Quickest Calculus: Class Use 9798357865137

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Quickest Calculus: Class Use First Edition, v7

Ann Vogel Gerck, B.A. Founder Planalto Research

Ed Gerck, Ph.D. CEO & Founder

Published by Planalto Research Mountain View, CA, USA

Editor: Ann Vogel Gerck Copyright © 2022 by Ed Gerck All rights reserved, worldwide.

Reproduction or translation of any part of this work beyond that permitted by the 1976 United States Copyright Act, such as Section 107 or 108, without the written permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permission Department, Planalto Research at 211 Hope St #793, Mountain View, CA, ZIP 94041, USA, or email [email protected]

Gerck, Ann V. and Gerck, Ed. Quickest Calculus: A Self-Study Guide With New Applications.

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WARRANTY AND SERVICES

This book comes with a WARRANTY, suitable for self-study. If you get the right answer but feel you need more practice, simply follow the directions for the MAYBE answer. There are no prizes, or endorsements for doing the book in the shortest time, and it is our truthful duty to tell you that. This book will be updated, importantly taking into account your feedback and of the community. Our WARRANTY is that the electronic version is only paid once -- you have a right to update your electronic version, permanently, at no cost, albeit your online connection and equipment. To motivate class use, this electronic version will be provided at US$20.00 each, with a 50% discount for paperback and hardcover editions. The electronic content is updatable freely, plus any connection cost. You just need a proof-of-purchase, also acceptable if it is for a printed copy. Planalto Research will also offer books and services that you can pay at a cost basis by buying this book, and other companies are welcome to partner. The first service will be an Exam, a comprehensive test -- with hard, unique, randomly selected questions, composing a time-limited comprehensive test, earning a valuable certificate to that test, numbered, that you can "hang on a wall", post online, present to an institution for credit, or use to win company employment. The time limit for the test is 1 hour. We can also list publicly online your full name, and grade, if you want it. There is no limit to how many times you take the Exam. This can be done in training mode, without any attribution, or in competition mode, recorded as evidence of your progress. Each Exam will be available on a cost basis, less than you pay for a single shot of espresso! Email [email protected]

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PREFACE For advanced students, the mathematical theory has been freely published, and is available also without cost at https://www.mdpi.com/2227-7390/11/1/68 Calculus, as well-known, was invented by Hindu mathematicians (led by Mādhava of Sangamagrāma) 250 years before Newton and Leibniz, and they did not use the fictions of infinitesimals, irrational numbers, or imaginary numbers. This is not well understood even in India today, providing more recognition to the genius of Madhāva. Calculus exists on discrete numbers, with no infinitesimals. Even middle-school children can learn calculus this way, following this book. There are no monsters to fear. The Hindus were guided by astronomy, nature, and used the same that we are doing here, as the properties in integer numbers, and obtained the same formulas and new results that we reproduce. We are just revisiting their results in modern language, and adding quantum mechanics as opening new avenues and results, such as using periodicity. Numbers, qua values, must obey physical laws -- and no one finds infinitesimals or imaginary numbers in nature, as this work indicates. We stop this issue here, being explored elsewhere, and understand that new mathematical methods build upon the old, even when not using them, and will be successful without any criticism. This is an edition for class use, mainly with mathematical subjects.

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Following the well-known principles of semiotics, a number can be consistently considered as a 1:1 mapping between a symbol and a value. The symbol can be arbitrary, subjective, even with no sound, but the value must be objective. Many different persons looking at our Sun may refer to it by different names, but all agree on 1 value: there is 1 Sun, no matter where it sets or rises at each time of the year. Thus, the value must exist in nature, to be objective, to be unique for all observers, humans and non-humans, friends or foes. Our definitions of numbers are objective, when following values, not symbols. One means, then, objective values, not subjective symbols, when one talks generally about numbers. The definition of integer numbers (set Z) can be chosen to be the known definition by Kronecker, and includes 0. The definition of natural numbers (set N) follows from integers, and excludes 0. A rational number (set Q) is defined as a ratio of integers, excluding 0 in the denominator. This defines the sets N, Z, and Q, the only type of numbers (as defined in Martin-Löf Type Theory) needed to use in calculus, algebra, and arithmetic. Periodicity in numbers can provide prime number factoring (Peter Shor, 1994). This is very inspiring to solve an otherwise difficult problem in Number Theory. This connects physics in quantum mechanics (QM) with Number Theory in "pure" mathematics, using a "wormhole" to connect

