is here, as is in the situation of .9999... equaling something. If we teach students the correct and true numbers of mathematics as Decimals, we avoid that crazy silly thought that .999...=1 for it is not equal to 1. I remember myself in High School, that the issue of what is .9999... never came up. It came up for, the first time was a freshman in college, studying Calculus. But more on that in High School do I cover the issue of .9999...., for this issue if far over the heads of Grade School, and I bring it up, only, because, to show how important it is to teach decimals, rather than fractions because that issue of .9999.... has its roots in the problem of 1/3, what is it? Is 1/3, that of .3333..... So, so, save these problems for later years.

Summary of Decimals:
So I had to catch the students of age 8, so as to avoid teaching these students falsehoods and later having to tell them-- what you learned is phony and here is the real truth.

So, by this lesson, I want the teacher to see if all his/her students can do Decimals with a fairly good degree of accuracy, maybe not Long Division, but at least add, subtract, multiply. If my above is too difficult, then tone it down until the students get a good idea of what they have to do for add, subtract, multiply, divide.

The Long Division is the most difficult one, so here, get what the Common Core has found to work well in teaching add, subtract, multiply, divide of decimals.

Now, the students are going to see decimals come up and come up again and again, each time adding on more insights and increased understanding. So although I have just 3 lessons on decimals, the classroom of 8 year olds maybe spending 3 or 4 months on Decimals.

AP

[Zelos Malum]

And we want you to learn proper mathematics, but you sure as hell ain't never gonna do that.

Post by Archimedes Plutonium
So I had to catch the students of age 8, so as to avoid teaching these students falsehoods and later having to tell them-- what you learned is phony and here is the real truth.

So you'd teach them garbage in favour of things known to work? Arsehole

[Archimedes Plutonium]

Lesson4 Logic, and, or Re: _Textbook :: TEACHING TRUE MATHEMATICS, 2018

Review of Decimals, for teacher and student::

Alright, still in Grade School with 8 year olds. We just taught them decimals but every year, here after we teach them more about decimals, for that was just a starting point of decimal teaching. The lesson of mathematics for all students, for they all need to know what numbers are, to count, to manage money. Not only do future scientists, engineers have to master decimals, but all people who never go on in science or engineering need numbers and those numbers are decimals.

When you say math, the first thing to come to mind is numbers, and that means decimals.

So, start them at age 8 with decimals. And this book will further review decimals years after 8 years old.

Now I do not mind the students doing the math of decimal operations and having a calculator handy. Only when doing quizzes and tests, they cannot have a calculator. And remind the students that they cannot cheat because they have to show work, not just a answer. And on tests, they must show their work, such as in multiplication or long division, so tell them, calculators will not help on tests or quizzes.

Now I gave some problems in last lesson. I do not know whether those are too hard or just right for your class of 8 year olds. So the teacher may want to adjust. I reckon it will take 4 months focused on Decimals, so the teacher has his/her work cut out. Use the best Common Core on teaching decimals.

I did say, let us cut back severely on Fractions. Fractions are not numbers but a division exercise.

We do not call
       ____________
137 | 1.000000  

a number, but rather it is a division that will lead to a number. This is one of the reasons I am starting with 8 year olds. For I do not want the 8 year old or younger be taught the phony notion that 1/137 is a number. It is not a number, but a division, a long division. And although after you divide you get a number--- a decimal number. The symbols of 1/137 is not a number. Now the 8 year olds need not know that now, in this book, even though they can easily read my words. But, before they get into the last 3 years of High School, they must know by then, that fractions are not numbers, but long-division exercises. The division ends up being a number, but before the number is reached, it is not a number.

You cannot order and count fractions. You can order and count Decimals and it is for this reason that Decimals are the Only Numbers.

To help make this clear, suppose cars that run are numbers, well we have all kinds of cars running on roads. But now, suppose someone drops off a large box containing car parts, disassembled. Now, maybe a good mechanic can put the parts all together to make a running car. But in the box it is not a car for it cannot run. Same thing in mathematics. Fractions are not numbers but a invitation to divide numbers. And although a number after the division is produced, it is not a number until the division ends. So fractions are not numbers, but decimals are numbers. And it is for this reason I start this book at age 8. I do not want the student's mind polluted with bad math, with wrong math at such an early age.

So cut the teaching of fractions by half, and that half devote to teaching better, decimals. The history of this problem stretches back centuries, and the major part of the problem was in writing division.

We easily can see that

5 -:-  3 is not mistaken for a number but is a division process that will lead to a decimal number answer

And, I have asked that the symbol -:- be removed out of all of mathematics for it is just too confusing as to what is being divided into what. So, delete the symbol altogether out of math.

Math can get by with the symbols of
____
|

and the symbol / for division

Likewise we do not mistake
   ________
3 | 5.000   is not a number but a plea to divide

Likewise

5/3 is not a number but a plea to divide.

Unfortunately, because we see just a line drawn between 5 and 3, that math goofballs thought 5/3 is a number. So the convenience, the sheer brevity, the sheer simple design of a slanted line /

The / between two numbers, that goofballs of logic, thought 5/3 is a number itself. It is not; it is only a plea to do the operation of divide. What comes out of the division is a number.

So, fractions are not numbers. Fractions are division. Decimals are numbers and we will learn, decimals are the only numbers in all of mathematics. That is why these first lessons are about Decimals, for if you want to go straight, true clear in math, you must learn Decimals.

Now, today in this lesson we do Logic. We start Logic, we start a new topic, which is a relief to some 8 year olds who spent the last 3 or 4 months learning decimals. Ask the students if they ever watched Star Trek and Dr. Spock. Well, logic is what Dr. Spock was proud of, and it was his specialty. Even though it was science fiction, we see what logic is about-- thinking cool, thinking straight, thinking clear, thinking correctly.

And today we learn two connectors of Logic, the And, and the Or.

So, teacher, do about a page full of And statements and a page full of Or statements.

Something like this:

The boy with the ball and the girl with the bat went to the backyard field to play.

That can be broken down into

The boy with the ball went to the backyard field to play.

The girl with the bat went to the backyard field to play.

So we stick an AND in those two sentences to produce our original sentence. Ask the class what is And like? Is it add, subtract, multiply, divide?

A Or example.

Either it rains at noon time today or it is sunshine at noon today.

So, ask the class what that OR is like, is it add, subtract, multiply, divide

Show them it is subtract, for at Noon time, if it is raining it is not sunny and you subtracted sunshine. But if it is sunny, it is not raining and so you subtracted rain.

Now many examples come from math itself.

2 AND 3 is what? It is 5

7 OR 10 is what? Well if you have 7 not 10 you subtracted 10, and if you have 10 not 7 you subtracted 7. Or is one of two items but never both. Or is subtraction.

AND is the joining together of items. AND is add, OR is subtract, remove.

So have a sheet full of AND examples and another sheet full of OR examples.

Warning to Teacher: please spend considerable time on composing AND, OR statements, especially OR statements, for although we may think this is easy material, in fact, it is difficult to do nice pure logical statements. So prepare the night before especially well and cautiously.

AP

[Archimedes Plutonium]

#5 Lesson, AND//OR connectors in Logic Re: _Textbook :: TEACHING TRUE MATHEMATICS, 2018

Alright the first 4 lessons were mostly about Decimals and took about 4 months to cover them. As I said before, let us cut back on fractions, and with that extra time devote to Decimals. Now fractions according to Common Core Curriculum are desirable for eventually doing percentages. But I suspect that a cutback in fractions is not going to harm the development of percentages at all.

Now this textbook is unique from all others in another facet, another way. In that I am constantly talking to not only the student but to the teacher also. I know of no textbook in the world where the teacher and the student are being lectured to. This is good practice, for it is pragmatic to make the best teaching possible to students.

Now this text is written logically. I am not shy in stating that I was blessed with a very logical mind and I want to share that blessing to teachers and students, devoting much of my life to giving science and the best written math and physics books possible.

So, do not think for a moment that the topics are not connected. They are connected as best as I can, as best as logically possible. So the topics for 8 year olds is Decimal, Logic, Graphing, and Sigma error. And those four topics will be repeated every year for about the next 6 years of the students education, until the student is say well into High School. But each year of repeating Decimal, Logic, Graphing, Sigma-error there will be less time spent on these four topics.

The Decimals-- because basically those are all the Numbers of math that exist. The Logic-- well, we want students to think straight and think clearly and think correctly. The Graphing is a mix of numbers but is also the foundation of geometry. The student will take geometry later in High School and learn all about Plane Geometry and how to do simple proofs of math. But the Graphing that the students begin to learn at age 8 will carry them through geometry which leads to calculus.

So much of the Common Core is really "irrelevant math" such as the over hype on fractions, the doing of quadratic equations, solving for x as in 2x^2 +3x - 10. Much of math education is irrelevant side show without a focus of attention on the four main issues -- Numbers as decimals, and Logic-- think straight and clear and correct, Graphing-geometry with numbers, and Sigma-error which is the limitations of mathematics, for so much of science by 2018 is a quantity of uncertainty, the errors not only in measurement, but in science and math itself. Sigma comes from a branch of mathematics called statistics and probability. Now, I do not know if the 8 year old student can go through Harold Jacobs book Mathematics, A Human Endeavor, 1970. How many 8 year olds can go through that book without a teacher, and how many can comprehend that book? It has a chapter on Probability and one on Statistics. It even has a chapter on Logic, only Jacobs calls it Deductive Reasoning. But I am sure this book by Jacobs is too much for 8 year olds, and probably a good book for 16 year olds to learn without a teacher accompanying the book.

So my focus is those four Decimals, Logic, Graphing, Sigma error. And by the time High School rolls around for the student where they take a year of Geometry, a year of Algebra, perhaps 2 years of Algebra, perhaps Calculus in the final year, before some students head off to College. And when they take Plane Geometry, this book will have just a few lessons to be supplemented by some excellent other books-- Jacobs has a good geometry text besides his Mathematics, A Human Endeavor. Now the teacher will have a lot of work cut out for them to supplement my text in High School.

In this lesson I want to go over again AND of Logic and OR of Logic. Here again, give the students a whole page of AND statements. Another whole page of OR statements and let them figure out how the AND is adding, and then, how the OR is subtracting by choosing one item and dismissing the other item.

You be surprised at how well students at this young age 8 pickup on knowledge. When I was a teacher, my favorite classes were these young students for their sole interest in life at that age was learning. A little older and then body hormones kick in and the students are interested in boy girl, girl boy too much, and easily distracted.

So the lessons on Decimals took about 4 months. The lessons on Logic, And with Or, is only going to take a week, maybe 2 weeks. Not much.

Then we move on to Graphing, and the graphing is going to bring back the Decimal numbers again. And is going to cost the School in graph paper. So, either the School provides graph paper, or the students bring their own graph paper. If a poor district, better have the school provide.

There was an example of Either Or. Either the School provides the graph paper, or, the student brings their own graph paper.

Now in addition to Graph paper, have them bring a pencil with good eraser, or even a gummy eraser that does not tear up the graph paper so they can use the graph paper over and over. If careful, a student can use one graph paper that lasts for weeks even a month. Erase gently, and do not write the numbers or lines too dark or too thick.

Also, now the student needs a ruler.

So, next lesson is Graphing.

AP

[Archimedes Plutonium]

#6 Lesson, Graphing Re: _Textbook :: TEACHING TRUE MATHEMATICS, 2018

Alright, time to move on to Lesson6, Graphing

For Grade School into start of High School I want a theme package of four major topics-- Numbers, decimals mostly, Logic, Graphing and Sigma error.

Those four form the basis of a good education in math, what all students need, even if they do not take math ever again after High School.

The Decimals are the numbers, the algebra of mathematics. The Logic is to think clear, correct and straight. The Graphing is geometry, a geometry with numbers. The error in math when using geometry and many measurements. So those four are the essential four of mathematics.

Now the first 6 lessons are for students age 8. I need to teach those at 8 years old the true math, so their minds are not polluted and have to re-teach them. Education is always horrendous if a teacher one year tells the students-- forget what your past teachers told you-- this is the straight up true. That is horror education. And trouble is, today in 2018, more than 75% of what is taught in schools is outright fakery, mind numbing propaganda, much of it reminding me of Soviet education when Stalin was in USSR, where the State teaches not what is true, but what pleases some people in power. About 75% of the math education in the USA today in my opinion, is not true math, but math that pleases those math professors and their publishing presses making them rich and filling their pockets with money.

So, I need to catch the student at an age in school, where money-math is totally cut out.

And I suspect age 8 is a good year to start so the student does not encounter the con math money machine and pollute their understanding of math, rather than enhance their math.

So, today, Lesson6, starts Graphing.

