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Archimedes Plutonium

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Nov 26, 2023, 7:17:30 PM11/26/23
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Kibo on drug spam flooding sci.math,sci.physics

Rensselaer Polytechnic Institute Physics dept Dr.Martin Schmidt (ee), Dr.Ivar Giaever
Dr.Vincent Meunier, Dr.Ethan Brown,Dr.Glenn Ciolek is Kibo Parry (Volney) running this spam machine out of Rensselaer, flooding sci.math and sci.physics with Indonesia slot spam and pretend Russian propaganda. Given that rpi dot edu address of Kibo.

Large Primes
by
James 'Kibo' Parry

May 26, 1989, 10:35:51 AM

What's the largest prime currenlty known? (All the information
I could dig up here was either fairly old or contradictory...)

james "kibo" parry | Some days you just can't get rid of a bomb.
kibo%pawl.r...@itsgw.rpi.edu (internet)
userfe0n@rpitsmts (bitnet) | Anything I say represents the opinion of
kibo%mts.r...@itsgw.rpi.edu | myself and not this computer.


> Dr.Bulent Yener,Dr.Donald Drew,Dr.William Siegmann, Rensselaer Polytech is this your spam??
>
> Kibo is this your and Rensselaer spam-- flood of slot and other
>
> Kibo Parry Moroney Volney wrote:
> On Wednesday, December 6, 2017 at 12:30:22 AM UTC-6, Michael Moroney wrote:
> > Silly boy, that's off by more than 12.6 MeV, or 12% of the mass of a muon.
> > Hardly "exactly" 9 muons.
> > Wednesday, December 6, 2017 at 9:52:21 AM UTC-6, Michael Moroney wrote:
> > Or, 938.2720813/105.6583745 = 8.88024338572. A proton is about the mass
> > of 8.88 muons, not 9. About 12% short.
>
>
> Why Volney?? Because they stop short of completing the Water Electrolysis Experiment by only looking at volume, when they are meant to weigh the mass of hydrogen versus oxygen?? Such shoddy minds in experimental physics and chemistry.
>
> Rensselaer Polytechnic Institute Physics dept Dr.Martin Schmidt (ee), Dr.Ivar Giaever
> Vincent Meunier, Ethan Brown, Glenn Ciolek, Julian S. Georg, Joel T. Giedt, Yong Sung Kim, Gyorgy Korniss, Toh-Ming Lu, Charles Martin, Joseph Darryl Michael, Heidi Jo Newberg, Moussa N'Gom, Peter Persans, John Schroeder, Michael Shur, Shawn-Yu Lin, Humberto Terrones, Gwo Ching Wang, Morris A Washington, Esther A. Wertz, Christian M. Wetzel, Ingrid Wilke, Shengbai Zhang
>
> Rensselaer math department
> Donald Schwendeman, Jeffrey Banks, Kristin Bennett, Mohamed Boudjelkha, Joseph Ecker, William Henshaw, Isom Herron, Mark H Holmes, David Isaacson, Elizabeth Kam, Ashwani Kapila, Maya Kiehl, Gregor Kovacic, Peter Kramer, Gina Kucinski, Rongjie Lai, Fengyan Li, Chjan Lim, Yuri V Lvov, Harry McLaughlin, John E. Mitchell, Bruce Piper, David A Schmidt, Daniel Stevenson, Yangyang Xu, Bulent Yener, Donald Drew, William Siegmann
>
>
>
>
> Kibo loves TEACHING TRUE MATHEMATICS for Xmas stocking stuffers.
>
> TEACHING TRUE MATHEMATICS, AP seeks the super easiest calculus possible on Earth-- polynomials as the only valid functions-- thus, and therefore, making derivative and integral as easy as Power Rule- 14 year olds master calculus. Because the Power Rule is merely add or subtract 1 from exponent so we can teach calculus in High School.
>
> Only Math textbooks with the true numbers of mathematics-- Decimal Grid Numbers, not the insane silly Reals & Complex with their crank crackpot imaginary b.s.
>
> I doubt the two math cranks Andrew Wiles and Terence Tao will ever understand mathematics for they continue to refuse to admit to even the most simple truths of mathematics-- slant cut of cone is Oval, not ellipse. A cylinder slant cut is ellipse, never cone.
>
> On Tuesday, September 26, 2023 at 4:21:58 PM UTC-5, Volney wrote:
> > The punishment will continue until morale improves.
> >
>
>
> Kibo Parry Volney, if you had studied under TEACHING TRUE MATHEMATICS, probably today would understand what a correct percentage was instead of your failureship. And likely Dr. Wiles if not blind in his eyes had studied under TEACHING TRUE MATHEMATICS, would know slant cut of cylinder is truly a ellipse but not of cone for that slant cut is a oval.
>
> Old Math is in a world of hurt for it does not even have the correct numbers of mathematics. Old Math was arrogant and ignorant starting year 1900 when Quantum Mechanics in physics took off and it means the world is discrete and not continuous. Yet the foolish bozos of Old Math stuck with their continuous Reals and even had the idiotic notion of going further out on the limb of madness with Cohen's continuum hypothesis, while Quantum Mechanics gave us a new age in physics with their discrete world. One would think the idiots of Old Math would finally look at physics and pay attention and learn something. No. They never did. And so today in October of 2023 we still have idiots of math teaching calculus with never a valid proof of Fundamental Theorem of Calculus, because Reals are not the true numbers of mathematics, the Decimal Grid Number System is the true numbers of math for they are discrete, and they make calculus, a billion, perhaps a trillion times easier to study , to learn to understand. In fact, we TEACHING TRUE MATHEMATICS, teaches calculus to 13 and 14 year olds. Because calculus is as easy as add or subtract 1 from the exponent.
>
> TEACHING TRUE MATHEMATICS the fake calculus of Thomas Hales, Andrew Wiles, Ken Ribet, Ruth Charney, Terence Tao, John Stillwell with their fake "limit analysis" for a true proof of Fundamental Theorem of Calculus (FTC) has to be a geometry proof for the integral is area under a graphed function. This is why only a polynomial can be a valid function of math, for the polynomial is a function of the straightline Y --> mx + b. All the other so called functions have no straightline-- they are curves of continuum and cannot give a proof of the fundamental theorem of calculus.
>
> The proof of FTC needs a empty space Discrete Geometry from one point to the next point so as to allow for the construction of a midpoint between point A to point B and thus to hinge up from A at the midpoint and to determine the next point B in the derivative. This is why Calculus is so enormously a tool for physics, as point A predicts point B.