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these different universes. They possess only one common reality, connected by the “wormhole”: the natural numbers (the set N). The laws that work for N, are common to all uses of N. Reality wins over any logically-assumed result, using TT. We now use the above to present in this book an absolutely exact formulation of calculus, using the sets N, Z, and Q. In mathematics, “measurement theory” involves types of infinite point sets and their extent in variously-defined spaces. However, in daily usage, when we speak of taking a “measurement” it is always intended to mean “of some physical, real-world quantity”. Nature appears what we find in “nature”, and “calculate” is what we can demonstrate theoretically, maybe as a “jump”, albeit firmly based on experimental science. Infinity is not a value and is not found in physics, but we do not need to run away from it. It can be consistently used in mathematics, such as in the principle of finite induction, in continued fractions, and in series. Vectors, multivectors, and complex numbers are not used in this book. This book uses numbers as rational scalar quantities only. This can be easily extended to multivectors though, following Grassmann and Clifford. Vectors are restricted to 2D, and not considered trustworthy in physics, hence not to be trusted in mathematics in the 21st century. This book shows that deductions in current mathematics are not exact, because Nature appears to us as not continuous -- one 6

would be pretending to measure mathematically, what does not exist in nature, physically. Physically, not only rational numbers are the only numbers measurable, but they are also the only numbers produced. No production is continuous. Nature appears given by digital numbers, everywhere we look, not by continuous numbers. Continuous numbers cannot be constructed, or are produced. This allows us to follow a short teaching route, already pioneered by Pestalozzi: observe, and learn. Nature becomes our teacher, also in mathematics, in trust, using priming (see Frame 2). This makes it possible to cover calculus in mathematics, with no physics, when mathematics can become the "common denominator'' of all sciences -- often called "the queen of sciences". We believe with this book that the education of mathematics should be in harmony with nature, and be useful to all sciences. Biology, for example, can use this book in order to better explain mitosis and meiosis, accepting a discontinuous change, albeit with zero physics needed and current usage in mathematics. A lesser claim also seems easier to present, and would expand to more applications of this book, as some readers have suggested. This makes it easier to discuss this book with middle-school students, psychologists, physicians, bankers, veterinarians, dentists, English teachers, anthropologists, and stay-at-home-moms – useful to people not yet in more advanced 7

mathematics! No one has to be math-averse, or show math-phobia. The multiple connections between mathematics and all sciences, though, work as “checks and balances” on what one may imagine. This is not Boolean logic. We call this the Holographic Principle (HP), and disarms Kurt Gödel's uncertainty. Moving to increase rigor, to absolute accuracy in measurements, this book shows that one can change calculus to rational numbers using the set Q, keeping the same formulas, and smooth graphs -- while making calculus easy and intuitive in the century we live in. And one finds many new applications that were being obscured by those seemingly “undetectable” and small measurement errors! You will understand not only calculus better, but Engineering, Physics , Biology, and Science, better. Ed Gerck, Ph.D Mountain View, California

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In Memoriam

Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī, Omar Khayyam (1048–1131) Hermann Günther Grassmann (1809-1877) William Kingdon Clifford (1845-1879) Tom Mike Apostol (1923-2016) A. Brandão d’Oliveira (1946-2019) Leopold Kronecker (1823-1891)

"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk."

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Calculus Considerations

John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." According to Tom M. Apostol [1.5], there seems to be no general agreement as to what should constitute a first course in calculus and analytical geometry. This book intends to resolve the issue. Some people think that the only way to really understand calculus is to start off with a thorough treatment of the mathematical real-number system [that, as revealed in this book, only exists in the large scale, using Universality, and using artificial continuity], and develop the subject step by step in a logical and [supposedly] rigorous fashion. Others, disappointed with the lacks of success of the first approach [1.7-8], argue that calculus is primarily a tool for engineers and physicists; they believe the course should stress applications of calculus by appealing to intuition and by extensive drill on problems which develop manipulative skills [sidestepping any questions on infinitesimals and Cauchy epsilon-deltas]. Instead, we follow a third route. Calculus is viewed in this book as an inventive and deductive science, as a branch of pure mathematics that is necessarily connected with Computers, TT, and QM, in a HP.

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Calculus has strong roots in physical problems and derives much of its power and beauty from the variety of its applications. It is possible to combine a strong theoretical development with sound training in technique; and this book represents an attempt to strike a sensible balance between the two, using Universality. Mathematical real-numbers, the usual basis of calculus, are seen in this book as macroscopic interpolations, axiomatically continuous, approximately valid albeit artificial. They work in relative accuracy -- in spite of the possibly limitless digits of a representation. They have several flaws. And there is no need to “justify” mathematical real-numbers or calculus with ghostly fictions -- an idea of infinitesimals, or that microscopic continuity would exist. We do not see them in nature. We just have to observe. This book provides physical results that one can verify by experiments with nature -- by just observing. The macroscopic interpolation of mathematical real-numbers becomes now a secondary result that can even be approximately valid while axiomatically independent, albeit keeping the primary result rigorous, based on the set Q. The irrational numbers are considered in this book not as somewhat “mysterious”, or as a “pariah” among numbers, but as approximated as well as desired using the set Q, following the Hurwitz Theorem [2.1]). There are 0 (zero) irrational numbers in Q, which we use as a rigorous basis of calculus, achieving smooth graphs in Q.