Now either the teacher supplies each student with graph paper (poor school districts) or the student's parents supply their student with these materials. I prefer the white graph paper where all the blocks are uniform the same. Some graph paper is green tinted and where they make bold lines and non-bold lines. The thing I hate about these green tinted, is they make the bold lines after every 5 blocks. Why could they not be more logical. Since numbers are all Decimals, make the bold lines after every 10 blocks. Did any graph paper publishing company ever produce such?

So here, is an example for students 8 years old, to see that Logic is so very very important. Because if you make bold lines on graph paper, why in the world would you chose every 5 small blocks. Decimals are 10 not 5.

So, furnish the student with Graph paper and with pencil and gummy erasure that does not tear up the paper, and pencil sharpener-- those small plastic thumb size sharpeners, and with ruler. Tell the kids that they will use the eraser many times and not to make too dark of spots on the graph paper so they can reuse the paper over and over again.

Now, for first Graphing, teacher, have the students write as many numbers as possible for each block on the bottom and going up the side.

On my white all uniform blocks Graph Paper, no bold lines but all the same, I can go out to 42 along the bottom and 32 along the side, looking like this:

32
31
30
29
28
.
.
.
.
.
4
3
2
1
0 1, 2, 3, 4, 5, 6, . . .                  38, 39, 40, 41, 42

Teacher, spend a nice full day just teaching graphs for the first time to these students.

Make sure they write numbers small, for there is little room. Go around to each student helping them
do this clean and properly. There are no commas, for I only wrote them since the above is not a graph paper.

And have the students write them lightly because there is going to be many erasing and you need to save on graph paper.

Now I wish we had paper that was 100 blocks high and 100 blocks long, but our paper is too small to have that.

I prefer the Decimal representation even in graphing. So, teacher, try to get a overhead projector with a graph of 100 blocks to supplement this teaching, and tell the students, even though their paper goes from 32 to 42, what we really wanted was 100 by 100.

AP

[Archimedes Plutonium]

#7 Lesson, Graphing, coordinate points x,y axis Re: _Textbook :: TEACHING TRUE MATHEMATICS, 2018

Alright, so my graph paper is 42 along the bottom and 32 blocks high. Now the students graph paper may be different so the teacher is going to have to take an inspection of each students graph paper so that they can cope with the x-axis number line and the y-axis number line.

So, teach which is the x-axis and which is the y-axis. The x-axis runs along the bottom. The y-axis runs from bottom to top.

Teacher, teach the bottom row is called the x-axis and draw a arrow  ---->. Teach the column that goes up 32 blocks is called the y-axis and draw a arrow

^
|
|
|

Now, teach the meaning of a coordinate point. (x, y)

For example, do these coordinate points and plot them.



(1,1)

(2,3)

(5, 10)

(6, 20)

(0, 0)

Notice I deliberately did (0,0) last, thinking that this may stump some young 8 year olds.

By plotting them, let them make an "x" mark on the graph where the coordinate point lies. Or, if they write it small, just write the point such as (6,20).

Now I believe in having fun in learning. So have the teacher, make pictures of things and then find out coordinate points so that the students can connect and find out what the picture was when they connect by a straightline from one point to the next point, such as a house. Those game books where young people connect the dots and find out what the figure is. Like a star, like a car, like a house.

The prime lesson here is Coordinate Points and x-axis, y-axis.

Spend several days on plotting points. Always reviewing the key concepts, of x-axis, y-axis, coordinate points (x,y) like (1,2).

Summary:: Learn what x-axis is, what y-axis is. Learn what coordinate point is, such as (1, 3). Learn how to plot those points given, then connect points with a ruler.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
#7 Lesson, Graphing, coordinate points x,y axis Re: _Textbook :: TEACHING TRUE MATHEMATICS, 2018
Alright, so my graph paper is 42 along the bottom and 32 blocks high. Now the students graph paper may be different so the teacher is going to have to take an inspection of each students graph paper so that they can cope with the x-axis number line and the y-axis number line.
So, teach which is the x-axis and which is the y-axis. The x-axis runs along the bottom. The y-axis runs from bottom to top.
Teacher, teach the bottom row is called the x-axis and draw a arrow  ---->. Teach the column that goes up 32 blocks is called the y-axis and draw a arrow
^
|
|
|
Now, teach the meaning of a coordinate point. (x, y)
For example, do these coordinate points and plot them.
(1,1)
(2,3)
(5, 10)
(6, 20)
(0, 0)
Notice I deliberately did (0,0) last, thinking that this may stump some young 8 year olds.
By plotting them, let them make an "x" mark on the graph where the coordinate point lies. Or, if they write it small, just write the point such as (6,20).
Now I believe in having fun in learning. So have the teacher, make pictures of things and then find out coordinate points so that the students can connect and find out what the picture was when they connect by a straightline from one point to the next point, such as a house. Those game books where young people connect the dots and find out what the figure is. Like a star, like a car, like a house.

No, I take that back, in that some goofy figures would more confuse students then help in learning.

So I want math figures like square, rectangle, parallelogram be drawn. So introduce the 8 year old for the first time what a square, rectangle, parallelogram, rhombus, trapezoid are.

Post by Archimedes Plutonium
The prime lesson here is Coordinate Points and x-axis, y-axis.
Spend several days on plotting points. Always reviewing the key concepts, of x-axis, y-axis, coordinate points (x,y) like (1,2).
Summary:: Learn what x-axis is, what y-axis is. Learn what coordinate point is, such as (1, 3). Learn how to plot those points given, then connect points with a ruler.
AP

[Archimedes Plutonium]

#7 Lesson, Graphing, x-axis, y-axis, coordinate points, TEACHING TRUE MATHEMATICS, 2018

#7 Lesson, Graphing, coordinate points x,y axis Re: _Textbook :: TEACHING TRUE MATHEMATICS, 2018

Alright, so my graph paper is 42 along the bottom and 32 blocks high. Now the students graph paper may be different so the teacher is going to have to take an inspection of each students graph paper so that they can cope with the x-axis number line and the y-axis number line.

So, teach which is the x-axis and which is the y-axis. The x-axis runs along the bottom. The y-axis runs from bottom to top.

Teacher, teach the bottom row is called the x-axis and draw a arrow  ---->. Teach the column that goes up 32 blocks is called the y-axis and draw a arrow

^
|
|
|

Now, teach the meaning of a coordinate point. (x, y)

For example, do these coordinate points and plot them.



(1,1)

(2,3)

(5, 10)

(6, 20)

(0, 0)

Notice I deliberately did (0,0) last, thinking that this may stump some young 8 year olds.

By plotting them, let them make an "x" mark on the graph where the coordinate point lies. Or, if they write it small, just write the point such as (6,20).

Now have the teacher, make pictures of things such as triangle, square, rectangle, parallelogram, rhombus, trapezoid connect by a straightline from one point to the next point. The teacher has to do this on the blackboard so the students themselves connect the correct points.

The prime lesson here is Coordinate Points and x-axis, y-axis.

Spend several days on plotting points. Always reviewing the key concepts, of x-axis, y-axis, coordinate points (x,y) like (1,2).

Summary:: Learn what x-axis is, what y-axis is. Learn what coordinate point is, such as (1, 3). Learn how to plot those points given, then connect points with a ruler.

AP

[Archimedes Plutonium]

#8 Lesson

Alright in the Common Core, I do not know when they teach right-angle or angles for the first time. So I am going to presume 8 year olds are taught angles for the first time. And here we introduce students to the most common figures in geometry.

Such things as a triangle, square, a rectangle, a parallelogram, a rhombus, a trapezoid.

Find out the coordinate points of figures, then have the students plot them, then connect points with a ruler to find out what the coordinate points formed.

Now my graph paper allows for 42 blocks by 32 blocks and I have plotted some of these figures. However, if your students have smaller graph paper, then the teacher is going to have to assign different points for the below exercises.

Right triangle (1,1) (1,4) (5,1) plot the points and have the students connect with a ruler, but have the teacher show each of these on the blackboard first. And here introduce what a Right-Angle is, a 90 degree angle, introduce the term perpendicular, means right-angle.

Right Angle: teach where two line rays, call it a "line ray", where you have two line rays and they meet at a vertex point |__ and the angle is the amount of space between the two line rays.

Not a Right triangle: (7,3) (10,6) (12,3) and show the students that the angles are less than 90 degrees

Square: (2,6) (6,6) (6,10) (2,10), teacher show this on the blackboard first and show them how to connect with ruler. Discuss the four angles.

Rectangle: (13,4) (17,4) (17,11) (13,11). Discuss the angles and then discuss what is different from the square and rectangle-- the square sides are all four of them equal to each other. Whereas the rectangle has two of the four equal to each other, the opposite sides are equal to each other.

Trapezoid: (13,15) (17,15) (17,20) (13,17) discuss what is different about a trapezoid from a rectangle

Parallelogram: (19,14) (25,14) (28,17) (22,17) discuss what is different between a rectangle and parallelogram, note especially there is no right angle

Rhombus: (21,4) (26,4) (30,8) (25,8) discuss what is different between a rhombus and square and what is different between a rhombus and parallelogram.

Now this lesson is to practice in plotting points, then connecting points with a straightline segment. We learn a right-angle and we just eyeball in what is not a right angle, for these are 8 year olds and we need not go into depth about angles.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
#8 Lesson

Now I write this textbook on mathematics because I found so many errors in Old Math, that the best thing to do is completely throw Old Math out, and so I am faced with presenting a new textbook on mathematics starting with 8 year olds to teach them TRUE mathematics, not that fakery, that marrs and pollutes their minds for the rest of their lives. I have to intervene in teaching so the minds of youngsters are not corrupted.

Throw out all the Old Math textbooks, it is alright to cut and paste some Old Math, but most is too far corrupt.

Now I want to teach Philosophy also, in writing this text and in living life. My favorite philosophy was Pragmatism which I learned as senior in High School. Now before High School ends for the student, I want to teach them a bit of philosophy, for philosophy comes in handy in mathematics. It comes in handy in what math I teach, the decision as to what math is good math and what is side show math. Sort of like going to the grocery store and deciding-- that is good food and nay, I want to avoid that food.

PHILOSOPHY of Math teaching::

Here I want some principles to guide me

Data, Information, Inputs

Understanding, moving forward from data

Knowledge, memory of understanding, memory of moving forward, action on understanding

Wisdom, reflection of knowledge, and pondering whether there was better action moving forward, evaluation of knowledge, assessment of knowledge, judging of knowledge.

Now, the reason I write this and include this snapshot of philosophy, is that I made a Wisdom choice of teaching Decimals, Logic, Graphing , Sigma-error. I made a choice of those four major topics, and said, pare down on teaching fractions. And increase teaching of Logic and logic analysis, and teach sigma error, something unheard of before. Decrease teaching algebra quadratic equations and polynomials, decrease teaching geometry congruence.

So much of mathematics that is taught today, is either outright fakery or is useless to a student becoming a scientist or engineer.

For example, much of algebra is that of chasing after numbers that end up being equal to 0. Something like this x^3 +5x^2 +2x +8 = 0. Totally useless, and in fact, fakery math. Because well, imagine, you have 0 you have nothing so to speak, and who cares what numbers you dither about, dither here, dither there to make them equal to zero, to nothing. You see, math like that, is created because hundreds and thousands of years ago, a few men played games with math, like someone playing a crossword puzzle, and which future teachers would be able to make money by teaching that nonsense math, money by printing textbooks on that nonsense math and many jobs by people teaching that nonsense worthless math to you.

Think for a moment. Does it make a difference in science to know what numbers multiplied, divided, added, subtracted equals 0 in x^3 +5x^2 +2x +8 = 0

It is poor bad and fake math, that I aim to steer clear of in this textbook. Math that is taught in school, only because of huge money interests invested in teaching that fake math. Most of the numbers that college math professors teach-- rationals, irrationals, transcendentals, sqrt-1, all of them are fakery math. Fake math there, because a entire industry of teaching fakery makes money.

Fake math is there, because money is there, and we see that in the fact that all textbooks of Old Math say an ellipse is a conic when it never was a conic but rather a cylinder section.

Fake math as in trigonometry that says a sine and cosine are Sinusoid waves, when in truth they are semicircle waves. Yet it is too costly for math teachers to demand truth in mathematics, for they have to print new books and retrain teachers to teach true math.

The list goes on and on of fake math, and so, that is why I write TEACHING TRUE MATHEMATICS.

AP

[Bill]

Post by Archimedes Plutonium

Post by Archimedes Plutonium
#8 Lesson

Now I write this textbook on mathematics because I found so many errors in Old Math, that the best thing to do is completely throw Old Math out, and so I am faced with presenting a new textbook on mathematics starting with 8 year olds to teach them TRUE mathematics, not that fakery, that marrs and pollutes their minds for the rest of their lives. I have to intervene in teaching so the minds of youngsters are not corrupted.

You may be able to force your kids to read it, but I don't think it will
adopted by the public schools. Big waste of time, no?