>
> Discrete Geometry is required for the proof of FTC and that requires the true numbers of mathematics be Decimal Grid Numbers, for they cannot be the continuum idiocy of Reals and Complex.
>
> To make a half circle function in True Math, we have to go out to something like 10^6 Grid to make the points close enough together for the function visual to start looking like a half circle. But still there are holes in between one point and the next point to allow the existence of calculus.
>
> On a downward slope function, we have a different graphics than the usual upward slope function. For the upward slope requires the midpoint in the empty space to predict the next point of the thin rectangle that occupies that empty space (see the graphics below and in my books TEACHING TRUE MATHEMATICS). In a downward slope function graph we still have those thin rectangles occupy the empty space for integral but we do not need to construct the midpoint, we simply shave away a right triangle that reveals-- predicts point B starting from point A on the other side.
>
>
>
> TEACHING TRUE MATHEMATICS, AP seeks the super easiest calculus possible on Earth-- polynomials as the only valid functions-- thus, and therefore, making derivative and integral as easy as Power Rule- 14 year olds master calculus. Because the Power Rule is merely add or subtract 1 from exponent so we can teach calculus in High School.
>
> Old Math makes and keeps Calculus as classroom torture chambers with their 1,000s of different functions yet the polynomial is the only valid function of math, and makes it super super easy to learn calculus
>
> TEACHING TRUE MATHEMATICS, AP seeks the super easiest calculus possible on Earth-- polynomials as the only valid functions-- thus, and therefore, making derivative and integral as easy as Power Rule- 14 year olds master calculus.
>
> If you come to me with a pathetic non polynomial especially that ugly trig functions, I have you go home and convert your nonsense to a polynomial. The Lagrange interpolation converts stupid nonfunctions like trig, into valid functions of polynomials.
>
> TEACHING TRUE MATHEMATICS textbooks, makes calculus as easy as adding or subtracting 1 from exponent--only valid functions are polynomials contrast with mainstream--vomiting during exams, torture chamber and nervous breakdown by sado-masochist teachers. Old Math is thousands of different kook functions with thousands of different rules. AP Calculus is one function-- the polynomial for we care about truth in math, not on whether kooks of math become rich and famous off the suffering-backs of students put through a torture chamber that is present day calculus. If you come to math with a function that is not a polynomial, you have to convert it to a polynomial. Once converted, calculus is super super easy. But math professors seem to enjoy torturing students, not teaching them. Psychology teaches us that when a kook goes through a torture chamber and comes out of it as a math professor-- they want to be vindictive and sado masochists and love to torture others and put them through the same torture chamber that they went through. AP says-- stop this cycle of torture and teach TRUE CORRECT MATH.
>
> TEACHING TRUE MATHEMATICS textbooks, makes calculus as easy as adding or subtracting 1 from exponent--only valid functions are polynomials contrast with mainstream--vomiting during exams, torture chamber and nervous breakdown by sado-masochist teachers. Old Math is thousands of different kook functions with thousands of different rules. AP Calculus is one function-- the polynomial for we care about truth in math, not on whether kooks of math become rich and famous off the suffering of students put through a torture chamber that is present day calculus. If you come to math with a function that is not a polynomial, you have to convert it to a polynomial. Once converted, calculus is super super easy. But math professors seem to enjoy torturing students, not teaching them.
>
> Old Math calculus textbooks like Stewart are more than 1,000 pages long and they need that because they have a mindless thousand different functions and no valid proof of Fundamental Theorem of Calculus. AP's calculus is less than 300 pages, because we have a valid geometry proof of Fundamental Theorem of Calculus which demands the only valid function of math be a polynomial function. We can teach calculus in Junior High School for the calculus is reduced to adding or subtracting 1 from the exponent. The only hard part of calculus in New Math is to convert the boneheaded function into a polynomial that was brought to the table by the boneheaded math professor who thinks that a function does not need to be a polynomial.
>
> AP calculus transforms the calculus classroom. It is no longer vomiting during exams. No longer a torture chamber for our students of youth, and no longer a nightmare nor nervous breakdown for our youthful students, who, all they ever wanted was the truth of mathematics.
>
> Teaches that derivative predicts next point of function graph--silly Old Math has derivative as tangent to function graph unable to predict. The great power of Calculus is integral is area under function graph thus physics energy, and its prediction power of the derivative to predict the next future point of function graph thus making the derivative a "law of physics as predictor". Stupid Old Math makes the derivative a tangent line, while New Math makes the derivative the predictor of next point of function graph. No wonder no-one in Old Math could do a geometry, let alone a valid proof of Fundamental Theorem of Calculus, for no-one in Old Math even had the mind to realize Calculus predicts the future point in the derivative.
>
>
> TEACHING TRUE MATHEMATICS-- only math textbooks with a valid proof of Fundamental Theorem of Calculus--teaches that derivative predicts next point of function graph--silly Old Math has derivative as tangent to function graph unable to predict. This is why calculus is so important for physics, like a law of physics-- predicts the future given nearby point, predicts the next point. And of course the integral tells us the energy. Silly stupid Old Math understood the integral as area under the function graph curve, but were stupid silly as to the understanding of derivative-- predict the next point as seen in this illustration:
>
>
> From this rectangle of the integral with points A, midpoint then B
>
>
> ______
> | |
> | |
> | |
> ---------
>
>
> To this trapezoid with points A, m, B
>
> B
> /|
> / |
> m /----|
> / |
> | |
> |____|
>
>
> The trapezoid roof has to be a straight-line segment (the derivative)
> so that it can be hinged at m, and swiveled down to form rectangle for
> integral.
>
> Or going in reverse. From rectangle, the right triangle predicts the next successor point of function graph curve of B, from that of midpoint m and initial point of function graph A.
>
>
> My 134th published book
>
> Introduction to TEACHING TRUE MATHEMATICS: Volume 1 for ages 5 through 26, math textbook series, book 1 Kindle Edition
> by Archimedes Plutonium (Author)
>
> The 134th book of AP, and belatedly late, for I had already written the series of TEACHING TRUE MATHEMATICS in a 7 volume, 8 book set. This would be the first book in that 8 book set (one of the books is a companion book to 1st year college). But I suppose that I needed to write the full series before I could write the Introduction and know what I had to talk about and talk about in a logical progression order. Sounds paradoxical in a sense, that I needed to write the full series first and then go back and write the Introduction. But in another sense, hard to write an introduction on something you have not really fully done and completed. For example to know what is error filled Old Math and to list those errors in a logical order requires me to write the full 7 volumes in order to list in order the mistakes.