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In conventional mathematics, students are taught to subdivide an interval and to keep doing it, until they reach an interval as close as they desire to 0, and to think that any remaining error would then have to become negligible. This would mean that an infinitesimal exists, as close to zero as one wants, albeit not 0. This is absurd, while it does not somehow “vilify” infinitesimals. The process simply would soon pass the size of molecules, atoms, and even unseen particles, such as quarks. That is not physically possible, but it was imagined possible in the 17/18th century. What is going on? There was no malice, “fake news”, or “conspiracy theory”. Life has shown us different realities since the 17/18th century, e.g., with resonance [7.1] giving conditions for prime numbers existence, and quantum potentials [7.13] giving conditions for prime number separation, and more [7.2-10]. This follows a familiar process, where a solution is easier to find when an equation is seen through a connection as shown below, taken from [5.2].

Fig.(1.1) Method for easy solution of difficult problems.

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This is an exact, “click-mathematics”, with pieces that fit together like Lego. But seeking a different solution only for prominence or solipsistic purposes, has been an unfortunate metaphor in academic works [1.4 Foreword, pp.1-38] , and that afflicts students [1.7] as well as teachers [1.8]. No one is looking for “octopuses on Mars”. Here, the set Q offers a trustworthy basis for calculus, without inducing any metric: all members of the set Q are objective, and exact -- with no error, with absolute accuracy. Friends and foes can agree on the set Q, no one can influence, and anyone can play. We can face new challenges, such as QC. By adding a metric function, one has access to the artificial, approximate, mathematical real-numbers, but loses accuracy and speed of calculation, and “hides” applications. Mathematical real-numbers were considered having physical significance in the 17/18th century. This was within the limits of physical measurement then, in an apparently continuous scale. In the 21st century, however, better measurements show a quite different picture: Nature appears made by “grainy” numbers, everywhere one looks, and works like Lego, with exact fitting. In summary: ● This book uses the connection found by Peter Shor in 1994 [7.1] in Number Theory, using the common set N to create a “wormhole” with QM -- where the laws governing the set N are the same in physics and in mathematics; ● We use the set N as giving the recurring basis for the set Q, infinitely extensible, which we use as the basis of a new approach to calculus in this book; 13

● This book defines “measure”, “calculate”, and “nature”, as in an experimental science; ● This book revisits calculus, eliminating infinitesimals, microscopic continuity, Cauchy epsilon-deltas, and Cauchy accumulation points; ● All familiar pairs of differentials/integrals are reaffirmed, now rigorously, and all graphs are infinitely smooth, when seen under any magnification; ● One can differentiate discontinuous functions; ● The fundamental theorem of calculus is used to simplify Integral calculus; ● The mathematical real-numbers are kept as a macroscopic, axiomatically continuous, an HP achievement of many researchers since the 17/18th century, albeit imprecise; and ● Calculus becomes like Lego, with differential forms. Anything constructed can be taken apart again, and the pieces reused to make new things. Creativity is empowered. This book promises to be a paradigm shift that can help you save time, with many shortcuts. You will understand calculus better, as an inventive and rigorous science. With this book, discontinuous functions can now be differentiated, and GR does not have to be continuous and can finally follow QM. Mathematics no longer needs apologies or fear in the 21st century [1.8-9].

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CONTENTS

WARRANTY AND SERVICES……………………………………………………………….3 Preface…………………………………………………………………………………………………..4 In Memoriam…………………………………………………………………………………………9 Calculus Considerations…………………………………………………………………..10 Contents………………………………………………………………………………………………15 List of Abbreviations………………………………………………………………………….16 Chapter 1: Preliminaries……………………………………………………………………17 Chapter 2: Number Systems…………………………………………………………….42 Chapter 3: Set Theory, Logic, Functions, and Calculator………………58 Chapter 4: Universality……………………………………………………………………….86 Chapter 5: Differential Calculus……………………………………………………..108 Chapter 6: Integral Calculus……………………………………………………………133 ACKNOWLEDGEMENTS……………………………………………….…………………149

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List of Abbreviations AAIS — Arithmetic, Algebra, Infinite Series AES — Advanced Encryption Standard CAD — Computer Aided Design CS — Computer Science DFT — Discrete Fourier Transform DVD — Digital Video Disc FFT — Fast Fourier Transform FIF — Finite Integer Field FUD — Fear, Uncertainty, and Doubt GF — Galois Field GR — General Relativity HP — Holographic Principle LEM — Law of the Excluded Middle QC — Quantum Computing QM — Quantum Mechanics QP — Quantum Properties Sets of Numbers: N — Natural numbers Z — Integer numbers over N Q — Rational numbers over Z G — Gaussian numbers over Q (not used here) (unnamed) — Irrational numbers (not used here) R — Mathematical real-numbers (not used here) C — Mathematical complex numbers over R (not used here) … SR — Special Relativity TT — Martin-Löf Type Theory

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CHAPTER 1: Preliminaries Experimental science in the 21st century allows this book to stand on the shoulders of giants, some mentioned above, evolving from unconventional mathematical methods that were first formed even in the 17/18th centuries, and before. 1__________________________________________________ In this chapter 1, a few preliminaries are presented. The plan of the book is laid out, and some elementary concepts are reviewed as needed. By the end of the chapter 3 you will be familiar with: ● Defining mathematical functions, in both discrete and continuous models. ● Different logic models, including general logic, binary or Boolean logic, and 3 state logic. ● Graphs of functions, using rectangular Cartesian Coordinates, and their method of construction. ● The properties of the most widely used functions: linear and quadratic functions, trigonometric functions, inverse function, exponentials, and logarithms. ● Use of inexpensive 21st century calculators, in your pocket – a cell phone or a tablet. You can also use your computer. And, use them to learn by mimicry. ● Use all your 21st century knowledge, already learned.