Post by Archimedes Plutonium
Throw out all the Old Math textbooks, it is alright to cut and paste some Old Math, but most is too far corrupt.
Now I want to teach Philosophy also, in writing this text and in living life. My favorite philosophy was Pragmatism which I learned as senior in High School. Now before High School ends for the student, I want to teach them a bit of philosophy, for philosophy comes in handy in mathematics. It comes in handy in what math I teach, the decision as to what math is good math and what is side show math. Sort of like going to the grocery store and deciding-- that is good food and nay, I want to avoid that food.
Here I want some principles to guide me
Data, Information, Inputs
Understanding, moving forward from data
Knowledge, memory of understanding, memory of moving forward, action on understanding
Wisdom, reflection of knowledge, and pondering whether there was better action moving forward, evaluation of knowledge, assessment of knowledge, judging of knowledge.
Now, the reason I write this and include this snapshot of philosophy, is that I made a Wisdom choice of teaching Decimals, Logic, Graphing , Sigma-error. I made a choice of those four major topics, and said, pare down on teaching fractions. And increase teaching of Logic and logic analysis, and teach sigma error, something unheard of before. Decrease teaching algebra quadratic equations and polynomials, decrease teaching geometry congruence.
So much of mathematics that is taught today, is either outright fakery or is useless to a student becoming a scientist or engineer.
For example, much of algebra is that of chasing after numbers that end up being equal to 0. Something like this x^3 +5x^2 +2x +8 = 0. Totally useless, and in fact, fakery math. Because well, imagine, you have 0 you have nothing so to speak, and who cares what numbers you dither about, dither here, dither there to make them equal to zero, to nothing. You see, math like that, is created because hundreds and thousands of years ago, a few men played games with math, like someone playing a crossword puzzle, and which future teachers would be able to make money by teaching that nonsense math, money by printing textbooks on that nonsense math and many jobs by people teaching that nonsense worthless math to you.
Think for a moment. Does it make a difference in science to know what numbers multiplied, divided, added, subtracted equals 0 in x^3 +5x^2 +2x +8 = 0
It is poor bad and fake math, that I aim to steer clear of in this textbook. Math that is taught in school, only because of huge money interests invested in teaching that fake math. Most of the numbers that college math professors teach-- rationals, irrationals, transcendentals, sqrt-1, all of them are fakery math. Fake math there, because a entire industry of teaching fakery makes money.
Fake math is there, because money is there, and we see that in the fact that all textbooks of Old Math say an ellipse is a conic when it never was a conic but rather a cylinder section.
Fake math as in trigonometry that says a sine and cosine are Sinusoid waves, when in truth they are semicircle waves. Yet it is too costly for math teachers to demand truth in mathematics, for they have to print new books and retrain teachers to teach true math.
The list goes on and on of fake math, and so, that is why I write TEACHING TRUE MATHEMATICS.
AP

[a***@gmail.com]

Post by Bill

Post by Archimedes Plutonium

Post by Archimedes Plutonium
#8 Lesson

Now I write this textbook on mathematics because I found so many errors in Old Math, that the best thing to do is completely throw Old Math out, and so I am faced with presenting a new textbook on mathematics starting with 8 year olds to teach them TRUE mathematics, not that fakery, that marrs and pollutes their minds for the rest of their lives. I have to intervene in teaching so the minds of youngsters are not corrupted.

You may be able to force your kids to read it, but I don't think it will
adopted by the public schools. Big waste of time, no?

I don't think he has any kids, and that's a Good Thing.

[Archimedes Plutonium]

#9 Lesson, Graphing: figures of geometry; circle TEACHING TRUE MATHEMATICS, 2018



#9 Lesson

Alright in the Common Core, I do not know when they teach circle for the first time. So I am going to presume 8 year olds are taught circles for the first time. The students are going to need their graph paper, pencil and eraser, and a compass (teacher be wary since a compass has a needle like end, and do not want anyone getting hurt).

Now do a x-axis from 0 to 15, and do a y-axis from 0 to 15. Remind students of the x and y axis. The x-axis is horizontal, going across the bottom and the y-axis is vertical going upwards.

Now, find the point (8,7) and measure a distance of 7 units for the compass, radius. From the point (8,7) with the compass, make a circle that is radius 7, diameter 14.

Teacher: show the students this construction on the blackboard, before they do it on their graph paper.

Now, for the rest of the class period, have the students write out the coordinate points of the circle.

And, we are going to teach Sigma Error here.

(1,7) is the first point I get off the circle, and moving on to 2 on the x-axis we have two new points.

(2, 3.4) now, here, teacher, explain how we get the .4 in 3.4

and we must not forget the other point (2, 10.6)

Explanation:

Say we have a block of the graph as this where the circle curve passes through that block
__
|__|

And our circle curve goes through that block

We divide the block into 10 fine marks evenly spaced on side and on bottom of block and we guess where the curve lands or passes through.

__
|-__|

To, me, it looks like it landed .4, but some may say .3, others may say .5.

The next two points is when we have a 3 on the x-axis.

And eyeballing the two points in, I get

(3, 2) and (3, 12)

The next two points is when we have a 4 on x-axis.

(4, 1.3) and (4, 12.7)

Teacher, have the students go all the way out to 15 on the x-axis and while they are doing that, walk around the classroom seeing if each student is coping with the job at hand.

AP

[Archimedes Plutonium]

#10 Lesson, Sigma-error in math TEACHING TRUE MATHEMATICS, 2018

#10 Lesson

Alright, this is new to mathematics, totally new, for it teaches that math is internally having error. We always thought math was perfectly error free, but here we teach that math, like physics has uncertainty, internal error. And math even has a whole subject on error, but never realized that the error was a part of mathematics itself. It is called Statistics. And much of science deals with statistics, even physics is inundated with statistical mechanics.

So, it is best to start teaching the error of math, rather than have young people, with 8 year olds, thinking math is perfect, when far from it, math has internal errors.

And I cannot think of a better "first lesson of sigma-error" than in reading the coordinate points of a circle. In the last lesson of #9 we learned how to write the coordinate points of a circle drawn by a compass (teacher watch the students carefully for a compass has a sharp pointed end).

Post by Archimedes Plutonium
Alright in the Common Core, I do not know when they teach circle for the first time. So I am going to presume 8 year olds are taught circles for the first time. The students are going to need their graph paper, pencil and eraser, and a compass (teacher be wary since a compass has a needle like end, and do not want anyone getting hurt).
Now do a x-axis from 0 to 15, and do a y-axis from 0 to 15. Remind students of the x and y axis. The x-axis is horizontal, going across the bottom and the y-axis is vertical going upwards.

Teacher, review the circle drawn last time.

Post by Archimedes Plutonium
Now, find the point (8,7) and measure a distance of 7 units for the compass, radius. From the point (8,7) with the compass, make a circle that is radius 7, diameter 14.

Review that circle again with radius 7.

Post by Archimedes Plutonium
Teacher: show the students this construction on the blackboard, before they do it on their graph paper.
Now, for the rest of the class period, have the students write out the coordinate points of the circle.
And, we are going to teach Sigma Error here.
(1,7) is the first point I get off the circle, and moving on to 2 on the x-axis we have two new points.
(2, 3.4) now, here, teacher, explain how we get the .4 in 3.4

So, teacher, here we teach Sigma error. Because the point (2, 3.4) the ".4" we had to eyeball in, and some students are going to guess .5 not .4, some will guess .3 and not .4

So, teacher, draw on the blackboard a magnified unit block and hatch mark in so you have 10 equal spaces below and above, like this

--
--
--
--
--
--
--
--
--
-- | | | | | | | | | |

Pretend that is a Magnified block that the graph curve goes through of the circle
4
--
--
--
--
--
--
--X
--
--
-- | | | | | | | | | |
3

And our circle curve went through that block from 3 to 4 at about .4

Post by Archimedes Plutonium
and we must not forget the other point (2, 10.6)
Say we have a block of the graph as this where the circle curve passes through that block
__
|__|
And our circle curve goes through that block

So teacher, here, what you are teaching is how math can be imprecise, imperfect and that it depends on your eyeballing in what is the point, is it (2, 3.4) or some would say it was (2, 3.3)

Post by Archimedes Plutonium
We divide the block into 10 fine marks evenly spaced on side and on bottom of block and we guess where the curve lands or passes through.
__
|-__|
To, me, it looks like it landed .4, but some may say .3, others may say .5.

Teacher, now go around and list on the blackboard what each student eyeballed that point in as.

Make a list of who got 3.2, who got 3.3, who got 3.4, who got 3.5

Post by Archimedes Plutonium
The next two points is when we have a 3 on the x-axis.
And eyeballing the two points in, I get
(3, 2) and (3, 12)
The next two points is when we have a 4 on x-axis.
(4, 1.3) and (4, 12.7)
Teacher, have the students go all the way out to 15 on the x-axis and while they are doing that, walk around the classroom seeing if each student is coping with the job at hand.

Now, as exercise, the teacher can do a new circle, with a new radius, say radius 8 with a new center, say center (8,8) and see how the students figure out what the coordinate points are, and check out the Sigma error.

Now these are 8 year olds, so we only want to teach them SIGMA ERROR to get acquainted with the idea of error in math. We do not want to go into any detail or depth of error, just an error because we had to judge or decide where a point was, inbetween 10 hatch marks. Remember, decimals are ten, so we have ten hatch marks to decide.

It is important to teach Sigma Error early in math, because it removes that flawed perception that mathematics is always "perfect with no error". But, even numbers multiplied have a sigma error. For the square root of 2 is, and have your students of 8 years old, do this exercise. The square root of a number, teach them, the square root of a number, is what number when multiplied by itself equals 2.

So, to the class, say, is 1 multiplied by itself, is that 2? Is 2, multiplied by itself equal to 2? No, so it must be in between 1 and 2, lies the square root of 2. So, now, anyone guess? Let us guess 1.5 is the square root of 2.

1.5
x1.5
--------
75
15
_____
2.25

Check to see if all the students remember how to multiply decimals

So, is 1.5 the square root of 2? No, it is too large, so let us try 1.4

1.4
x 1.4
--------
56
14
-------
1.96

So, is 1.4, the square root of 2? No, it is too small and needs a tiny bit more. So we try 1.41

1.41
x 1.41
---------
141
564
141
------
1.9881

So, is 1.41, the square root of 2? No, it is too small and needs a tiny bit more. So we try 1.42

Now, teacher this is a pretty way of ending a year of teaching math to 8 year olds for it ends by reviewing multiplication of Decimals, and decimals understanding is of prime major concern for youngsters of this age. Hopefully the 8 year olds have a running start with decimal Long Division, and if not, well, 9 and 10 year olds will be taught more and more of Long Division, until they are fluent in it. But, back to this square root of 2.

1.42
x 1.42
---------
284
568
142
------
2.0164

Now, here the teacher is going to have to review the fact that we are in 10 Grid, and in 10 Grid, 1.42 is the square root of 2 because the only numbers that exist in 10 Grid have one digit to the right of the decimal point so the digits of "164" are Sigma Error of math, for in 10 Grid, the square root of 2 is precisely 1.42.

--- END, end of a one year teaching of 8 year olds ---

AP

[Archimedes Plutonium]

#10 Lesson, SIGMA error in math, TEACHING TRUE MATHEMATICS, 2018


#10 Lesson, Sigma-error in math TEACHING TRUE MATHEMATICS, 2018

#10 Lesson

Alright, this is new to mathematics, totally new, for it teaches that math is internally having error. We always thought math was perfectly error free, but here we teach that math, like physics has uncertainty, internal error. And math even has a whole subject on error, but never realized that the error was a part of mathematics itself. It is called Statistics. And much of science deals with statistics, even physics is inundated with statistical mechanics.

So, it is best to start teaching the error of math, rather than have young people, with 8 year olds, thinking math is perfect, when far from it, math has internal errors.

And I cannot think of a better "first lesson of sigma-error" than in reading the coordinate points of a circle. In the last lesson of #9 we learned how to write the coordinate points of a circle drawn by a compass (teacher watch the students carefully for a compass has a sharp pointed end).

Post by Archimedes Plutonium
Alright in the Common Core, I do not know when they teach circle for the first time. So I am going to presume 8 year olds are taught circles for the first time. The students are going to need their graph paper, pencil and eraser, and a compass (teacher be wary since a compass has a needle like end, and do not want anyone getting hurt).
Now do a x-axis from 0 to 15, and do a y-axis from 0 to 15. Remind students of the x and y axis. The x-axis is horizontal, going across the bottom and the y-axis is vertical going upwards.

Teacher, review the circle drawn last time.

Post by Archimedes Plutonium
Now, find the point (8,7) and measure a distance of 7 units for the compass, radius. From the point (8,7) with the compass, make a circle that is radius 7, diameter 14.

Review that circle again with radius 7.