>
> Cover Picture: Mathematics begins with counting, with numbers, with quantity. But counting numbers needs geometry for something to count in the first place. So here in this picture of the generalized Hydrogen atom of chemistry and physics is a torus geometry of 8 rings of a proton torus and one ring where my fingers are, is a equator ring that is the muon and thrusting through the proton torus at the equator of the torus. So we count 9 rings in all. So math is created by atoms and math numbers exist because atoms have many geometry figures to count. And geometry exists because atoms have shapes and different figures.
>
> Product details
> • ASIN ‏ : ‎ B08K2XQB4M
> • Publication date ‏ : ‎ September 24, 2020
> • Language ‏ : ‎ English
> • File size ‏ : ‎ 576 KB
> • Text-to-Speech ‏ : ‎ Enabled
> • Screen Reader ‏ : ‎ Supported
> • Enhanced typesetting ‏ : ‎ Enabled
> • X-Ray ‏ : ‎ Not Enabled
> • Word Wise ‏ : ‎ Not Enabled
> • Print length ‏ : ‎ 23 pages
> • Lending ‏ : ‎ Enabled
> • Best Sellers Rank: #224,974 in Kindle Store (See Top 100 in Kindle Store)
> ◦ #3 in 45-Minute Science & Math Short Reads
> ◦ #23 in Calculus (Kindle Store)
> ◦ #182 in Calculus (Books)
>
>
>
> #5-2, My 45th published book.
>
> TEACHING TRUE MATHEMATICS: Volume 2 for ages 5 to 18, math textbook series, book 2
> by Archimedes Plutonium (Author) (Amazon Kindle edition)
>
> Last revision was 2NOV2020. And this is AP's 45th published book of science.
>
> Preface: Volume 2 takes the 5 year old student through to senior in High School for their math education.
>
> This is a textbook series in several volumes that carries every person through all his/her math education starting age 5 up to age 26. Volume 2 is for age 5 year old to that of senior in High School, that is needed to do both science and math. Every other math book is incidental to this series of Teaching True Mathematics.
>
> It is a journal-textbook because Amazon's Kindle offers me the ability to edit overnight, and to change the text, almost on a daily basis. A unique first in education textbooks-- almost a continual overnight editing. Adding new text, correcting text. Volume 2 takes the 5 year old student through to senior in High School for their math education. Volume 3 carries the Freshperson in College for their math calculus education.
>
> Cover Picture: The Numbers as Integers from 0 to 100, and 10 Grid when dividing by 10, and part of the 100 Grid when dividing by 100. Decimal Grid Numbers are the true numbers of mathematics. The Reals, the rationals & irrationals, the algebraic & transcendentals, the imaginary & Complex, and the negative-numbers are all fake numbers. For, to be a true number, you have to "be counted" by mathematical induction. The smallest Grid system is the Decimal 10 Grid.
>
>
>
> Product details
> ASIN ‏ : ‎ B07RG7BVZW
> Publication date ‏ : ‎ May 2, 2019
> Language ‏ : ‎ English
> File size ‏ : ‎ 2024 KB
> Text-to-Speech ‏ : ‎ Enabled
> Screen Reader ‏ : ‎ Supported
> Enhanced typesetting ‏ : ‎ Enabled
> X-Ray ‏ : ‎ Not Enabled
> Word Wise ‏ : ‎ Not Enabled
> Print length ‏ : ‎ 423 pages
> Lending ‏ : ‎ Enabled
> Best Sellers Rank: #235,426 in Kindle Store (See Top 100 in Kindle Store)
> #15 in General Geometry
> #223 in Geometry & Topology (Books)
>
>
> #5-3, 55th published book
>
> TEACHING TRUE MATHEMATICS: Volume 3 for age 18-19, 1st year College Calculus, math textbook series, book 3 Kindle Edition
> by Archimedes Plutonium (Author)
>
> Last revision was 25Jun2021. And this is AP's 55th published book of science.
>
> Teaching True Mathematics, by Archimedes Plutonium 2019
>
> Preface: This is volume 3, book 3 of Teaching True Mathematics, designed for College Freshperson students, 1st year college students of age 18-19. It is the continuation of volume 2 for ages 5 through 18 years old.
>
> The main major topic is the AP-EM equations of electricity and magnetism, the mathematics for the laws of electricity and magnetism; what used to be called the Maxwell Equations of Physics. The 1st Year College Math has to prepare all students with the math for all the sciences. So 1st year college Math is like a huge intersection station that has to prepare students with the math they need to do the hard sciences such as physics, chemistry, biology, astronomy, geology, etc. What this means is, 1st year college is calculus that allows the student to work with electricity and magnetism. All the math that is needed to enable students to do electricity and magnetism. In Old Math before this textbook, those Old Math textbooks would end in 1/3 of the text about Arclength, vector space, div, curl, Line Integral, Green's, Stokes, Divergence theorem trying to reach and be able to teach Maxwell Equations. But sadly, barely any Old Math classroom reached that 1/3 ending of the textbook, and left all those college students without any math to tackle electricity and magnetism. And most of Old Math was just muddle headed wrong even if they covered the last 1/3 of the textbook. And that is totally unacceptable in science. This textbook fixes that huge hole and gap in Old Math education.
>
> And there is no way around it, that a course in 1st year College Calculus is going to do a lot of hands on experiment with electricity and magnetism, and is required of the students to buy a list of physics apparatus-- multimeter, galvanometer, coil, bar magnet, alligator clip wires, electromagnet, iron filing case, and possibly even a 12 volt transformer, all shown in the cover picture. The beginning of this textbook and the middle section all leads into the ending of this textbook-- we learn the AP-EM Equations and how to use those equations. And there is no escaping the fact that it has to be hands on physics experiments in the classroom of mathematics.
>
> But, do not be scared, for this is all easy easy easy. For if you passed and enjoyed Volume 2 TEACHING TRUE MATHEMATICS, then I promise you, you will not be stressed with Volume 3, for I go out of my way to make it clear and understandable.
>
> Warning: this is a Journal Textbook, meaning that I am constantly adding new material, constantly revising, constantly fixing mistakes or making things more clear. So if you read this book in August of 2019, chances are it is different when you read it in September 2019. Ebooks allow authors the freedom to improve their textbooks on a ongoing basis.
>
> The 1st year college math should be about the math that prepares any and all students for science, whether they branch out into physics, chemistry, biology, geology, astronomy, or math, they should have all the math in 1st year college that will carry them through those science studies. I make every attempt possible to make math easy to understand, easy to learn and hopefully fun.