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● To observe using mimicry as a fast and easy method of learning, through priming. An inexpensive 21st century hand-held calculator (see Chapter 3) can do all of calculus in this book, plus algebra, exponentials, trigonometric functions, logarithms, and more, and save you work, only using N – the set of natural numbers. The hardware uses only binary numbers, addition and encoding. Mathematically we can use the set Q, as finely as desired. This book does not miss anything to achieve a better future; we know that between any two mathematical real-numbers -- including irrational numbers, there is always a rational number, in fact, an infinitude of them. So anyone can do calculus, as you can observe, and the results in your calculator are physical evidence of that. No mathematical real-numbers are needed, or used, although coprocessors can emulate apparently continuous mathematical real-numbers. Surely, you can master the text without any calculator, 17th century style. But you would not be using the resources widely available in this century. The calculator can become more than a trusted teacher, helping this self-study book, it is also your laboratory. You will learn by observation of the calculator, not just by group/rote work, 17th century style.

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2___________________________________________________ PRIMING ____________________________________________________ Observation includes copying, and mimicry, and all are considered valid forms of learning. It is used in priming -- a cognitive phenomenon whereby exposure to one stimulus influences a response to a subsequent stimulus, even without conscious guidance or intention. Like Pestalozzi, an early educator, our method encourages sensory learning through use of “hands-on” activities in this book, and nature studies. Pestalozzi had envisioned schools should feel more homelike rather than institutions, and we favor self-study. He believed in schools where teachers actively engaged with students in learning by sensory experiences, where we favor the suggested calculator. Pestalozzi's method shows the encouragement of students needing an emotionally secure environment as the setting for a successful learning experience, which can be found more easily in self-study. To better use the contents of this book in a priming process, using modern cognitive psychology, we suggest you to: 1. 2. 3. 4. 5.

Put this study on your calendar. Read the preface, or another text, to inspire you. Write your questions and goals. Visualize reaching your goals. Listen to your subconscious mind, while you follow this book. 6. Annotate the answers you find to #3, and read out loud. 7. Repeat 2-6, until done.

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This method has been largely forgotten in academic teaching today, favoring speculative interests and bias on demanding rote work … for students. But we follow Pestalozzi, an early pioneer in teaching. Learning can become, again, fun. A ludic activity. That is how parents teach children. Mathematics can become intuitive in this style. However, leaving the calculator aside, and working by hand the numerical problems in this book, mathematically using arithmetic and algebra, will also help to increase your insight, as a test you can apply on yourself. 3___________________________________________________ UNIVERSALITY ____________________________________________________ One can see Universality in how waves appear. Macroscopically we feel them on a beach, very clearly. Their impact is measured by their amplitude. But microscopically, we see only molecules, atoms, and ions. Their impact is measured by their frequency. An unseen digital reality exists microscopically. A wave is a collective effect, one needs a certain amount of water, and it is a matter of scale. Both visions are right, “p” and “not p”, it is just not Boolean. But what does that have to do with calculus? Calculus has to do with numbers. Similarly, the same that happens with waves, happens with numbers. The set of numbers we use is the set Q, and their appearance is a matter of scale.

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For example, natural numbers are well-known to be used in ISO standards to measure the speed of light. The natural numbers are close enough from each other to measure it in meter/second. But they are too far apart to measure the speed of a car in the same units. But if we change the units to nanometer/second, they are OK. Instead of changing the units, people thought we can measure things in rational numbers. But some things are still not measurable. Like the diagonal of a unit square, as the Greeks famously discovered, with irrational numbers. People thought one can insert that into the set of mathematical real-numbers, and then one thinks that one can measure anything, from distance to stars to separation of atoms. Did we achieve continuity? No, each number is still a distinct number, not a blob. It is a matter of scale. An unseen digital reality appears microscopically. Can we work with absolute accuracy? A point of 0-dimensions, called a mathematical point, with no error, has been recommended to use since 300 BC. Why would we need it, is a story paved by new applications, some reported in [7.1-13] below. It is similar to using a short or a long ruler in drafting. You cannot draw a straight line with accuracy over ten meters when using a short ruler, the size of your palm. You will then need 3 or more lines to define a point, with some precision. Absolute accuracy eludes you. You cannot calculate an image by ray tracing.