Post by Archimedes Plutonium
Teacher: show the students this construction on the blackboard, before they do it on their graph paper.
Now, for the rest of the class period, have the students write out the coordinate points of the circle.
And, we are going to teach Sigma Error here.
(1,7) is the first point I get off the circle, and moving on to 2 on the x-axis we have two new points.
(2, 3.4) now, here, teacher, explain how we get the .4 in 3.4

So, teacher, here we teach Sigma error. Because the point (2, 3.4) the ".4" we had to eyeball in, and some students are going to guess .5 not .4, some will guess .3 and not .4

So, teacher, draw on the blackboard a magnified unit block and hatch mark in so you have 10 equal spaces below and above, like this
--
--
--
--
--
--
--
--
--
-- |  |  |  |  |  |  |  |  |  |

Pretend that is a Magnified block that the graph curve goes through of the circle
4
--
--
--
--
--
--
--X
--
--
-- |  |  |  |  |  |  |  |  |  |
3

And our circle curve went through that block from 3 to 4 at about .4

Post by Archimedes Plutonium
and we must not forget the other point (2, 10.6)
Say we have a block of the graph as this where the circle curve passes through that block
 __
|__|
And our circle curve goes through that block

So teacher, here, what you are teaching is how math can be imprecise, imperfect and that it depends on your eyeballing in what is the point, is it (2, 3.4) or some would say it was (2, 3.3)

Post by Archimedes Plutonium
We divide the block into 10 fine marks evenly spaced on side and on bottom of block and we guess where the curve lands or passes through.
  __
|-__|
To, me, it looks like it landed .4, but some may say .3, others may say .5.

Teacher, now go around and list on the blackboard what each student eyeballed that point in as.

Make a list of who got 3.2, who got 3.3, who got 3.4, who got 3.5

Post by Archimedes Plutonium
The next two points is when we have a 3 on the x-axis.
And eyeballing the two points in, I get
(3, 2) and (3, 12)
The next two points is when we have a 4 on x-axis.
(4, 1.3) and (4, 12.7)
Teacher, have the students go all the way out to 15 on the x-axis and while they are doing that, walk around the classroom seeing if each student is coping with the job at hand.

Now, as exercise, the teacher can do a new circle, with a new radius, say radius 8 with a new center, say center (8,8) and see how the students figure out what the coordinate points are, and check out the Sigma error.

Now these are 8 year olds, so we only want to teach them SIGMA ERROR to get acquainted with the idea of error in math. We do not want to go into any detail or depth of error, just an error because we had to judge or decide where a point was, in-between 10 hatch marks. Remember, decimals are ten, so we have ten hatch marks to decide.

It is important to teach Sigma Error early in math, because it removes that flawed perception that mathematics is always "perfect with no error". But, even numbers multiplied have a sigma error. For the square root of 2 has a Sigma error. And have your students of 8 years old, do this exercise. The square root of a number, teach them, the square root of a number, is what number when multiplied by itself equals that given number. In our case we want the square root of 2. Which means, what number when multiplied by itself is equal to 2. Teacher, show the students that the square root of 9 is what? Is 3 because 3x3=9. The square root of 4 is what, the square root of 16 is what? The square root of 25 is what? The square root of 36, of 49, of 64, of 81, of 100 is what?

So, to the class, say that we are going to find the square root of 2, and see what Sigma Error exists for the square root of 2.

So, to the class, say, is 1 multiplied by itself, is that 2? Is 2, multiplied by itself equal to 2? No, so it must be in between 1 and 2, lies the square root of 2. So, now, anyone guess? Let us guess 1.5 is the square root of 2.

  1.5
x1.5
--------
  75
15
_____
2.25

Check to see if all the students remember how to multiply decimals

So, is 1.5 the square root of 2? No, it is too large, so let us try 1.4

   1.4
x 1.4
--------
   56
 14
-------
1.96

So, is 1.4, the square root of 2? No, it is too small and needs a tiny bit more. So we try 1.41

   1.41
x 1.41
---------
    141
  564
141
------
1.9881

So, is 1.41, the square root of 2? No, it is too small and needs a tiny bit more. So we try 1.42

Now, teacher this is a pretty way of ending a year of teaching math to 8 year olds for it ends by reviewing multiplication of Decimals, and decimals understanding is of prime major concern for youngsters of this age. Hopefully the 8 year olds have a running start with decimal Long Division, and if not, well, 9 and 10 year olds will be taught more and more of Long Division, until they are fluent in it. But, back to this square root of 2.

   1.42
x 1.42
---------
    284
  568
142
------
2.0164

Now, here the teacher is going to have to review the fact that we are in 10 Grid, and in 10 Grid, 1.42 is the square root of 2 because the only numbers that exist in 10 Grid have one digit to the right of the decimal point so the digits of "164" are Sigma Error of math, for in 10 Grid, the square root of 2 is precisely 1.42.

--- END, end of a one year teaching of 8 year olds ---

AP

[Archimedes Plutonium]

Alright, now, I am very happy with the progress so far-- for me, that means Logical progress. My mind is unique over most other humans, in that I have a finely tuned logical mind, which means, I can seem to detect-- errors and weakness in flow of thought that most scientists never are able to detect-- witness:: in 1897, JJ Thomson discovered a .5MeV particle that he thought, and everyone else thought was the electron, and then by 2017, it was my logical mind that could see, the real electron was the 105 MeV particle, and the .5 MeV was a magnetic monopole particle.

A scientists, including mathematicians, greatest tool, is logical thought, yet, given 99% of mathematicians, they run on no logic whatsoever. Show 99% of mathematicians that the sine and cosine are semicircle waves, or that the oval is a conic section never the ellipse, or even just show them that 2 OR 2 = 4 is preposterous, and even a 12 year old street urchin knows 2 AND 2 is equal to 4. So, what I am saying, is, Logic in science is rare, very rare.

Now, I pause at this moment because I noticed something that could lead to bigger things. I noticed in the first lesson this chart of counting I made::

We start this book for 8 year old students and to Count and then from Counting learn "What is a number".

Have the students write out all the numbers from 0 to 100. They are Counting

Something like this::

90, 91, 92, 93, 94, 95 96, 97, 98, 99, 100
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
00, 01, 02, 03, 04, 05, 06, 07, 08, 09,

I noticed something else that we can do that same chart as this::

We start this book for 8 year old students and to Count and then from Counting learn "What is a number".

Have the students write out all the numbers from 0 to 100. They are Counting

Something like this::

91, 92, 93, 94, 95 96, 97, 98, 99, 100
81, 82, 83, 84, 85, 86, 87, 88, 89, 90
71, 72, 73, 74, 75, 76, 77, 78, 79, 80
61, 62, 63, 64, 65, 66, 67, 68, 69, 70
51, 52, 53, 54, 55, 56, 57, 58, 59, 60
41, 42, 43, 44, 45, 46, 47, 48, 49, 50
31, 32, 33, 34, 35, 36, 37, 38, 39, 40
21, 22, 23, 24, 25, 26, 27, 28, 29, 30
11, 12, 13, 14, 15, 16, 17, 18, 19, 20
01, 02, 03, 04, 05, 06, 07, 08, 09, 10
00

So, one chart is where 100 sticks out, and the other is where 0 sticks out.

Now, if that was a chart of Integers in 100 Grid, integers only and we were to do a function graph of Y= 1/x, in a sense, 0 is not a point of the Graph at all, and we do not worry about 1/0.

So, if we isolated 0 and see the 1st Quadrant Only as all nonzero numbers we would have helped in functions at 0.

But, I see the above as even more far reaching in theory. For the above if overlain by the Cartesian Coordinate system, we can see the above as a Theoretical Graph Paper.

For instance the point (8,7) as a circle center in one of the lessons given, that point would be the point 78.

So, what I am thinking is that likely, Mathematics itself for Graphing has and needs two simultaneous graphs. The points of vertices on a graph paper are Numbers in numerical order. So in Cartesian, our circle center at (8,7) is also the point that is 87 in numerical order.

Now, when we have an equation of semicircle we are using Cartesian numbers for the x and y, but, can we also write an equation of circle where the number 87 comes forth as that same center.

The function Y= x is a diagonal 45 degree straight line and in Cartesian is (0,0) (1,1) (2,2) etc. But in Counting Graph as above, Y= x would be 11, 22, 33, 44,....

So, I think I am at the cusp of doing something entirely new in Mathematics Graphing, where I compound graphing and Coordinate Systems as a bond between two different systems. Where I glue together a Cartesian with a Counting system to make a one whole better overall system.

And, perhaps we can write out functions that focus on the Counting Graph more than the Cartesian Graph.

I am encouraged by this from the Calculus, for the Calculus and functions are an Orderly trip along the x-axis, wanting to find a y-axis intersect. But the Counting graph is a Orderly trip throughout the entire plane of graphing.

Just some thoughts for Graduate College math in this textbook.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright, now, I am very happy with the progress so far-- for me, that means Logical progress. My mind is unique over most other humans, in that I have a finely tuned logical mind, which means, I can seem to detect-- errors and weakness in flow of thought that most scientists never are able to detect-- witness:: in 1897, JJ Thomson discovered a .5MeV particle that he thought, and everyone else thought was the electron, and then by 2017, it was my logical mind that could see, the real electron was the 105 MeV particle, and the .5 MeV was a magnetic monopole particle.
A scientists, including mathematicians, greatest tool, is logical thought, yet, given 99% of mathematicians, they run on no logic whatsoever. Show 99% of mathematicians that the sine and cosine are semicircle waves, or that the oval is a conic section never the ellipse, or even just show them that 2 OR 2 = 4 is preposterous, and even a 12 year old street urchin knows 2 AND 2 is equal to 4. So, what I am saying, is, Logic in science is rare, very rare.
We start this book for 8 year old students and to Count and then from Counting learn "What is a number".
Have the students write out all the numbers from 0 to 100. They are Counting
90, 91, 92, 93, 94, 95 96, 97, 98, 99, 100
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
00, 01, 02, 03, 04, 05, 06, 07, 08, 09,
We start this book for 8 year old students and to Count and then from Counting learn "What is a number".
Have the students write out all the numbers from 0 to 100. They are Counting
91, 92, 93, 94, 95 96, 97, 98, 99, 100
81, 82, 83, 84, 85, 86, 87, 88, 89, 90
71, 72, 73, 74, 75, 76, 77, 78, 79, 80
61, 62, 63, 64, 65, 66, 67, 68, 69, 70
51, 52, 53, 54, 55, 56, 57, 58, 59, 60
41, 42, 43, 44, 45, 46, 47, 48, 49, 50
31, 32, 33, 34, 35, 36, 37, 38, 39, 40
21, 22, 23, 24, 25, 26, 27, 28, 29, 30
11, 12, 13, 14, 15, 16, 17, 18, 19, 20
01, 02, 03, 04, 05, 06, 07, 08, 09, 10
00
So, one chart is where 100 sticks out, and the other is where 0 sticks out.
Now, if that was a chart of Integers in 100 Grid, integers only and we were to do a function graph of Y= 1/x, in a sense, 0 is not a point of the Graph at all, and we do not worry about 1/0.
So, if we isolated 0 and see the 1st Quadrant Only as all nonzero numbers we would have helped in functions at 0.
But, I see the above as even more far reaching in theory. For the above if overlain by the Cartesian Coordinate system, we can see the above as a Theoretical Graph Paper.
For instance the point (8,7) as a circle center in one of the lessons given, that point would be the point 78.