>
> Product details
> • ASIN ‏ : ‎ B07WN9RVXD
> • Publication date ‏ : ‎ August 16, 2019
> • Language ‏ : ‎ English
> • File size ‏ : ‎ 1390 KB
> • Simultaneous device usage ‏ : ‎ Unlimited
> • Text-to-Speech ‏ : ‎ Enabled
> • Screen Reader ‏ : ‎ Supported
> • Enhanced typesetting ‏ : ‎ Enabled
> • X-Ray ‏ : ‎ Not Enabled
> • Word Wise ‏ : ‎ Not Enabled
> • Print length ‏ : ‎ 236 pages
> • Lending ‏ : ‎ Enabled
> • Best Sellers Rank: #1,377,070 in Kindle Store (See Top 100 in Kindle Store)
> ◦ #411 in Calculus (Kindle Store)
> ◦ #2,480 in Calculus (Books)
>
>
>
> #5-4, 56th published book
>
> COLLEGE CALCULUS GUIDE to help students recognize math professor spam from math truth & reality// math textbook series, book 4 Kindle Edition
>
> by Archimedes Plutonium (Author)
>
>
> #1 New Releasein 15-Minute Science & Math Short Reads
>
>
> This textbook is the companion guide book to AP's Teaching True Mathematics, 1st year College. It is realized that Old Math will take a long time in removing their fake math, so in the interim period, this Guide book is designed to speed up the process of removing fake Calculus out of the education system, the fewer students we punish with forcing them with fake Calculus, the better we are.
> Cover Picture: This book is part comedy, for when you cannot reason with math professors that they have many errors to fix, that 90% of their Calculus is in error, you end up resorting to comedy, making fun of them, to prod them to fix their errors. To prod them to "do right by the students of the world" not their entrenched propaganda.
> Length: 54 pages
>
>
> Product details
> File Size: 1035 KB
> Print Length: 64 pages
> Simultaneous Device Usage: Unlimited
> Publication Date: August 18, 2019
> Sold by: Amazon.com Services LLC
> Language: English
> ASIN: B07WNGLQ85
> Text-to-Speech: Enabled
> X-Ray: 
Not Enabled 

> Word Wise: Not Enabled
> Lending: Enabled
> Enhanced Typesetting: Enabled
> Amazon Best Sellers Rank: #253,425 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
> #38 in 90-Minute Science & Math Short Reads
> #318 in Calculus (Books)
> #48 in Calculus (Kindle Store)
> 

> #5-5, 72nd published book
>
> TEACHING TRUE MATHEMATICS: Volume 4 for age 19-20 Sophomore-year College, math textbook series, book 5 Kindle Edition
> by Archimedes Plutonium (Author)
>
> Preface: This is volume 4, book 5 of Teaching True Mathematics, designed for College Sophomore-year students, students of age 19-20. It is the continuation of volume 3 in the end-goal of learning how to do the mathematics of electricity and magnetism, because everything in physics is nothing but atoms and atoms are nothing but electricity and magnetism. To know math, you have to know physics. We learned the Calculus of 2nd dimension and applied it to the equations of physics for electricity and magnetism. But we did not learn the calculus of those equations for 3rd dimension. So, you can say that Sophomore year College math is devoted to 3D Calculus. This sophomore year college we fill in all the calculus, and we start over on all of Geometry, for geometry needs a modern day revision. And pardon me for this book is mostly reading, and the students doing less calculations. The classroom of this textbook has the teacher go through page by page to get the students comprehending and understanding of what is being taught. There are many hands on experiments also.
>
> Cover Picture shows some toruses, some round some square, torus of rings, thin strips of rings or squares and shows them laid flat. That is Calculus of 3rd dimension that lays a ring in a torus to be flat in 2nd dimension.
> Length: 105 pages
>
> Product details
> • ASIN ‏ : ‎ B0828M34VL
> • Publication date ‏ : ‎ December 2, 2019
> • Language ‏ : ‎ English
> • File size ‏ : ‎ 952 KB
> • Text-to-Speech ‏ : ‎ Enabled
> • Screen Reader ‏ : ‎ Supported
> • Enhanced typesetting ‏ : ‎ Enabled
> • X-Ray ‏ : ‎ Not Enabled
> • Word Wise ‏ : ‎ Not Enabled
> • Print length ‏ : ‎ 105 pages
> • Lending ‏ : ‎ Enabled
> • Best Sellers Rank: #242,037 in Kindle Store (See Top 100 in Kindle Store)
> ◦ #36 in Calculus (Kindle Store)
> ◦ #219 in Calculus (Books)
>
>
> #5-6, 75th published book
>
> TEACHING TRUE MATHEMATICS: Volume 5 for age 20-21 Junior-year of College, math textbook series, book 6 Kindle Edition
> by Archimedes Plutonium 2019
>
> This is volume 5, book 6 of Teaching True Mathematics, designed for College Junior-year students, students of age 20-21. In first year college Calculus we learned calculus of the 2nd dimension and applied it to the equations of physics for electricity and magnetism. And in sophomore year we learned calculus of 3rd dimension to complete our study of the mathematics needed to do the physics of electricity and magnetism. Now, junior year college, we move onto something different, for we focus mostly on logic now and especially the logic of what is called the "mathematical proof". Much of what the student has learned about mathematics so far has been given to her or him as stated knowledge, accept it as true because I say so. But now we are going to do math proofs. Oh, yes, we did prove a few items here and there, such as why the Decimal Grid Number system is so special, such as the Pythagorean Theorem, such as the Fundamental Theorem of Calculus with its right-triangle hinged up or down. But many ideas we did not prove, we just stated them and expected all students to believe them true. And you are now juniors in college and we are going to start to prove many of those ideas and teach you "what is a math proof". Personally, I myself feel that the math proof is overrated, over hyped. But the math proof is important for one reason-- it makes you better scientists of knowing what is true and what is a shaky idea. A math proof is the same as "thinking straight and thinking clearly". And all scientists need to think straight and think clearly. But before we get to the Mathematics Proof, we have to do Probability and Statistics. What you learned in Grade School, then High School, then College, called Sigma Error, now becomes Probability and Statistics. It is important because all sciences including mathematics needs and uses Probability and Statistics. So, our job for junior-year of college mathematics is all cut out and ahead for us, no time to waste, let us get going.