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But, use a 21st century CAD. You can now use absolute accuracy, doing away with the need for trial-and-error. We need absolute accuracy in calculus. One can no longer work with faulty calculus [1.4-9], that requires continuity before measuring .... a lack of the same … continuity. In spite of its past centuries of FUD [1.4 Foreword, pp.1-38] and many other pages in [1.4], and [1.5], and difficult accounts in [1.5], [1.7-9], the reader does not need them. It is just not possible to have a consistent theory of microscopic continuity, infinitesimals, mathematical hyper-real-numbers, or ultra-filters, when microscopic reality is the opposite. Insisting on them, leaves calculus ghostly and subjective, and hides new results that now can be yours. This book opens a cornucopia of new, consistent, results, with a sample at [7.1-13], and promises more for QC and quantum cryptography. Calculus in the 21st century does not have to be a particularly difficult subject, and we can use our 21st century knowledge to great advantage. With diligence, you can learn its basic ideas fairly quickly, and you already should know most of them, in daily life observing in the 21st century. This book will get you started in calculus using any set of numbers, in self-study. This book can interest you in mathematics more, save your career in college, and avoid many nights of rote work. After working through it, you ought to be able to handle many problems and you should be prepared to learn more elaborate techniques that can surprise others, whenever you need them - this is your “book of magic”.

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4___________________________________________________ CALCULATOR ____________________________________________________ Remember that the important word now is observing, And we hope you will find that much of the work is now fun to do, and builds up your thinking in other subjects – especially physics, biology, and computer science. Even an anthropologist can work in somewhat equal terms, with a physicist; and an English teacher can view verbs as functions in mathematics. We can all be math-friendly, learn programming, learn more and more mathematics in the 21st century, and see no cause for numerophobia any more. Most of your observations will be from your private teacher, the calculator in your cell phone or tablet. Most of your work will be answering questions and observing the calculator solve problems, where you learn naturally through mimicry. The main observation is that, no one needs to read a manual any more. We do not need a “better help” file, we need simpler instructions, as a customer said. The particular route you will follow will depend on your answers. Your reward for doing a problem correctly is your own immediate progress, and to go straight on to new material. This applies a principle well-known in teaching: to provide immediate gratification to the student. On the other hand, if you make an error, you can know it promptly -- work with the calculator on your own, and/or the solution will be explained in this book, and you usually will get additional problems to see whether you have caught on, or you can search. 23

In any case, you will always be able to check your answers immediately with the book and/or the calculator after doing a problem. In the end, you will become a proficient, exact calculator yourself! 5___________________________________________________ GENERAL LOGIC ___________________________________________________ Many of the problems have multiple choice answers, showing practically that we cannot deal only with Boolean choices, binary, where the only answers possible are 1 (yes) or 0 (no), obeying the absolute rule of the LEM. Where is the maybe? Humans do not obey the LEM, though, as parents of teenagers soon learn. Given two propositions, 'p' or 'not p', general logic can accept both at the same time, e.g., with the connective 'and', but binary logic does not allow it. We use up to three logic choices. These are sufficient to open any number of possibilities: Yes, No, or Maybe. Many cultures, including in the U.S., in the UK, Brazil, China, Japan, Korea, Germany, and France, use indeterminate states in their daily language, in practical examples, such as “err…”, "umnn" “imph”, “huh”, “né!” -- or “não é?" -- and “daí”. In traffic lights, the use of 3 states is standard (Green, Yellow, Red). In mathematics, students soon observe: it seems that only YES or NO are possible. The MAYBE seems to indicate relative precision, indeterminacy. It is considered OK, though, as an intermediate

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state. We are looking for absolute accuracy, as a final answer, but we accept MAYBE as a valid logical state. The final answers in this book are always: YES or NO/MAYBE or, YES/MAYBE or NO. In Science, we note that Yes means “not yet false”, and No means “could be true”. This is how the scientific method should be seen, leaving room for indetermination as a way to be precise. This book follows the same route, with MAYBE. 6___________________________________________________ Since many of the challenges in analysis can be done to any desired accuracy already, using the older paradigm of microscopic continuity, one can say that there is no practical use whatsoever for an exact solution of these problems, as one might be tempted to think. However, one is essentially relying on assumptions, and that is an untrustworthy method. What if …? The Dunning-Kruger effect applies. The Dunning-Kruger effect occurs when a person's lack of knowledge and skills in a certain area cause them to overestimate their own competence. This is a first effect, a second effect also causes those who excel in a given area to think the task is simple for everyone, and underestimate their relative abilities as well. Using, instead, a factual reality, this will provide an absolute accuracy that one can rely absolutely on. Besides, one now has access to surprises -- new and easier results.