Error-- 87, not 78

Post by Archimedes Plutonium
So, what I am thinking is that likely, Mathematics itself for Graphing has and needs two simultaneous graphs. The points of vertices on a graph paper are Numbers in numerical order. So in Cartesian, our circle center at (8,7) is also the point that is 87 in numerical order.
Now, when we have an equation of semicircle we are using Cartesian numbers for the x and y, but, can we also write an equation of circle where the number 87 comes forth as that same center.
The function Y= x is a diagonal 45 degree straight line and in Cartesian is (0,0) (1,1) (2,2) etc. But in Counting Graph as above, Y= x would be 11, 22, 33, 44,....
So, I think I am at the cusp of doing something entirely new in Mathematics Graphing, where I compound graphing and Coordinate Systems as a bond between two different systems. Where I glue together a Cartesian with a Counting system to make a one whole better overall system.
And, perhaps we can write out functions that focus on the Counting Graph more than the Cartesian Graph.
I am encouraged by this from the Calculus, for the Calculus and functions are an Orderly trip along the x-axis, wanting to find a y-axis intersect. But the Counting graph is a Orderly trip throughout the entire plane of graphing.
Just some thoughts for Graduate College math in this textbook.
AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright, now, I am very happy with the progress so far-- for me, that means Logical progress. My mind is unique over most other humans, in that I have a finely tuned logical mind, which means, I can seem to detect-- errors and weakness in flow of thought that most scientists never are able to detect-- witness:: in 1897, JJ Thomson discovered a .5MeV particle that he thought, and everyone else thought was the electron, and then by 2017, it was my logical mind that could see, the real electron was the 105 MeV particle, and the .5 MeV was a magnetic monopole particle.
A scientists, including mathematicians, greatest tool, is logical thought, yet, given 99% of mathematicians, they run on no logic whatsoever. Show 99% of mathematicians that the sine and cosine are semicircle waves, or that the oval is a conic section never the ellipse, or even just show them that 2 OR 2 = 4 is preposterous, and even a 12 year old street urchin knows 2 AND 2 is equal to 4. So, what I am saying, is, Logic in science is rare, very rare.
We start this book for 8 year old students and to Count and then from Counting learn "What is a number".
Have the students write out all the numbers from 0 to 100. They are Counting
90, 91, 92, 93, 94, 95 96, 97, 98, 99, 100
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
00, 01, 02, 03, 04, 05, 06, 07, 08, 09,
We start this book for 8 year old students and to Count and then from Counting learn "What is a number".
Have the students write out all the numbers from 0 to 100. They are Counting
91, 92, 93, 94, 95 96, 97, 98, 99, 100
81, 82, 83, 84, 85, 86, 87, 88, 89, 90
71, 72, 73, 74, 75, 76, 77, 78, 79, 80
61, 62, 63, 64, 65, 66, 67, 68, 69, 70
51, 52, 53, 54, 55, 56, 57, 58, 59, 60
41, 42, 43, 44, 45, 46, 47, 48, 49, 50
31, 32, 33, 34, 35, 36, 37, 38, 39, 40
21, 22, 23, 24, 25, 26, 27, 28, 29, 30
11, 12, 13, 14, 15, 16, 17, 18, 19, 20
01, 02, 03, 04, 05, 06, 07, 08, 09, 10
00
So, one chart is where 100 sticks out, and the other is where 0 sticks out.
Now, if that was a chart of Integers in 100 Grid, integers only and we were to do a function graph of Y= 1/x, in a sense, 0 is not a point of the Graph at all, and we do not worry about 1/0.
So, if we isolated 0 and see the 1st Quadrant Only as all nonzero numbers we would have helped in functions at 0.
But, I see the above as even more far reaching in theory. For the above if overlain by the Cartesian Coordinate system, we can see the above as a Theoretical Graph Paper.
For instance the point (8,7) as a circle center in one of the lessons given, that point would be the point 87.
So, what I am thinking is that likely, Mathematics itself for Graphing has and needs two simultaneous graphs. The points of vertices on a graph paper are Numbers in numerical order. So in Cartesian, our circle center at (8,7) is also the point that is 87 in numerical order.
Now, when we have an equation of semicircle we are using Cartesian numbers for the x and y, but, can we also write an equation of circle where the number 87 comes forth as that same center.
The function Y= x is a diagonal 45 degree straight line and in Cartesian is (0,0) (1,1) (2,2) etc. But in Counting Graph as above, Y= x would be 11, 22, 33, 44,....
So, I think I am at the cusp of doing something entirely new in Mathematics Graphing, where I compound graphing and Coordinate Systems as a bond between two different systems. Where I glue together a Cartesian with a Counting system to make a one whole better overall system.
And, perhaps we can write out functions that focus on the Counting Graph more than the Cartesian Graph.
I am encouraged by this from the Calculus, for the Calculus and functions are an Orderly trip along the x-axis, wanting to find a y-axis intersect. But the Counting graph is a Orderly trip throughout the entire plane of graphing.
Just some thoughts for Graduate College math in this textbook.

You see, with Counting Graph, the numbers become vertices of the blocks in the graph paper.

So, what I am looking to do, is see if by Combining Cartesian Coordinate System with that of a Counting of Vertices Coordinate System, combining the two into one, whether there is an improvement to Analytic Geometry, or Calculus. For example in graphing a circle that is in the middle of 100 Grid would be the counted number 55 and the Cartesian (5,5).

If this idea has no practical use, well, it remains just a idea.

Now, I recall that how DesCartes discovered his famous coordinate system had a practical use. He was observing a fly walk across the ceiling in the room he was sleeping in, and thought to himself, the distance from the corner to the fly in terms of a number along one wall and a number along the perpendicular wall, told him every movement of the fly.

In my case, of Counting Numbers being vertices in a linear order, we simply state a number where the fly is.

91, 92, 93, 94, 95 96, 97, 98, 99, 100
81, 82, 83, 84, 85, 86, 87, 88, 89, 90
71, 72, 73, 74, 75, 76, 77, 78, 79, 80
61, 62, 63, 64, 65, 66, 67, 68, 69, 70
51, 52, 53, 54, 55, 56, 57, 58, 59, 60
41, 42, 43, 44, 45, 46, 47, 48, 49, 50
31, 32, 33, 34, 35, 36, 37, 38, 39, 40
21, 22, 23, 24, 25, 26, 27, 28, 29, 30
11, 12, 13, 14, 15, 16, 17, 18, 19, 20
01, 02, 03, 04, 05, 06, 07, 08, 09, 10
00

So, if the fly is at 22, we picture in our minds where the fly is. If the fly is in the far right corner, it is 10. If the fly is center of the ceiling it is at 55. Only, hold on a minute, I need 00 for 55 to be center.

AP

[a***@gmail.com]

Post by Archimedes Plutonium

Post by Archimedes Plutonium
Alright, now, I am very happy with the progress so far-- for me, that means Logical progress. My mind is unique over most other humans, in that I have a finely tuned logical mind, which means, I can seem to detect-- errors and weakness in flow of thought that most scientists never are able to detect-- witness:: in 1897, JJ Thomson discovered a .5MeV particle that he thought, and everyone else thought was the electron, and then by 2017, it was my logical mind that could see, the real electron was the 105 MeV particle, and the .5 MeV was a magnetic monopole particle.
A scientists, including mathematicians, greatest tool, is logical thought, yet, given 99% of mathematicians, they run on no logic whatsoever. Show 99% of mathematicians that the sine and cosine are semicircle waves, or that the oval is a conic section never the ellipse, or even just show them that 2 OR 2 = 4 is preposterous, and even a 12 year old street urchin knows 2 AND 2 is equal to 4. So, what I am saying, is, Logic in science is rare, very rare.
We start this book for 8 year old students and to Count and then from Counting learn "What is a number".
Have the students write out all the numbers from 0 to 100. They are Counting
90, 91, 92, 93, 94, 95 96, 97, 98, 99, 100
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
00, 01, 02, 03, 04, 05, 06, 07, 08, 09,
We start this book for 8 year old students and to Count and then from Counting learn "What is a number".
Have the students write out all the numbers from 0 to 100. They are Counting
91, 92, 93, 94, 95 96, 97, 98, 99, 100
81, 82, 83, 84, 85, 86, 87, 88, 89, 90
71, 72, 73, 74, 75, 76, 77, 78, 79, 80
61, 62, 63, 64, 65, 66, 67, 68, 69, 70
51, 52, 53, 54, 55, 56, 57, 58, 59, 60
41, 42, 43, 44, 45, 46, 47, 48, 49, 50
31, 32, 33, 34, 35, 36, 37, 38, 39, 40
21, 22, 23, 24, 25, 26, 27, 28, 29, 30
11, 12, 13, 14, 15, 16, 17, 18, 19, 20
01, 02, 03, 04, 05, 06, 07, 08, 09, 10
00
So, one chart is where 100 sticks out, and the other is where 0 sticks out.
Now, if that was a chart of Integers in 100 Grid, integers only and we were to do a function graph of Y= 1/x, in a sense, 0 is not a point of the Graph at all, and we do not worry about 1/0.
So, if we isolated 0 and see the 1st Quadrant Only as all nonzero numbers we would have helped in functions at 0.
But, I see the above as even more far reaching in theory. For the above if overlain by the Cartesian Coordinate system, we can see the above as a Theoretical Graph Paper.
For instance the point (8,7) as a circle center in one of the lessons given, that point would be the point 87.
So, what I am thinking is that likely, Mathematics itself for Graphing has and needs two simultaneous graphs. The points of vertices on a graph paper are Numbers in numerical order. So in Cartesian, our circle center at (8,7) is also the point that is 87 in numerical order.
Now, when we have an equation of semicircle we are using Cartesian numbers for the x and y, but, can we also write an equation of circle where the number 87 comes forth as that same center.
The function Y= x is a diagonal 45 degree straight line and in Cartesian is (0,0) (1,1) (2,2) etc. But in Counting Graph as above, Y= x would be 11, 22, 33, 44,....
So, I think I am at the cusp of doing something entirely new in Mathematics Graphing, where I compound graphing and Coordinate Systems as a bond between two different systems. Where I glue together a Cartesian with a Counting system to make a one whole better overall system.
And, perhaps we can write out functions that focus on the Counting Graph more than the Cartesian Graph.
I am encouraged by this from the Calculus, for the Calculus and functions are an Orderly trip along the x-axis, wanting to find a y-axis intersect. But the Counting graph is a Orderly trip throughout the entire plane of graphing.
Just some thoughts for Graduate College math in this textbook.

You see, with Counting Graph, the numbers become vertices of the blocks in the graph paper.
So, what I am looking to do, is see if by Combining Cartesian Coordinate System with that of a Counting of Vertices Coordinate System, combining the two into one, whether there is an improvement to Analytic Geometry, or Calculus. For example in graphing a circle that is in the middle of 100 Grid would be the counted number 55 and the Cartesian (5,5).
If this idea has no practical use, well, it remains just a idea.
Now, I recall that how DesCartes discovered his famous coordinate system had a practical use. He was observing a fly walk across the ceiling in the room he was sleeping in, and thought to himself, the distance from the corner to the fly in terms of a number along one wall and a number along the perpendicular wall, told him every movement of the fly.
In my case, of Counting Numbers being vertices in a linear order, we simply state a number where the fly is.
91, 92, 93, 94, 95 96, 97, 98, 99, 100
81, 82, 83, 84, 85, 86, 87, 88, 89, 90
71, 72, 73, 74, 75, 76, 77, 78, 79, 80
61, 62, 63, 64, 65, 66, 67, 68, 69, 70
51, 52, 53, 54, 55, 56, 57, 58, 59, 60
41, 42, 43, 44, 45, 46, 47, 48, 49, 50
31, 32, 33, 34, 35, 36, 37, 38, 39, 40
21, 22, 23, 24, 25, 26, 27, 28, 29, 30
11, 12, 13, 14, 15, 16, 17, 18, 19, 20
01, 02, 03, 04, 05, 06, 07, 08, 09, 10
00
So, if the fly is at 22, we picture in our minds where the fly is. If the fly is in the far right corner, it is 10. If the fly is center of the ceiling it is at 55. Only, hold on a minute, I need 00 for 55 to be center.
AP

Uh-oh. It looks like poor Archie the Pooh is starting to forget even how to count. If 55 is the center of the ceiling, what is 95? I know, it's an infinite irrational number! Or is it that Archie is infinitely irrational?

[Archimedes Plutonium]

Post by Archimedes Plutonium
If this idea has no practical use, well, it remains just a idea.
Now, I recall that how DesCartes discovered his famous coordinate system had a practical use. He was observing a fly walk across the ceiling in the room he was sleeping in, and thought to himself, the distance from the corner to the fly in terms of a number along one wall and a number along the perpendicular wall, told him every movement of the fly.
In my case, of Counting Numbers being vertices in a linear order, we simply state a number where the fly is.

So, let us explore if All Graphing employs two Coordinate Systems simultaneously

Post by Archimedes Plutonium
91, 92, 93, 94, 95 96, 97, 98, 99, 100
81, 82, 83, 84, 85, 86, 87, 88, 89, 90
71, 72, 73, 74, 75, 76, 77, 78, 79, 80
61, 62, 63, 64, 65, 66, 67, 68, 69, 70
51, 52, 53, 54, 55, 56, 57, 58, 59, 60
41, 42, 43, 44, 45, 46, 47, 48, 49, 50
31, 32, 33, 34, 35, 36, 37, 38, 39, 40
21, 22, 23, 24, 25, 26, 27, 28, 29, 30
11, 12, 13, 14, 15, 16, 17, 18, 19, 20
01, 02, 03, 04, 05, 06, 07, 08, 09, 10
00
So, if the fly is at 22, we picture in our minds where the fly is. If the fly is in the far right corner, it is 10. If the fly is center of the ceiling it is at 55. Only, hold on a minute, I need 00 for 55 to be center.

Instead of 00 being the oddball, let us take 100 as the oddball

90, 91, 92, 93, 94, 95 96, 97, 98, 99, 100
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
00, 01, 02, 03, 04, 05, 06, 07, 08, 09,

Now, for a function like Y=1, it would be 10,11,12,..19. A function like Y =7 would be 70,71,..79

A function like Y = x would be 00, 11, 22, 33,..99

Curiously a function like Y = -x would be 90, 81, 72, 63, 54, 45, 36, 27, 09, the Rule of 9

But now, that is 100 Grid Integers only, what about 10 Grid integers only and 1000 Grid integers only.