>
> Cover Picture: is a sample of the Array Proof, a proof the ellipse is not a conic but rather a cylinder cut wherein the oval is the slant cut of a cone, not the ellipse.
>
> Length: 175 pages
>
>
> Product details
> ASIN : B0836F1YF6
> Publication date : December 26, 2019
> Language : English
> File size : 741 KB
> Text-to-Speech : Enabled
> Screen Reader : Supported
> Enhanced typesetting : Enabled
> X-Ray : Not Enabled
> Word Wise : Not Enabled
> Print length : 175 pages
> Lending : Enabled
> Best Sellers Rank: #3,768,255 in Kindle Store (See Top 100 in Kindle Store)
> ◦ #3,591 in Probability & Statistics (Kindle Store)
> ◦ #19,091 in Probability & Statistics (Books)
>
>
>
> #5-7, 89th published book
>
> TEACHING TRUE MATHEMATICS: Volume 6 for age 21-22 Senior-year of College, math textbook series, book 7 Kindle Edition
> by Archimedes Plutonium 2020
>
> Last revision was 6Feb2021.
> Preface: This is the last year of College for mathematics and we have to mostly summarize all of mathematics as best we can. And set a new pattern to prepare students going on to math graduate school. A new pattern of work habits, because graduate school is more of research and explore on your own. So in this final year, I am going to eliminate tests, and have it mostly done as homework assignments.
>
> Cover Picture: Again and again, many times in math, the mind is not good enough alone to think straight and clear, and you need tools to hands-on see how it works. Here is a collection of tools for this senior year college classes. There is a pencil, clipboard, graph paper, compass, divider, protractor, slide-ruler. And for this year we spend a lot of time on the parallelepiped, showing my wood model, and showing my erector set model held together by wire loops in the corners. The plastic square is there only to hold up the erector set model.
>
> Length: 110 pages
>
> Product details
> ASIN ‏ : ‎ B084V11BGY
> Publication date ‏ : ‎ February 15, 2020
> Language ‏ : ‎ English
> File size ‏ : ‎ 826 KB
> Text-to-Speech ‏ : ‎ Enabled
> Screen Reader ‏ : ‎ Supported
> Enhanced typesetting ‏ : ‎ Enabled
> X-Ray ‏ : ‎ Not Enabled
> Word Wise ‏ : ‎ Enabled
> Print length ‏ : ‎ 110 pages
> Lending ‏ : ‎ Enabled
> Best Sellers Rank: #224,965 in Kindle Store (See Top 100 in Kindle Store)
> ◦ #345 in Mathematics (Kindle Store)
> ◦ #373 in Physics (Kindle Store)
> ◦ #2,256 in Physics (Books)
>
> #5-8, 90th published book
>
> TEACHING TRUE MATHEMATICS: Volume 7 for age 22-26 Graduate school, math textbook series, book 8 Kindle Edition
> by Archimedes Plutonium 2020
>
> Last revised 1NOV2020. This was AP's 90th published book of science.
>
> Preface: This is College Graduate School mathematics. Congratulations, you made it this far. To me, graduate school is mostly research, research mathematics and that means also physics. So it is going to be difficult to do math without physics. Of course, we focus on the mathematics of these research projects.
>
> My textbook for Graduate school is just a template and the professors teaching the graduate students are free of course to follow their own projects, but in terms of being physics and math combined. What I list below is a template for possible projects.
>
> So, in the below projects, I list 36 possible research projects that a graduate student my like to undertake, or partake. I list those 36 projects with a set of parentheses like this (1), (2), (3), etc. Not to be confused with the chapters listing as 1), 2), 3), etc. I list 36 projects but the professor can offer his/her own list, and I expect students with their professor, to pick a project and to monitor the student as to his/her progresses through the research. I have listed each project then cited some of my own research into these projects, below each project is an entry. Those entries are just a help or helper in getting started or acquainted with the project. The entry has a date time group and a newsgroup that I posted to such as sci.math or plutonium-atom-universe Google newsgroups. Again the entry is just a help or helper in getting started.
>
> Now instead of picking one or two projects for your Graduate years of study, some may select all 36 projects where you write a short paper on each project. Some may be bored with just one or two projects and opt for all 36.
>
> Cover Picture: A photo by my iphone of a page on Permutations of the Jacobs book Mathematics: A Human Endeavor, 1970. One of the best textbooks ever written in Old Math, not for its contents because there are many errors, but for its teaching style. It is extremely rare to find a math textbook written for the student to learn. Probably because math professors rarely learned how to teach in the first place; only learned how to unintentionally obfuscate. The page I photographed is important because it is the interface between geometry's perimeter or surface area versus geometry's area or volume, respectively. Or, an interface of pure numbers with that of geometry. But I have more to say on this below.
> Length: 296 pages
>
> Product details
> ASIN ‏ : ‎ B085DF8R7V
> Publication date ‏ : ‎ March 1, 2020
> Language ‏ : ‎ English
> File size ‏ : ‎ 828 KB
> Text-to-Speech ‏ : ‎ Enabled
> Screen Reader ‏ : ‎ Supported
> Enhanced typesetting ‏ : ‎ Enabled
> X-Ray ‏ : ‎ Not Enabled
> Word Wise ‏ : ‎ Not Enabled
> Print length ‏ : ‎ 296 pages
> Lending ‏ : ‎ Enabled
> Best Sellers Rank: #224,981 in Kindle Store (See Top 100 in Kindle Store)
> ◦ #13 in General Geometry
> ◦ #213 in Geometry & Topology (Books)
>
>
> #5-9, 221st published book
>
> An Education Ladder Guideline for teaching mathematics and a Test to see if you are cut out to be a mathematician//Teaching True Mathematics
> by Archimedes Plutonium (Author) (Amazon's Kindle)
>
> Preface: This book is written to improve math education in school and at home. Trouble is, you cannot improve math education if the professors of mathematics have much of their teachings in error. So I write this book mostly as a test for math professors because to shine a light on math professor failure is the best way to improve math teaching, and thereby improve school curriculums especially colleges and universities. But others, such as laypersons are welcomed to join in. And it is the laypersons and students that will make the greatest amount of use of this book because math professors are usually stubborn and idiotic and hard to change for the better. And so when students and laypersons keep asking questions of their math professors, their brainwashing and thus poor teaching, they eventually come around to the truth and then change their bad behavior and bad misunderstanding; to proper true mathematics.