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7___________________________________________________ To the workflow we use in this book. Choose an answer by circling your choice. The correct answers can be found in the next Frames. Some questions must be answered with text. Space for these is indicated by a blank, and you will be referred to another Frame for the correct answer. If you get the right answer but feel you need more practice, simply follow the directions for a MAYBE answer, or the wrong answer (could be YES or NO). Please also read the page on Warranty and Services that are available to help you in a self-study setting, and update -- more is to be available over time. The impossibility of using Cauchy epsilon-deltas or infinitesimals are so out of the 17th/18th century mind-set, as older attempts [1.4] and [1.5] show until today, that many otherwise competent researchers, may flatly deny some problems as impossible, but can quickly adapt to the methods in this book. For example in Chapter 6, when we note that the "mean value" theorem has a flaw, the "consolation" is that it is true in Universality. Many might be absolutely immune to persuasion, and can be seen as in the earlier phase of the Dunning-Kruger effect. But if you want the fastest and most secure route to analysis and beyond — and get to 21st century applications --- then this book will help you overcome it, gradually. You will learn to reason using nature, while using an absolutely rigorous approach in calculus to confirm with zero error. All you need is to be a citizen of the 21st century, know algebra, and have an understanding of polynomials — that is, the

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equivalent of a middle-school education with a love for algebra — to use this book. Welcome, enterprising young students! However, this book speaks mainly to older students -- college and university students, who need to know analysis ASAP with no counter-intuitive thinking, and applicable to computers, making the reader ready for more advanced study. This book is alive – it cites resources online (that may change in time), grows and self-updates, and you can access it in an updated electronic format at any time, with proof-of-purchase -also of a printed copy. See the page on Warranty and Services. The edition and version number are printed near the title, you can ask Amazon for a freely updated version, or tell us (see Warranty). 8___________________________________________________ OUTLINE ____________________________________________________ In case you want to know what's ahead, here is a brief outline of the book: it begins with a list of abbreviations used; this first chapter is a review, which will also be useful later on; Chapter 2 is on number systems; Chapter 3 is on discrete functions, trigonometry, and the recommended use of an inexpensive digital calculator that can do trigonometric functions, logarithms, graphs, algebra, differentials, integrals, and more -- for your phone or tablet. The hardware works only with natural numbers (like this book) and yet is exact; Chapter 4 is on Universality; Chapter 5 is on Differential calculus; Chapter 6 covers Integral calculus using observation of Chapter 5, which ends this book.

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More will be available over time, email to [email protected] A word of caution about the next frames. Since we must start with some definitions, the first section has to be somewhat more formal, but using examples for more clarity. We believe in abstract methods, but only after an example is understood. Here, your calculator will be more useful as a guide, furnishing examples at will, and you can get more used to it through mimicry, using priming to learn. First we review the definitions of set theory and functions. If you are already familiar with this, and with the idea of independent variables (domain) and dependent variables (image), you could skip it. (In fact, in the first four chapters there is ample opportunity to observe, to skip it, or fast-read if you already know the material). On the other hand, some of the material may be new to you, or promises a new angle, and a bit of time spent on review can be a good thing. If this is all clear to you, in a first reading you can Skip now to Chapter 2, and return later. Otherwise, move to the next Frame. You should write your notes below, and firm your questions. 9___________________________________________________ Mathematics always considers a point as having 0th dimensions. This is consistent with a rigorous treatment, as followed here. A drafting in CAD also uses a mathematical point with 0 dimensions, consistent with Mathematical usage and a rigorous 28

treatment. It has been useful to consider such a mathematical point, existing as something absolutely accurate, clearly defined, with 0th dimensions, since Euclid in 300 BC. The number line is digital in the 21st century -- different from Descartes -- and we use that smoothly in graphs, using Cartesian Coordinates, as we will see in Chapter 2, exemplifying Fermat’s Last Theorem. Numbers are still scalars. Linear multivectors can be formed from any origin point as explained by Grassmann [1.1], [1.2], and [1.3], but will not be used in this book. We will consider only scalars, with 0 dimensions, this simplifies [1.4-5]. Mathematical real-numbers were invented to provide a macroscopic interpolation between any two points, albeit approximately. This approximate interpolation existed mainly because it was useful before computers. Indeed, in the 21st century, one can use the mathematical real-numbers as an interpolation to pretend continuity -- but the mathematical real-numbers and the continuity thus obtained are artificial -- they have no existence by themselves, and are not rigorous. This idea, furthermore, of a number existing in 0th dimensions, can be considered to exist as a physical image projection of a point on a screen, or as an archetype in our timeless consciousness. The words, "scalars have 0-dimension", are considered in [1.3], and are used here. Therefore, absolute accuracy as a number exists, and is described as scalars [1.3]. We will use this to our advantage -calculus now has a firm base, and no metric function -- so it has easier use, absolute accuracy, becomes an inventive and 29

deductive science, with less guess/rote work, and does not “hide” applications. It has many more applications, and is ready for QC and the 21st century. In mathematics, there is no longer any objective use of "convergence" or "limit", nor "accumulation" points. These concepts were once fancied in calculus due to lack of resolution in methods. This used intersubjectivity, and forced relative accuracy. But each natural point is already isolated -- surrounded by "nothingness" (see Chapter 2). There is no uncertainty in the natural numbers, thus in every number system based on N. These are seen as functions of N (see Chapter 2); the domain of the set N is each a mathematical point, objective, absolute, digital, isolated, rigorous, with a separation of exactly 1 (see Chapter 2), and, therefore, so is the image -- the only aspect that scales with the function is the amount of separation. The idea of mathematical real-numbers and macroscopic continuity is useful as an interpolation, approximately, but they must follow these ideas from the set N, as well, as R and C include N, Z, and Q [1.4-5]. One could also use different interpolations, while one uses the euclidean metric in mathematical real-numbers, even if unsaid. The idea of macroscopic continuity was possibly due to the intersubjective, relative accuracy in conventional methods, and lack of resolution, where superposition and overload could not be resolved, and one confused a jot for a point, a visibly continuous line (as Descartes proposed) for what looked like continuous numbers, that we cannot even write.