Here this becomes perhaps really great, because maybe, just maybe we can tie together circle with square in a beautiful connection. For 100 Grid integers only, 10000 Grid integers only and all higher even numbered are squares.

But what are the odd Grids, 10, 1000, 100000, etc etc?

Could they be circles? But circles do not exist, so could they be grids of regular polygons?

10 Grid integers only has 10 members not counting 0, if we count 0 and ignore the 10 it still has 10 members. But the feature of 100 Grid integers only is vertices of the little blocks that make a square.
What can the 10 members of 10 Grid vertices be?

Well, a square of 10 would be 3.16 by 3.16, for 1000 Grid square it is 31.6 by 31.6. Trouble is 3.16 and 31.6 are not integers and it makes no order-sense to have 31 rows and 31 columns. So the geometry of the Odd Grids Integers, must be different than that of squares.

Naturally, it looks like a rectangle for 10 Grid Integers would be a 1 by 10 rectangle, and 1000 Grid Integers only would be a 10 by 100 rectangle

But, can we do even better than a rectangle? Can these Odd Grids of Integers Only be circles or regular polygons?

10 Grid, Integers only as

7 8 9

4 5 6

1 2 3

So the 5 is a circle center or center of octagon

Now for the 1000 Grid where 31 makes sense as a Number that ends a row or column.

So here we either have the points of a circle whose radius is 31.6 or the vertices of a regular polygon.

AP

[a***@gmail.com]

Post by Archimedes Plutonium

Post by Archimedes Plutonium
If this idea has no practical use, well, it remains just a idea.
Now, I recall that how DesCartes discovered his famous coordinate system had a practical use. He was observing a fly walk across the ceiling in the room he was sleeping in, and thought to himself, the distance from the corner to the fly in terms of a number along one wall and a number along the perpendicular wall, told him every movement of the fly.
In my case, of Counting Numbers being vertices in a linear order, we simply state a number where the fly is.

So, let us explore if All Graphing employs two Coordinate Systems simultaneously

Post by Archimedes Plutonium
91, 92, 93, 94, 95 96, 97, 98, 99, 100
81, 82, 83, 84, 85, 86, 87, 88, 89, 90
71, 72, 73, 74, 75, 76, 77, 78, 79, 80
61, 62, 63, 64, 65, 66, 67, 68, 69, 70
51, 52, 53, 54, 55, 56, 57, 58, 59, 60
41, 42, 43, 44, 45, 46, 47, 48, 49, 50
31, 32, 33, 34, 35, 36, 37, 38, 39, 40
21, 22, 23, 24, 25, 26, 27, 28, 29, 30
11, 12, 13, 14, 15, 16, 17, 18, 19, 20
01, 02, 03, 04, 05, 06, 07, 08, 09, 10
00
So, if the fly is at 22, we picture in our minds where the fly is. If the fly is in the far right corner, it is 10. If the fly is center of the ceiling it is at 55. Only, hold on a minute, I need 00 for 55 to be center.

Instead of 00 being the oddball, let us take 100 as the oddball
90, 91, 92, 93, 94, 95 96, 97, 98, 99, 100
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
00, 01, 02, 03, 04, 05, 06, 07, 08, 09,
Now, for a function like Y=1, it would be 10,11,12,..19. A function like Y =7 would be 70,71,..79
A function like Y = x would be 00, 11, 22, 33,..99
Curiously a function like Y = -x would be 90, 81, 72, 63, 54, 45, 36, 27, 09, the Rule of 9
But now, that is 100 Grid Integers only, what about 10 Grid integers only and 1000 Grid integers only.
Here this becomes perhaps really great, because maybe, just maybe we can tie together circle with square in a beautiful connection. For 100 Grid integers only, 10000 Grid integers only and all higher even numbered are squares.
But what are the odd Grids, 10, 1000, 100000, etc etc?
Could they be circles? But circles do not exist, so could they be grids of regular polygons?
10 Grid integers only has 10 members not counting 0, if we count 0 and ignore the 10 it still has 10 members. But the feature of 100 Grid integers only is vertices of the little blocks that make a square.
What can the 10 members of 10 Grid vertices be?
Well, a square of 10 would be 3.16 by 3.16, for 1000 Grid square it is 31.6 by 31.6. Trouble is 3.16 and 31.6 are not integers and it makes no order-sense to have 31 rows and 31 columns. So the geometry of the Odd Grids Integers, must be different than that of squares.
Naturally, it looks like a rectangle for 10 Grid Integers would be a 1 by 10 rectangle, and 1000 Grid Integers only would be a 10 by 100 rectangle
But, can we do even better than a rectangle? Can these Odd Grids of Integers Only be circles or regular polygons?
10 Grid, Integers only as
7 8 9
4 5 6
1 2 3
So the 5 is a circle center or center of octagon
Now for the 1000 Grid where 31 makes sense as a Number that ends a row or column.
So here we either have the points of a circle whose radius is 31.6 or the vertices of a regular polygon.
AP

Archie is on the verge of a great discovery: odd numbers aren't the same kind of numbers as even numbers. This could have tremendous impact on arithmetic as well as geometry. One or another of these sets of umbers will have to be discarded. Will it be the evens, since even grids have odd centers, or will it be the odd grids that have even centers? You obviously can't have both. Counting is about to become another bit of old fake math. No one with logic would ever count.

[Archimedes Plutonium]

So for 10 Grid integers we have

7 8 9
4 5 6
1 2 3

For 1000 Grid integers we have

968, 969, 970, 971, ... 998, 999
937, 938, 939, 940, ... 966, 967
.
.
.
.
32, 33, 34, 35, .... 61, 62
1, 2, 3, 4, .... 30, 31

Where there are 31 numbers in a row and 32 rows with 8 extra numbers

All of which forming a circle. The extra row is the diameter and center of the circle, or regular polygon.

For 100000 Grid we have 316 numbers per row and have 316 rows with 144 extra

So, the even Grids are squares, leaving the odd Grids to be rectangles, but we look to see if those Odd Grids are circles or regular polygons, where a row of numbers forms the side of a regular polygon. If so, then the 10 Grid would be a 3-gon, the 1000 Grid would be a 32-gon and a 100000 Grid would be a 316-gon.

It makes sense, that graphing does not need something outside of math itself-- a lattice Grid of vertices in order to graph. It seems commonsense, that Math itself has a Natural Graph Grid System alongside the Cartesian coordinate system, but that requires a square for even Grids and a circle or regular polygon for odd Grids.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
So for 10 Grid integers we have
7 8 9
4 5 6
1 2 3

Now I am looking, looking all over the place for a way to make a Graph Vertices Grid System out of pure numbers.

We already have the Cartesian Coordinate system, it works splendidly (provided we do 1st Quadrant Only-- or, realize that negatives are not quantity but direction)

But, now, trouble is, with Graphing, math should have built inside itself a Vertices Grid System. Math should have a Graph already built inside itself by the sheer counting of numbers.

This works great for 100 Grid integers for they are a square block and each number is a vertex. However for 10 grid, 1000 grid and all odd numbered grids, they do not form a Square vertex graph paper. They form a rectangle, or, what I suspect some round shaped figure with vertices.

Post by Archimedes Plutonium
For 1000 Grid integers we have
968, 969, 970, 971, ... 998, 999
937, 938, 939, 940, ... 966, 967
.
.
.
.
32, 33, 34, 35, .... 61, 62
1, 2, 3, 4, .... 30, 31

Now here I tried making use of 31.6 from square root of 1000, so I get a square of approx 31 by 31 missing 39. And looking for some roundish shaped figures as a Grid Vertices Graph. I look at pentagonal numbers and Euler's generalized pentagonal numbers-- wouldn't you know it, he has 100 but only 1001.

So, I look at triangular numbers, but do not like all those 60 degrees.

Post by Archimedes Plutonium
Where there are 31 numbers in a row and 32 rows with 8 extra numbers
All of which forming a circle. The extra row is the diameter and center of the circle, or regular polygon.
For 100000 Grid we have 316 numbers per row and have 316 rows with 144 extra
So, the even Grids are squares, leaving the odd Grids to be rectangles, but we look to see if those Odd Grids are circles or regular polygons, where a row of numbers forms the side of a regular polygon. If so, then the 10 Grid would be a 3-gon, the 1000 Grid would be a 32-gon and a 100000 Grid would be a 316-gon.
It makes sense, that graphing does not need something outside of math itself-- a lattice Grid of vertices in order to graph. It seems commonsense, that Math itself has a Natural Graph Grid System alongside the Cartesian coordinate system, but that requires a square for even Grids and a circle or regular polygon for odd Grids.

Next, I look for hexagonal numbers. And maybe, just maybe, right triangles. For consider that if we are allowed two legs of a right triangle to be added together, so in 3,4,5 we have 3+3+4 = 10 for 10 Grid, and we have that for all other decimal clean-pure-numbers 10, 100, 1000, etc

So, then, if the solution is RIGHT TRIANGLE, and we can consider right triangles as somewhat Going Around, although not as smooth as a circle or Regular Polygon. And here, the visual problem is how do we place ten numbers as dots to form a right triangle


1
21
389
4567

*
**
***
****

Looking good, looking good, but need to check on hexagonal numbers and even the ellipse

AP

[Archimedes Plutonium]

I was searching through Oeis and stumbled on the perfect floor plan of maths internal graph for odd numbered 10, 1000, 100000 clean-pure-numbers

--- quoting oeis ---

ne from 0 in the direction 0,1,... - Floor van Lamoen, Jul 21 2001. The spiral begins:
85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96
.
a(n) = (3n-2)(3n-1)(3n)/((3n-1) + (3n-2) + (3n)), i.e., (the product of three conse

[Archimedes Plutonium]

Let me try this one, see if the format comes out better,,,,




85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96

AP

[j4n bur53]

Where is infinity?

Post by Archimedes Plutonium
Let me try this one, see if the format comes out better,,,,
85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96
AP

[Archimedes Plutonium]

Now here is a Vertices Graph and consider each number as a vertex of graphing paper
90, 91, 92, 93, 94, 95 96, 97, 98, 99, 100
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
00, 01, 02, 03, 04, 05, 06, 07, 08, 09,

Now, here are those same numbers out to 100, but in a rotational order of a hexagon (I prefer a octagon) but let us see if we can graph a function with these rotational numbers instead of our plain old usual graph of tiny block squares with vertices

85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96--97--98--99--100

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
Now here is a Vertices Graph and consider each number as a vertex of graphing paper
90, 91, 92, 93, 94, 95 96, 97, 98, 99, 100
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
00, 01, 02, 03, 04, 05, 06, 07, 08, 09,
Now, here are those same numbers out to 100, but in a rotational order of a hexagon (I prefer a octagon) but let us see if we can graph a function with these rotational numbers instead of our plain old usual graph of tiny block squares with vertices
85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96--97--98--99--100
AP

No, I am going to walk back all this stuff about a second coordinate system.

All Grid systems , 10, 100, 1000, etc etc have squares. What I was forgetting about is that the fractions go along with the integers. Integers only cause a rift.

So for 10 Grid, .1, .2, .3, . . ., 9.8, 9.9, 10 leaves 100 units of .1 and thus is 100 by 100 square, and utterly no problem with graph vertices.

Likewise the 1000 Grid is .001, .002, . . . , 999.999, 1000 and is a square that is 10^6 by 10^6

No need of a second Graph system

AP

[Archimedes Plutonium]

Alright, I abandon the second graph but this spiral has me wondering how I can get a spiral such that 10 is a vertex on one winding and 100 a vertex on another winding and whether 1000 can be a vertex on a future winding. So vertex for all three 10, 100, 1000, and let us throw in 10000, all vertices in a specific winding-- is it possible-- with symmetry of course.



85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96

In the above, the 10 is a vertex, but the 100 will not be a vertex. Are the numbers able to give that? Perhaps a octagon winding, or a pentagon winding.

Now, I have not thrown out the possibility that such spirals cannot do a coordinate system. For one aspect of them is that if you intersect two planes of the above sharing the same 0 as origin, that you thence have gotten rid of negative quadrants in 3rd dimension, and you have a solid geometry coordinate system. But the catch is, that you do not have coordinate points such as (1,3,5) for xyz in third dimension, but rather, say you wanted to plot (1,3,5) you end up connecting the 1 in the x plane, with the 3 in the y plane, connecting those two segments with the 5 in the z plane.

I am trying to see that in my mind, but far too difficult.

So in this new fangled contraption of a new coordinate system, you have three planes of the above say hexagon all sharing origin at 0 and being x,y,z planes.

Now to plot a point such as (1, 6, 2) is not a new point but a strange curve of a three pronged line segments joined together.