>
> Cover Picture: Is my iphone photograph of a rubber washer inside a plastic cone. The washer is at a steep slant angle to the cone perpendicular. Notice the washer near the apex is fully touching the side of the cone, but the washer directed towards the base has not yet cut through the side of the cone, and you can see a rainbow or a crescent shape of area where the washer will intersect the side of the cone, (where my two finger are), making a total figure of a Oval, never the ellipse. I was taking this picture as one person, so I had the iphone camera in one hand and the cone in another hand, and had to use a rubber washer to stay in place. The same green plastic cone used in this picture appears in both of my published books of the proof slant cut of cone is oval, never the ellipse.
>
> My 3rd published book with the same green cone on cover.
> AP's Proof-Ellipse was never a Conic Section // Math proof series, book 1 Kindle Edition
> by Archimedes Plutonium (Author)
>
> My 68th published book with the same green cone on cover.
> Proofs Ellipse is never a Conic section, always a Cylinder section and a Well Defined Oval definition//Student teaches professor series, book 5 Kindle Edition
> by Archimedes Plutonium (Author)
>
> Product details
> • ASIN ‏ : ‎ B0BQDYMYKQ
> • Publication date ‏ : ‎ December 16, 2022
> • Language ‏ : ‎ English
> • File size ‏ : ‎ 551 KB
> • Text-to-Speech ‏ : ‎ Enabled
> • Screen Reader ‏ : ‎ Supported
> • Enhanced typesetting ‏ : ‎ Enabled
> • X-Ray ‏ : ‎ Not Enabled
> • Word Wise ‏ : ‎ Not Enabled
> • Sticky notes ‏ : ‎ On Kindle Scribe
> • Print length ‏ : ‎ 65 pages
>
>
>
> #5-10, 160th published book
>
> MATHOPEDIA-- List of 82 fakes and mistakes of Old Math// mathematics & logic
> by Archimedes Plutonium
>
> Preface:
> A Mathopedia is like a special type of encyclopedia on the subject of mathematics. It is about the assessment of the worth of mathematics and the subject material of mathematics. It is a overall examination and a evaluation of mathematics and its topics.
>
> The ordering of Mathopedia is not a alphabetic ordering, nor does it have a index. The ordering is purely that of importance at beginning and importance at end.
>
> The greatest use of Mathopedia is a guide to students of what not to waste your time on and what to focus most of your time. I know so many college classes in mathematics are just a total waste of time, waste of valuable time for the class is math fakery. I know because I have been there.
>
> Now I am going to cite various reference sources of AP books if anyone wants more details and can be seen in the Appendix at the end of the book.
>
> I suppose, going forward, mathematics should always have a mathopedia, where major parts of mathematics as a science are held under scrutiny and question as to correctness. In past history we have called these incidents as "doubters of the mainstream". Yet math, like physics, can have no permanent mainstream, since there is always question of correctness in physics, there then corresponds questions of correctness in mathematics (because math is a subset of physics). What I mean is that each future generation corrects some mistakes of past mathematics. If anyone is unsure of what I am saying here, both math and physics need constant correcting, of that which never belonged in science. This then converges with the logic-philosophy of Pragmatism (see AP's book of logic on Pragmatism).
>
> Product details
> • ASIN ‏ : ‎ B09MZTLRL5 and ASIN ‏ : ‎ B09ZWFLKHC
> • Publication date ‏ : ‎ December 2, 2021
> • Product details
> • ASIN ‏ : ‎ B09ZWFLKHC
> • Publication date ‏ : ‎ May 8, 2022
> • Language ‏ : ‎ English
> • File size ‏ : ‎ 1154 KB
> • Text-to-Speech ‏ : ‎ Enabled
> • Screen Reader ‏ : ‎ Supported
> • Enhanced typesetting ‏ : ‎ Enabled
> • X-Ray ‏ : ‎ Not Enabled
> • Word Wise ‏ : ‎ Not Enabled
> • Sticky notes ‏ : ‎ On Kindle Scribe
> • Print length ‏ : ‎ 71 pages
>
>
>
> y z
> | /
> | /
> |/______ x
>
> Read my recent posts in peace and quiet.
> https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe
> Archimedes Plutonium
> Archimedes Plutonium's profile photo
> Archimedes Plutonium
> 2:12 AM (15 hours ago)
> 
> 
> 
> to
> Alright I come to realize I have no graphic explanation for the proof of the Fundamental Theorem of Calculus for a downward slope function graph. I gave a proof for the upward slope function.
>
> We start with the integral rectangle in the Cell, a specific cell of the function graph. In 10 Decimal Grid there are exactly 100 cells for each number interval, say from 0 to 0.1, then the next cell is 0.1 to 0.2. The midpoint in each cell belongs to a number in the next higher Grid System, the 100 Grid. So the midpoint of cell 1.1 to 1.2 is 1.15 as midpoint.
>
> Now the integral in that cell of 1.1 to 1.2 is a rectangle and say our function is x^2 --> Y. So the function graph is (1.1, 1.21) and (1.2, 1.44). Now we are strictly in 10 Grid borrowing from 100 Grid.
>
> So say this is our Integral rectangle in cell 1.1 to 1.2.
>
> _____
> | |
> | |
> | |
> | |
> _____
> 1.1 1.2
>
> More later,...
>
> What I am getting at is that in a upward slope the right triangle whose tip is 1.44 hinged at the midpoint 1.15 predicts that future point in the derivative as the right triangle hypotenuse.
>
> But the geometry is different for a downward slope function such as 10 -x --> Y. In this case we have the rectangle integral, but instead of hinging up the right triangle to predict the next point of the function graph, we totally remove the right triangle from the graph and the missing right-triangle is the successor point.
>
> Teaches that derivative predicts next point of function graph--silly Old Math has derivative as tangent to function graph unable to predict. The great power of Calculus is integral is area under function graph thus physics energy, and its prediction power of the derivative to predict the next future point of function graph thus making the derivative a "law of physics as predictor". Stupid Old Math makes the derivative a tangent line, while New Math makes the derivative the predictor of next point of function graph. No wonder no-one in Old Math could do a geometry, let alone a valid proof of Fundamental Theorem of Calculus, for no-one in Old Math even had the mind to realize Calculus predicts the future point in the derivative.