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Everything works precisely with absolute accuracy, however, using points from the set N, and yet you do not see or can measure those points macroscopically when using a fine enough spacing in the set Q. Even irrational numbers can be approximated as finely as desired by the set Q (Hurwitz Theorem [2.1]). Then, the error is actually 0 as one considers that any measurement must be a rational number. As the production of values is always in the set Q, so must be their measurement. Nothing in such a mathematical model is random, or stochastic -or the universe would be accumulating errors in 13.8 billion years. No one could equate 0.999... with 1, 1.999… with 2, etc., for 13.8 billion years, and live with impunity! The ancient Mayans used only integers in millennia of valid astronomy predictions. The ancient Greeks used only integers with gears, in the Antikythera Mechanism, also for millennia of valid astronomy predictions. Both showed it, without using mathematical real-numbers, mathematical decimal complex numbers, irrational numbers, physical laws, or any model for the phenomena, such as planets, stars, black-holes, or galaxies. 10__________________________________________________ MATHEMATICAL FIELD ____________________________________________________ Infinite mathematical real-numbers are called a "mathematical field", but modular arithmetic can also do precisely all four arithmetic operations (+-×÷) on a FIF -- a finite set of integer numbers as a mathematical field (explained below). This is the mathematical property used by the ancient Mayans, the Greeks, and in 21st century cryptography.

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Modular arithmetic is now the basis of the 21st century AES, in cryptography using a FIF, and the results of a FIF using all four operations (+-×÷) of arithmetic are shown to be complete as well. This can all be made more rigorous now. In mathematics, a ‘mathematical field’ is a technical language that must be respected exactly – it means any set of elements that satisfies the field axioms for both addition and multiplication, and is a commutative division algebra. An archaic name for a field is rational domain. The French term for a field is corps and the German word is Körper, both meaning “body” in English. It is an important, unifying concept. The group of integers modulo p, where p is a prime number, is denoted in mathematics by Z/Zp. It is well-known that Z/Zp: (1) is an abelian group under addition; (2) is associative and has an identity element under multiplication; (3) is distributive with respect to addition, under multiplication; (4) is a mathematical field. With Z/Zp one can precisely do all four arithmetic operations (+-×÷) using discrete, modular arithmetic -- as well as using familiar, supposedly continuous, mathematical real-numbers. A mathematical field with a finite number of members (the mathematical real-numbers do not have a finite number of elements) is known as a Galois field (we do not prefer using this term, see later why). For each prime power, in the Galois model, there exists exactly one (up to an isomorphism) finite field GF (pn), also written as F(p), where the order p of any finite field is always a prime, or a power n of a prime. The advent of 128-bit instructions, such as Intel’s Streaming SIMD Extensions, allows one to perform Galois Field arithmetic of prime 32

order 2n natively. This is much faster, because natively, in hardware. One can forthrightly detail this in the art, such as the SIMD instructions to multiply regions of bytes by constants in F(w) for w (as 2n) in 4, 8, 16, 32, and growing. Today, we also use more complex extensions of the prime finite field F(p). The initial field F(p) used at the lowest level of the construct is frequently called the basic finite field with respect to the extension. This explanation should be thought through, to denote what will be written more generally as a set of the finite integers as a field (FIF). One implicitly understands in the symbol FIF, finite set of integers with many p, each one called Z/Zp, p being a prime or a power of a prime, isomorphism, self-similarity, fields in mathematics, and Galois fields of order p and n power, denoted as GF(pn), for many p. By definition, any FIF ends in a number M of numbers. These numbers can be put in a 1:1 correspondence with the integers mod p, where M =< p, This includes possibly vacant states with the integers mod p, and allows one to build a finite field in mathematics, using integers, although the set of integer numbers themselves, and M, do not form a field. We call this “the algebraic method”, and it is used in this book. We name this construction FIF for short, as “finite integer field”. No such name presently exists in mathematics, which avoids confusion. A FIF can include unlimited Z/Zp, with different p, and numbers that do not form a field. This extends Z/Zp, and Galois fields.