Well, on second thought, maybe I can plot this in 2nd dimension. So we have that hexagon plane
and we have a function say Y = 2x +1 would be 1->3, 2->5, 3->7

And plotting that in the hexagon 2nd dimension would be



85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 *5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 *3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 *7 *1---*2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96

That would entail three line segments, the 1to3 criss crossing the 2 to 5, crisscrossing the 2 to 5

But I do not know if math is in need of functions where function yields not points but yields entire line segments. Does math need functions that yields line segments instead of points?

But then, perhaps volume in 3rd dimension is served by line segments instead of points.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright, I abandon the second graph but this spiral has me wondering how I can get a spiral such that 10 is a vertex on one winding and 100 a vertex on another winding and whether 1000 can be a vertex on a future winding. So vertex for all three 10, 100, 1000, and let us throw in 10000, all vertices in a specific winding-- is it possible-- with symmetry of course.
85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96
In the above, the 10 is a vertex, but the 100 will not be a vertex. Are the numbers able to give that? Perhaps a octagon winding, or a pentagon winding.
Now, I have not thrown out the possibility that such spirals cannot do a coordinate system. For one aspect of them is that if you intersect two planes of the above sharing the same 0 as origin, that you thence have gotten rid of negative quadrants in 3rd dimension, and you have a solid geometry coordinate system. But the catch is, that you do not have coordinate points such as (1,3,5) for xyz in third dimension, but rather, say you wanted to plot (1,3,5) you end up connecting the 1 in the x plane, with the 3 in the y plane, connecting those two segments with the 5 in the z plane.
I am trying to see that in my mind, but far too difficult.
So in this new fangled contraption of a new coordinate system, you have three planes of the above say hexagon all sharing origin at 0 and being x,y,z planes.
Now to plot a point such as (1, 6, 2) is not a new point but a strange curve of a three pronged line segments joined together.
Well, on second thought, maybe I can plot this in 2nd dimension. So we have that hexagon plane
and we have a function say Y = 2x +1 would be 1->3, 2->5, 3->7
And plotting that in the hexagon 2nd dimension would be
85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 *5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 *3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 *7 *1---*2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96
That would entail three line segments, the 1to3 criss crossing the 2 to 5, crisscrossing the 2 to 5
But I do not know if math is in need of functions where function yields not points but yields entire line segments. Does math need functions that yields line segments instead of points?
But then, perhaps volume in 3rd dimension is served by line segments instead of points.

Alright, some progress and some insights on this.

I finally figured out that the Graph Paper, of the tiny blocks forming Grid System has only regular polygons of a 4, 8, 16, etc sequence. The Graph paper does not allow for pentagon, hexagon series because some vertices do not exist.

Now, if I started in the middle of a sheet of fresh graph paper, I can circle around in a spiral, consuming all the vertex points, and I get the number 10 to lie on the same diameter line as that of 2, 1, 5.

I have not yet circled around to reach 100, but am curious if the 100 mark lies on the same diameter line as 10, 2, 1, 5.

But, the number 17 now lies on that rotation, and 17 does not fit well with 5, 1, 2, 10.

Oops, now swung that rotation around and the next number after 10 is 26. So that diameter line now holds 17, 5, 1, 2, 10, 26.

Nope, I could not wait, so finished out the rotation to 100, and the diameter line finishes out to
65, 37, 17, 5, 1, 2, 10, 26, 50, 82

The 100 mark ended the as the perfect square with the diameter line just above the 10-line. So the 100 diameter line is 100, 64, 36, 16, 4 , 3, 11, 27, 51, 83

Now, already I see a pattern emerging is that the 1/2 of a diameter line is odd, other half is even. And this pattern is also, sometimes, in the columns, of alternating even and odd.

Now the center 0 is not a vertice but is in the middle of the tiniest block surrounded by 1,2,3,4 vertices.

AP

[Archimedes Plutonium]

Now I am going back to bed, soon, but, before I go, I was wondering whether in graphing in Cartesian Coordinate system, whether that maintains associativity and commutativity.

What I mean in 3rd dimension, say we are given a point (1, 2, 3). So, I start with x-axis go out to 1 then up to 2 on y axis, now I drift up 3 notches on z axis.

So, is it associative? If I started on z axis with (3, 2, 1) which is (z, y, x) do I end up landing back at point 1 on x-axis.

So, I am at 3 on z axis, and have to go 2 on y axis, then I need to go 1 over on x-axis.

So, in my first walk of (1,2,3) I end up perched 3 units above the spot (1,2)

In my second walk I end up in the same place.

So, it appears that graphing upholds associativity and commutativity.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
Now I am going back to bed, soon, but, before I go, I was wondering whether in graphing in Cartesian Coordinate system, whether that maintains associativity and commutativity.
What I mean in 3rd dimension, say we are given a point (1, 2, 3). So, I start with x-axis go out to 1 then up to 2 on y axis, now I drift up 3 notches on z axis.

Now, often, one good question leads to another good one.

Post by Archimedes Plutonium
So, is it associative? If I started on z axis with (3, 2, 1) which is (z, y, x) do I end up landing back at point 1 on x-axis.
So, I am at 3 on z axis, and have to go 2 on y axis, then I need to go 1 over on x-axis.
So, in my first walk of (1,2,3) I end up perched 3 units above the spot (1,2)
In my second walk I end up in the same place.
So, it appears that graphing upholds associativity and commutativity.

Here we have to ask, is not commutative just an absurd redundancy. If something of add or multiply is associative, then only a feeble knucklehead would want to include commutative.

I cannot think of a single item in all of math, where something is associative, but not commutative.

So, is not the concept commutative silly excess baggage. What is often called in math "true as a degenerate case".

AP

[Archimedes Plutonium]

On Saturday, February 24, 2018 at 12:29:35 PM UTC-6, Archimedes Plutonium wrote:
(snipped)

Post by Archimedes Plutonium

Post by Archimedes Plutonium
So, it appears that graphing upholds associativity and commutativity.

Here we have to ask, is not commutative just an absurd redundancy. If something of add or multiply is associative, then only a feeble knucklehead would want to include commutative.
I cannot think of a single item in all of math, where something is associative, but not commutative.
So, is not the concept commutative silly excess baggage. What is often called in math "true as a degenerate case".

So, here is a new question, did Old Math go screwy on commutative and associative properties? I mean, if you have associative, you have commutative and vice versa, and only because Old Math had no one with a good logical mind, that Old Math, had this as another error?

Now here is a worthless meathead in math and logic, who still thinks 2 OR 3 = 5 when a 10 year old kid knows better that 2 AND 3 = 5, but here is the meathead on distinct points.

Post by Archimedes Plutonium

Post by Archimedes Plutonium

PAGE58, 8-3, True Geometry / correcting axioms, 1by1 tool, angles of logarithmic spiral, conic sections unified regular polyhedra, Leaf-Triangle, Unit Basis Vector
The axioms that are in need of fixing is the axiom that between any two points lies a third new point.

The should be "between and any two DISTINCT points."

What a monsterous fool you are

OMG. You are serious. Stupid and proud of it.

And yet Mr Plutonium is right.  Two points are distinct (else they would
be one) and it is not necessary to say so.

Now here is another Logical fruitcake, where to him, symbols is everything, and understanding of nothing, what I call the Symbol Skunk Spray Syndrome-- a person who thinks math or logic is just symbol hieroglyphics writing or chickensh(t scratch marks on a piece of paper.

Post by Archimedes Plutonium
*∀L ∈ Formal_Systems
∀X
True(L, X) ↔ ∃Γ ⊆ Axioms(L) Provable(Γ, X) *

So, what is the truth about Associative and Commutative properties? Is the Commutative just excess baggage and not needed? Is the Commutative, just like given any two points in the plane, that a meathead thinks you have to insert "distinct" when logically, two points are distinct and no-one has to say any more about it.

So, here, the question over Associative Property and Commutative Property, are they just, one and the same, and you need no two distinct categories?

Is not a + b = b + a

just the same thing as

(a+b) + c = a + (b+ c)

So that in Old Math, only meatheads think there is a difference, a distinction, when in fact, both are one and the same.

In words, both the commutative and the associative, both say one and the same thing-- if you have a sequence of additions, it matters not, in which order you add them, it matters not. Both for addition and multiplication.

And only because Old Math had not a single person with a head of 1 gram of Logical mind, did Old Math have two types of properties, which they never realized was just one type of property.

Now, some stupid git of Old Math will concoct some artificial algebra with his git-mind, saying his example is a counterexample. But, well, the only examples are with numbers, not some crazy loon who thinks he can construct a counterexample.

Give me any string of positive numbers, and it never matters in what order you add them.

Give me any string of positive numbers, and, it never matters in what order you multiply them, for they always end up with the same product.

So, like the meathead who thinks you need to include "distinct" or the meathead with all his Skunk Spraying Symbols all over the page.

For math, there is one and only one Property for addition and multiplication, not two different ones.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
I finally figured out that the Graph Paper, of the tiny blocks forming Grid System has only regular polygons of a 4, 8, 16, etc sequence. The Graph paper does not allow for pentagon, hexagon series because some vertices do not exist.
Now, if I started in the middle of a sheet of fresh graph paper, I can circle around in a spiral, consuming all the vertex points, and I get the number 10 to lie on the same diameter line as that of 2, 1, 5.
I have not yet circled around to reach 100, but am curious if the 100 mark lies on the same diameter line as 10, 2, 1, 5.
But, the number 17 now lies on that rotation, and 17 does not fit well with 5, 1, 2, 10.
Oops, now swung that rotation around and the next number after 10 is 26. So that diameter line now holds 17, 5, 1, 2, 10, 26.
Nope, I could not wait, so finished out the rotation to 100, and the diameter line finishes out to
65, 37, 17, 5, 1, 2, 10, 26, 50, 82
The 100 mark ended the as the perfect square with the diameter line just above the 10-line. So the 100 diameter line is 100, 64, 36, 16, 4 , 3, 11, 27, 51, 83
Now, already I see a pattern emerging is that the 1/2 of a diameter line is odd, other half is even. And this pattern is also, sometimes, in the columns, of alternating even and odd.
Now the center 0 is not a vertice but is in the middle of the tiniest block surrounded by 1,2,3,4 vertices.

Alright a picture of the turning of numbers 0 to 100 into a spiral

96, 95, 94, 93, 92, 91 90, 89, 88, 87,
97, 61, 60, 59, 58, 57, 56, 55, 54, 86,
98, 62, 34, 33, 32, 31, 30, 29, 53, 85,
99, 63, 35, 15, 14,13,12, 28, 52, 84,
100, 64,36,16, 4, 3, 11, 27, 51, 83,
65, 37,17, 5, 1, 2, 10, 26, 50, 82,
66, 38, 18, 6, 7, 8, 9, 25, 49, 81,
67, 39, 19, 20,21,22,23,24, 48, 80,
68, 40, 41, 42, 43, 44, 45, 46, 47, 79,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78,

alright, here is that spiral

[Archimedes Plutonium]

So in math it really is redundant to have commutative and associative for both stem from a principle that adding or multiplying a given number of terms matters not on any order-- for the end result is the same regardless of order.

We even see this principle in logic where the combo of Equal-Not truth table is TTTT so no matter what order you involve each term the product is the same T. Then the connector And is Add with truth table TTTF. Here again does it matter what order you do the "and"? No, it only matters if one term is T then the entire string of terms is true.

Now with division the order matters with If-->then

But we are surprized that subtraction seems to not matter either and suspect this is another error of Old Math, for they would say 3 - 1 is not the same as 1 - 3, but here i would intercede and say there are issues with subtraction in that 3 - 1 is 3 + -1 and 1-3 is 1 + -3 and proceed as addition. So here we solve the subtraction by stating at the beginning if any negative numbers are present. Only instead of getting rid of negative numbers we get rid of subtraction in this case.

AP

[Archimedes Plutonium]

Alright Logic and Math work together to reveal inner truth

In math there are axioms of commutative and associative for addition and multiplication, but in logic these are derivable from the connectors of logic-- meaning-- they are not axioms but come from more basic fundamentals

Multiplication is the Equal allied with Not connector that is truth table TTTT. That means 3=3=3=3, or 3x3x3x3. It matters not where one starts to multiply, order is irrelevant in multiplication.

AND is add in Logic with table TTTF and here again the order in which we add is irrelevant because all that is needed is one T and the entire string of terms is T

But can Logic reveal whether division and subtraction obey commutative or associative? Division is If-then, with table TFUU and obviously that is "order dependent". If it rains then i get wet is not that of if i get wet, then it is raining.