> >
> >
> > TEACHING TRUE MATHEMATICS-- only math textbooks with a valid proof of Fundamental Theorem of Calculus--teaches that derivative predicts next point of function graph--silly Old Math has derivative as tangent to function graph unable to predict. This is why calculus is so important for physics, like a law of physics-- predicts the future given nearby point, predicts the next point. And of course the integral tells us the energy. Silly stupid Old Math understood the integral as area under the function graph curve, but were stupid silly as to the understanding of derivative-- predict the next point as seen in this illustration:
> >
> >
> > From this rectangle of the integral with points A, midpoint then B
> >
> >
> > ______
> > | |
> > | |
> > | |
> > ---------
> >
> >
> > To this trapezoid with points A, m, B
> >
> > B
> > /|
> > / |
> > m /----|
> > / |
> > | |
> > |____|
> >
> >
> > The trapezoid roof has to be a straight-line segment (the derivative)
> > so that it can be hinged at m, and swiveled down to form rectangle for
> > integral.
> >
> > Or going in reverse. From rectangle, the right triangle predicts the next successor point of function graph curve of B, from that of midpoint m and initial point of function graph A.
> >
>
> AP
> Archimedes Plutonium's profile photo
> Archimedes Plutonium
> 1:04 PM (4 hours ago)
> 
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> to
> In the case of a upward slope function, the derivative requires a midpoint in the integral rectangle for which the right triangle is hinged at the midpoint and raised to rest upon the 4 sided trapezoid that the rectangle becomes. Thus the vertex tip of right triangle predicts the next future point of the function graph by this vertex tip.
>
> However, a different situation arises as the function graph has a downward slope. There is no raising of a right triangle cut-out of the integral rectangle. And there is no need for a midpoint on top wall of the integral rectangle. For a downward slope Function Graph, we cut-away a right triangle and discard it. Here the vertex tip is below the level of the entering function graph and is predicted by the derivative.
>
> So there are two geometry accounting for the Fundamental Theorem of Calculus proof. There is the accounting of a function graph if the function has a upward slope and there is the accounting if the function graph is a downward slope. Both involve the Integral as a rectangle in a cell of whatever Grid System one is in. In 10 Grid there are 100 cells along the x-axis, in 100 Grid there are 100^2 cells. If the function is upward slope we need the midpoint of cell and the right triangle is hinged at that midpoint. If the function is downward slope, the right triangle is shaved off and discarded-- no midpoint needed and the resultant figure could end up being a rectangle becoming a triangle. In the upward slope function graph, the rectangle becomes a trapezoid, possibly even a triangle.
>
> AP
> Archimedes Plutonium's profile photo
> Archimedes Plutonium
> 3:32 PM (2 hours ago)
> 
> 
> 
> to
> So for an upward slope function, the Proof of Fundamental Theorem of Calculus would have the integral rectangle turned into this.
>
> > ______
> > | |
> > | |
> > | |
> > ---------
> >
> >
> > To this trapezoid with points A, m, B
> >
> > B
> > /|
> > / |
> > m /----|
> > / |
> > | |
> > |____|
> >
>
> While for a downward slope function, the Proof of Fundamental Theorem of Calculus would have the integral rectangle turned into this.
>
> ______
> |....... |
> |....... |
> |....... |
> ---------
>
>
> |\
> |...\
> |....... |
> ---------
>
> Where the right-triangle is now swiveled at midpoint but rather where a right triangle is cut-away from the Integral that is a rectangle and that right triangle is then discarded.
>
>
> Now two of the most interesting and fascinating downward slope functions in 10 Grid of 1st Quadrant Only would be the quarter circle and the tractrix.
>
> Many of us forget that functions are Sequence progressions, starting at 0 and moving through all 100 cells of the 10 Decimal Grid System.
>
> Here, I have in mind for the quarter circle a radius of 10 to be all inclusive of the 10 Grid.
>
> AP
>
> By insisting that the only valid function in the world is a polynomial function, we thus reduce Calculus to the ultra simple task of the Power Rule.
>
> So we have a function of x^3, the derivative by Power Rule is (3)x^2. The integral by Power Rule is (1/4)x^4, and to check to see if integral is correct, we take the derivative of (1/4)x^4 to see if it becomes x^3, and surely it does so.
>
> So what AP teaches math to the world, is that Calculus can be mastered by 13 and 14 year olds. Students just beginning High School.
>
> Impossible in Old Math because Old Math is filled with mistakes and errors and crazy idiotic and stupid math.
>
> In New Math, we clean house. We do not let creeps and kooks fill up math that causes students to have nightmares and nervous breakdowns and vomit before tests.
>
> In New Math, we think only of our young students, we do not think of kooks like Dr.Hales, Dr.Tao, Dr. Wiles trying to achieve fame and fortune at the expense of our young students-- who, all they wanted was to learn the truth of mathematics.
>
> If you run to a teacher of New Math with a function, and that function is not a polynomial, then the teacher is going to tell you "that is not a valid function, and you simply convert it to a polynomial".
>
> In AP math class in 9th grade USA, AP makes students of 13 and 14 year old master Calculus. Master calculus better, far better than 1st year college students in Old Math at any college or university across the globe.
>
> 14 year old students in AP math class master calculus and "have fun and joy" in math class.
>
> 19 or 20 year olds in colleges and universities go through nightmares, vomiting, and even nervous breakdowns in their learning calculus.
>
> I am not exaggerating here, but obvious observations of education of mathematics.
>
> No-one in math education cares about students in Old Math. No-one has ever Cleaned House of Old Math, but let the rotten fetid Old Math stench increase.
>
>
> Now I need to add more to the Power Rules of Calculus as we make Polynomials be the only valid functions of mathematics. If you come to math with a function not a polynomial, you are sent home to convert your silly contraption into a polynomial over a interval in 1st Quadrant Only, a interval of concern.
>
> But in all the years I did calculus, I seem to not have registered in my mind the geometrical significance of the Power Rules. What is the geometry of taking x^2 to the power rule of n(x^n-1) for derivative. Then what is the geometry significance of taking the integral power rule-- (1/(n+1)) (x^(n+1)).
>
> It seems to me that at one moment in time, that geometry stuck to my mind, but is now elusive, I cannot recall the geometry significance of either Power Rule when played out on x^n.
>
> Cavalieri 1598-1647
>
> So that if we start with a polynomial function such as x^2 -> Y, we instantly know from the power rules that the derivative is 2x and the integral is 1/3x^3.
>
> Derivative Power Rule of a polynomial x^n that the derivative is n(x^n-1).
>
> The Integral Power Rule is sort of the opposite of the derivative rule so for polynomial x^n that the integral is (1/(n+1)) (x^(n+1)).
>
> On Tuesday, September 5, 2023 at 3:00:37 AM UTC-5, Archimedes Plutonium wrote:
> > Now I need to add more to the Power Rules of Calculus as we make Polynomials be the only valid functions of mathematics. If you come to math with a function not a polynomial, you are sent home to convert your silly contraption into a polynomial over a interval in 1st Quadrant Only, a interval of concern.