Subtraction is the OR connector and not obvious as to order dependent. And it maybe the case that the math view of subtraction shows more detail than the logic view of subtraction.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright Logic and Math work together to reveal inner truth
In math there are axioms of commutative and associative for addition and multiplication, but in logic these are derivable from the connectors of logic-- meaning-- they are not axioms but come from more basic fundamentals
Multiplication is the Equal allied with Not connector that is truth table TTTT. That means 3=3=3=3, or 3x3x3x3. It matters not where one starts to multiply, order is irrelevant in multiplication.
AND is add in Logic with table TTTF and here again the order in which we add is irrelevant because all that is needed is one T and the entire string of terms is T
But can Logic reveal whether division and subtraction obey commutative or associative? Division is If-then, with table TFUU and obviously that is "order dependent". If it rains then i get wet is not that of if i get wet, then it is raining.

Now Math outright says that division is not commutative, for 1 -;- 2 is not 2 -:- 1

And Math is outright vocal that subtraction is not commutative for 1 - 2 is not 2 -1

But Logic has something different to say, for it agrees that if 1 then 2 is not going to be if 2 then 1,
however, logic would accept either 1 or 2, is the same as either 2 or 1, where Logic has a choice of picking 1 and discarding 2 or picking 2 and discarding 1. So subtraction in Logic is more general than subtraction in math.

So in math we have 2-1 = 1, while 1-2 = -1, noncommutative.

In Logic we have either2 or 1, is the same as either1 or 2, both are true since we pick a number and discard the other.

Is there a quick solution to this problem?

Well, if we say that the Logic subtraction as Either Or is more general than the Math subtraction, and, if we say that Logic Either Or when combined with a "order function" then we have a Logic Subtraction equivalent to a Math Subtraction.

Either I eat Spaghetti for Sunday dinner, or I eat a hamburger for Sunday dinner. This is not a time sensitive syllogism.

Either I marry the first date in life, or, I marry the person that is a later date in life. This is a either-or syllogism with a time sensitive issue involved.

The first syllogism involves no time sensitivity

The second involves a time function relative to Either A or B, where the B is dependent on A.

So, what I am wondering, is if Subtraction involved in mathematics is a Logical time dependent connector.

I am certain the reason IF-Then is noncommutative is because it is time Dependent.

And then, add and multiply are time independent.

But now, for Logic Either-Or, it appears we can have either or as time dependent or time independent.

The Truth table of OR is FTTF. The Truth table of IF THEN is TFUU

Now, having Logic OR being able to be time dependent or time independent is a good thing, a very good thing, because if I had a mathematical sentence of

9 -7 + 6 - 5 + 4 is it commutative or not?

Well, we can transform that to

9 + (-7) + 6 + (-5) + 4 and now we have your straightforward addition commutative where it matters not where you start, your final answer is always the same. And even if we had the last term subtract 40 instead of add 4.

So, where is this heading for? I think what is going to happen is that the subtract of Logic can be either time dependent or time independent, and that the subtraction in mathematics can also be manipulated.

Physics of course will weigh in and have the final answers to say.

AP

Post by Archimedes Plutonium
Subtraction is the OR connector and not obvious as to order dependent. And it maybe the case that the math view of subtraction shows more detail than the logic view of subtraction.
AP

[Archimedes Plutonium]

Alright this has caught me very much off guard th complexity of Subtraction, both in the Logic of OR and the subtraction of math, although not as complicated as the Logic OR.

In math we can instantly take a series of terms and if all of them are add apply the associative or commutative law for the answer is always the same no matter what order you add the terms, same goes for multiplication.

But now Old Math, the dumb fuddy duddy math says subtraction is noncommutative, which they said for division and they are correct about division not being commutative, but wrong about subtraction.

Why wrong about subtraction? Because say you had a list of these terms to do

9 - 5 + 7 - 40 + 3

And noone bothers to put paranthesis to alert what order thinking subtraction is noncommutative. But they are wrong for that series is the same as

9 + (-5) +7 + (-40) +3 and fully commutative for the answer that everyone gets regardless of order you do it, always comes out the same -26

And this is what the OR logic connector tells us also in its truth table FTTF, for it has enough wiggle room to say subtract 40 is rendered into add a negative 40, and so subtraction is commutative, once you transform awkward terms.

You see, when OR is false is when both variables are the same, TT or FF, no choice and no subtraction, and that translates for mathematics into the ability of - 40 becomes + (-40)

Upshot:: the logic subtraction OR is far far more general than is the Subtraction of Math. But the subtraction of math is more clear to understand.

AP

[Archimedes Plutonium]

Summary so far the two properties of Old Math Algebra of commutative and associative for add and multiply were redundant. You can throw out one of them, for either one implies the other. But, instead of keeping one of them, throw out both and state this property as such::

(1) Given any number of terms to add, or any number of terms to multiply, the answer is always the same no matter what order you do the terms.

(2) If you have a series of terms mixed with add and subtract, then first start by transforming all subtractions into add a negative number and then proceed as in (1)

(3) If you have division, then it requires an order indicator.

(4) Now algebra has distributive, and that remains valid and no changes.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
(1) Given any number of terms to add, or any number of terms to multiply, the answer is always the same no matter what order you do the terms.
(2) If you have a series of terms mixed with add and subtract, then first start by transforming all subtractions into add a negative number and then proceed as in (1)
(3) If you have division, then it requires an order indicator.
(4) Now algebra has distributive, and that remains valid and no changes.

Alright, let us examine up close the two axioms of Arithmetic, that ones about equality, that if you add something to one side of the equation, you must add an equal amount to the other side to retain equality. That Axiom

Then, the axiom that says, if you multiply by a factor, in an equation, you retain equality if you multiply every term of the equation by that very same factor. That axiom

For it is the second axiom of equality that is in essence the same as the Distributive Law. We can sort of picture that the distributive law is just a rehashing of the equality axiom of arithmetic when you multiply.

But, is those three axioms, add arithmetic, multiply arithmetic, and distributive, are those three axioms able to be found in LOGIC connectors.

Now, that is a challenge, a steep challenge, because, one thing that math has that logic does not have is geometry.

It appears that LOGIC has no two houses, arithmetic and geometry. It appears that Logic is just arithmetic, just language.

Because, when it comes down to it, the distributive law is just Pythagorean theorem with multiplication equality.

Have to think on whether Logic can have geometry-- maybe the exist and universal quantifiers are geometry?? Don't know as yet.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium

Post by Archimedes Plutonium
(1) Given any number of terms to add, or any number of terms to multiply, the answer is always the same no matter what order you do the terms.
(2) If you have a series of terms mixed with add and subtract, then first start by transforming all subtractions into add a negative number and then proceed as in (1)
(3) If you have division, then it requires an order indicator.
(4) Now algebra has distributive, and that remains valid and no changes.

I spent the whole day thinking about whether Logic should have a geometry side along with its analytic algebra of language. It could be that geometry is embedded inside language, where There Exists is a geometrical object and For Every is a geometrical object.

Post by Archimedes Plutonium
Alright, let us examine up close the two axioms of Arithmetic, that ones about equality, that if you add something to one side of the equation, you must add an equal amount to the other side to retain equality. That Axiom
Then, the axiom that says, if you multiply by a factor, in an equation, you retain equality if you multiply every term of the equation by that very same factor. That axiom

Easy to see the four operators of Arithmetic are the images of the 4 Logic Connectors. Even easy to see IF-Then is like the derivative of Calculus

But it is not easy to see how Pythagorean theorem can come out of Logic, something that is both geometrical and arithmetical.

Post by Archimedes Plutonium
For it is the second axiom of equality that is in essence the same as the Distributive Law. We can sort of picture that the distributive law is just a rehashing of the equality axiom of arithmetic when you multiply.

And it is hard to coax any more out of the Equal-Not Connector as multiplication. About the most we can see from Equal-Not is it is TTTT, the only connector of all truths, and that would be like the field completion of numbers under multiplication, or like the Universal Space of Probability theory, a all encompassing Space is the TTTT. And the connector of least "area" is the single T in IF THEN of TFUU.

But can we coax out of Equal-Not that of Distributive Law, or can we coax out of Equal-Not the Equation law that you must multiply every addition term by the factor to maintain equality?

Now the Addition law in Equations where to maintain equality, you must add equal amounts to both sides of the equation, is probably derivable by the AND Connector TTTF.

Post by Archimedes Plutonium
But, is those three axioms, add arithmetic, multiply arithmetic, and distributive, are those three axioms able to be found in LOGIC connectors.
Now, that is a challenge, a steep challenge, because, one thing that math has that logic does not have is geometry.
It appears that LOGIC has no two houses, arithmetic and geometry. It appears that Logic is just arithmetic, just language.

Don't know, how much does the There Exists and For Every, pack into Logic, can those two concepts pack all of geometry?

Post by Archimedes Plutonium
Because, when it comes down to it, the distributive law is just Pythagorean theorem with multiplication equality.
Have to think on whether Logic can have geometry-- maybe the exist and universal quantifiers are geometry?? Don't know as yet.

So have to do more thinking. Stuff like this comes when you don't force it, it comes when you are not expecting it.

AP

[Archimedes Plutonium]

It just dawned on me as i was heading for bed. Logic does have a house of geometry alongside its algebra of Language.
It is set theory which was mistakenly put into math, thinking it belonged in math.

Things like union intersection Cartesian product, Power set, etc.

Tomorrow i go and look for a pythagorean type theorem out of set theory. Now what is a perpendicular in set theory, and an angle?

AP

[Archimedes Plutonium]

Sorry, i could not sleep, too restless. By the way if you see a post by me using "i" instead of I, and often with spelling errors is because the post is from an iphone. I like iphone posts stretched back in my recliner, feet up on the heater.

Anyway, years ago i threw out p-adics, but today see a vital role for them.

You see, i am certain that Logic needs a house of geometry to be dual to its house of algebra of language. The 4 connectors with 2 quantifiers is not enough to compose logic. It needs an additional house that mirror reflects geometry. Logic is more vast than is mathematics.

What the geometry of logic is-- is set theory. Set theory was mistakenly placed into mathematics but math can get by totally without ever even mentioning set concepts. All numbers come from the root seeds of Peano axioms which does not need set theory. Algebra and geometry have equations and functions and series and sequences which define whether or not a number is a member.

So the entire bundle that was set theory is the geometry of Logic. Being the geometry, set theory must yield three items quickly-- pythagorean theorem, perpendicular and angles.

Thinking about that in bed made me realize that the way to achieve that is if set theory had a Spin or rotation in numbers. With a spin we can get a square and thus angles an thus a perpendicular.

So, do numbers have spin? Yes they do, for, if you take the 10 Grid where 10 is thought of as infinity border but you keep counting integers up to 100. Well, 100 is 0 and 99 is -1 and 98 is -2 and 97 is -3. So the infinity numbers after 10 loop back around to 0. Now, this is not some fantasy make up, but an actual trait of numbers themselves, that when you reach the border, and go beyond the numbers themselves spin around to 0.

That is what i need to have set theory be geometry, for a set is a collection of numbers and if large enough becomes a right-triangle and larger still a square. And the rotation of numbers yields angles and especially the right angle perpendicular.

So in life, often when you throw things out, you later see you really need them.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright a picture of the turning of numbers 0 to 100 into a spiral
96, 95, 94, 93, 92, 91 90, 89, 88, 87,
97, 61, 60, 59, 58, 57, 56, 55, 54, 86,
98, 62, 34, 33, 32, 31, 30, 29, 53, 85,
99, 63, 35, 15, 14,13,12, 28, 52, 84,
100, 64,36,16, 4, 3, 11, 27, 51, 83,
65, 37,17, 5, 1, 2, 10, 26, 50, 82,
66, 38, 18, 6, 7, 8, 9, 25, 49, 81,
67, 39, 19, 20,21,22,23,24, 48, 80,
68, 40, 41, 42, 43, 44, 45, 46, 47, 79,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78,

Alright, the above is a square spiral of a Graph Paper using vertices as counting numbers.

Now, 10 ends up being in the 1,2, 10 row and 100 ends up being in the 100, 64, 36, 16, 4, 3 row

The question is, what row does 1000, and 10000 end up being in? Anyone have an algorithm

AP

[Chris M. Thomasson]

Post by Archimedes Plutonium
Let me try this one, see if the format comes out better,,,,
85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96

Nice spiral. :^)

[Archimedes Plutonium]

Post by Chris M. Thomasson

Post by Archimedes Plutonium
Let me try this one, see if the format comes out better,,,,
85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96

Nice spiral. :^)

Glad you like it, wait till you see my dancing.

[Chris M. Thomasson]

Post by Archimedes Plutonium

Post by Chris M. Thomasson

Post by Archimedes Plutonium
Let me try this one, see if the format comes out better,,,,
85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ \ . / / / /
92 62 38 20 8---9--10 25 46 73
\ \ \ \ . / / /
93 63 39 21--22--23--24 45 72
\ \ \ . / /
94 64 40--41--42--43--44 71
\ \ . /
95 65--66--67--68--69--70
\ .
96

Nice spiral. :^)

Glad you like it, wait till you see my dancing.

:^D