> >
> > But in all the years I did calculus, I seem to not have registered in my mind the geometrical significance of the Power Rules. What is the geometry of taking x^2 to the power rule of n(x^n-1) for derivative. Then what is the geometry significance of taking the integral power rule-- (1/(n+1)) (x^(n+1)).
> >
> > It seems to me that at one moment in time, that geometry stuck to my mind, but is now elusive, I cannot recall the geometry significance of either Power Rule when played out on x^n.
> >
> > Cavalieri 1598-1647
> >
> > So that if we start with a polynomial function such as x^2 -> Y, we instantly know from the power rules that the derivative is 2x and the integral is 1/3x^3.
> >
> > Derivative Power Rule of a polynomial x^n that the derivative is n(x^n-1).
> >
> > The Integral Power Rule is sort of the opposite of the derivative rule so for polynomial x^n that the integral is (1/(n+1)) (x^(n+1)).
>
> Now I need to include the Cavalieri proof, a geometry proof that rectangles under a function graph such as Y--> x^2 yields the power rule formula (1/(n+1))(x^(n+1)) so for x^2 the integral is (1/3)x^3.
>
> I would think that showing Cavalieri's proof would be standard fare in all 1st year college calculus textbooks. To my surprise, not Stewart, not Apostol, not Fisher& Zieber, not Ellis & Gulick, not Strang, no-one is up to the task of showing how Cavalieri got that formula from summing rectangles.
>
> Morris Kline in volume 1 "Mathematical Thought" shows a picture.
>
> Stillwell in "Mathematics and its History" shows a picture.
>
> But it must be too difficult for college authors to replicate Cavalieri's proof of approximating rectangles for x^2.
>
> Now if I were back in the days of Cavalieri and tasked to find a formula, I would do rectangles and trial and error. First finding a formula for easy ones such as Y--> x, then Y-->x^2, then a third trial, Y--> 2x to see if the formula is good, sort of a math induction settling upon (1/(n+1))(x^(n+1)).
>
> But I am very disappointed that none of my college calculus books derives the formula (1/(n+1))(x^(n+1)) via approximation.
>
>
> There were no standards for math proof in the days of Cavalieri for his genius of deriving the Integral Power rule. Y--> x^n is integral (1/(n+1))(x^(n+1))
>
> So what I am going to do is prove (1/(n+1))(x^(n+1)) in New Math.
>
> I looked through the literature and there was no actual Old Math proof of (1/(n+1))(x^(n+1))
>
> This is worthy of a whole entire new book of itself.
>
> And the beauty is that it is a Mathematical Induction proof.
>
> And the beauty also is that functions are chains of straightline connections from one point to the next in Discrete Geometry.
>
> That means we no longer approximate the integral but actually derive the Integral from a Right Trapezoid whose area is 1/2(base_1 + base_2)(height).
>
> We see that in a function such as 3x becomes integral (1/2)(3)x^2 due to that right-trapezoid area.
>
> The right-trapezoid is such that its base_1 and base_2 are the Y points for cells of calculus in Decimal Grid Systems.
>
> Trouble in Old Math is when the "so called historian" reads a passage in old works, they become overgenerous in crediting a proof when none really existed -- Fermat, Cavalieri. And this is the reason that no-one in modern times who wrote a Calculus textbook features the Cavalieri Integral Power Rule, because there never was a proof, .... until now... a Mathematical Induction proof.
>
> AP, King of Science
>
> None of this is a proof of Cavalieri's integral power rule formula. Because Geometry is discrete and all curves in geometry are chains of straightline segments. The Internet boasts of some modern recent proofs of Cavalieri, but I suspect all those are bogus claims, being victims of computer graphics and no honest down to earth proof at all. I myself was a victim of computer graphics, for a computer can really spit out any image you ask it to spit out, such as hexagon tiling of sphere surface.
>
> --- quoting Wikipedia ---
> The modern proof is to use an antiderivative: the derivative of xn is shown to be nxn−1 – for non-negative integers. This is shown from the binomial formula and the definition of the derivative – and thus by the fundamental theorem of calculus the antiderivative is the integral. This method fails for
> ∫1/x dx
> which is undefined due to division by zero. The logarithm function, which is the actual antiderivative of 1/x, must be introduced and examined separately.
>
>
> The derivative
> (x^n)'=nx^{n-1} can be geometrized as the infinitesimal change in volume of the n-cube, which is the area of n faces, each of dimension n − 1.
> Integrating this picture – stacking the faces – geometrizes the fundamental theorem of calculus, yielding a decomposition of the n-cube into n pyramids, which is a geometric proof of Cavalieri's quadrature formula.
> For positive integers, this proof can be geometrized: if one considers the quantity xn as the volume of the n-cube (the hypercube in n dimensions), then the derivative is the change in the volume as the side length is changed – this is xn−1, which can be interpreted as the area of n faces, each of dimension n − 1 (fixing one vertex at the origin, these are the n faces not touching the vertex), corresponding to the cube increasing in size by growing in the direction of these faces – in the 3-dimensional case, adding 3 infinitesimally thin squares, one to each of these faces. Conversely, geometrizing the fundamental theorem of calculus, stacking up these infinitesimal (n − 1) cubes yields a (hyper)-pyramid, and n of these pyramids form the n-cube, which yields the formula. Further, there is an n-fold cyclic symmetry of the n-cube around the diagonal cycling these pyramids (for which a pyramid is a fundamental domain). In the case of the cube (3-cube), this is how the volume of a pyramid was originally rigorously established: the cube has 3-fold symmetry, with fundamental domain a pyramids, dividing the cube into 3 pyramids, corresponding to the fact that the volume of a pyramid is one third of the base times the height. This illustrates geometrically the equivalence between the quadrature of the parabola and the volume of a pyramid, which were computed classically by different means.
>
> Alternative proofs exist – for example, Fermat computed the area via an algebraic trick of dividing the domain into certain intervals of unequal length; alternatively, one can prove this by recognizing a symmetry of the graph y = xn under inhomogeneous dilation (by d in the x direction and dn in the y direction, algebraicizing the n dimensions of the y direction), or deriving the formula for all integer values by expand
> --- end quoting Wikipedia on Cavalieri's quadrature formula ---
>
>
> y z
> | /
> | /
> |/______ x
>
> Read my recent posts in peace and quiet.
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> Archimedes Plutonium
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