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Except in computer science, where “finite integer” already means a representation of integers in terms of a finite number of bits, versus an open-ended expression. This use can be disambiguated by context. It will not be used in this book. Any finite set, not just of numbers, can thus become a field using the process described in this book. We also denote this as a field, as one example of a FIF. Finite integers as such are used extensively in the study of cryptography, error-correcting codes, digital communication, network coding, and recently [7.2-10], in physics. Therefore, while no finite field is infinite in the original sense, there are infinitely (as defined) many elements and many different finite fields. These fields cannot be counted. Yet, they are all isomorphic, and self-similar. In particular, while many influential mathematicians may consider finite fields synonymous with Galois fields of a certain power n, such as GF(2n), and do not disambiguate the order p, we disagree with such use, for reasons shown elsewhere. Again, Life defines limits that are stronger than mathematics – the limits of existence in a physical universe, which can be more diverse than any mortal can consider. Following nature and Life, a FIF may include a mixture of different Galois fields, of different orders, such as GF(2n) and GF(3p). This cannot all be modeled by only one effective GF(2w), but people can approximate if desired. The essential components of parity, mirror symmetry, and continuity are not representable in this case.

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To change between such reference frames, a well-known theorem of topology [1.10] is used. This we call Topological Relationship, and says that a mapping between spaces of different dimensionality must be discontinuous, in that a continuous path in a higher space must map into a broken path in the lower space. The consequences here are multiple, and this is being explored as well. 11__________________________________________________ Macroscopic continuity has had a rich tradition in mathematics, and is taught today as a "justification" of calculus [1.4], even "explained", as in the Foreword in [1.4], also in [1.5] -- but all is treated in a necessary macroscopic scale, ghostly and subjectively. We will, instead, use macroscopic "continuity" as an interpolation, objective, which becomes self-explanatory. This is used in Chapter 5. Our 21st century science shows that, on a microscopic scale, nature has been revealed in the last century to be discrete, not continuous. This is our objective reality, qua measured reality, the ontology. But how does one pass from discrete to continuous? Or, vice versa? One can now understand how a cell becomes a person, with mathematics. Macroscopically, continuity turns out to be a subjective interpolation, and adding a metric function. This makes microscopic continuity to be clearly counter-intuitive, unnecessary, and a creator of other non rigorous situations. These aspects are revealed to be unnecessary and clutter-rich as well. The basis for any "justification" in the 21st century on microscopic assumptions, cannot continue to be just an opinion -- even by 35

influential personalities of the past, using antiquated equipment to measure reality. Albeit, using the set N today, continuity can be justified as an objective interpolation -- where many different interpolation metric functions, including non-euclidean, can be visualized -- providing intersubjective, relative accuracy in the macroscopic scale, while one can use absolute accuracy in the microscopic scale. This book harmonizes both views, using a new approach attempting to resolve the issue. This new approach is naturally offered in mathematics by the set N, of the natural numbers, in an approach known to be used even by vegetables, animals and illiterate persons, also in crystals and other plants. The natural numbers (N) are seen as an archetype, as a recurrent symbol or motif in literature, art, science, and mythology. No number system could be more fundamental, widespread, or better tied to nature. You will then be able to better understand physics, biology, and any science, as well as humanities. This book stands ready for self-study, so that you can progress at your own pace, and create your own metaphors. If this is clear to you, in a first reading you can Skip to Chapter 2. Otherwise, please continue reading, and use this space to write your notes, while you use imprinting to learn.

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12__________________________________________________ The set N offers 3 quantum properties (QP), defined in Chapter 2, and induces them to any other derived number system -- by function induction: An exhaustive rote/group work [1.6-7] has been found to be necessary to convince students that continuity -- hence, calculus -- is "right". But, as John Von Neumann once said, "Young man, in mathematics you don't understand things. You just get used to them." This book shows that calculus is not “right” in those terms, and rote/group work is not needed. Of course, we can understand things. We are past the Dunning-Kruger first and second effects. Continuity is not seen microscopically, but emerges on large scale terms. Continuity does not exist on the small scale, which is “grainy”. Continuity is a macroscopic interpolation, quite insensitive to the microscopic details. On the small scale, one only has the set N of natural numbers, and they allow objective, absolute accuracy, naturally, being discrete, isolated, and having values separated from each next value by 1 -- showing 3 QP, as defined in Chapter 2. Mathematics has believed in a number of mirages that were typical of the 17/18th century, whereas the imagined physical reality did not turn out to actually exist on the small scale. However, interpolation can still be used and provides a measure of macroscopic continuity. 37

In Ancient China, for example, not being able to dissect cadavers, one imagined organs that did not exist -- but yet patients could still be quantitatively treated, albeit imperfectly. There is no microscopic continuity in Life. It is fruitless to use microscopic continuity in mathematics, it is worse than trying to find “octopuses on Mars”. Microscopic continuity was, however, imagined to exist in good faith, not as a joke, still even to today in mathematics. It is an interpolation that creates other interpolations, but can be useful macroscopically -- when one can ignore the "graininess" of the small scale. Calculus is now easy and absolutely accurate, supports mathematical real-numbers more effectively, and stands ready to help the remarkable progress that certainly will be made, in science and technology, during the following centuries. Calculus is ready for the digital future!

CHAPTER REFERENCES [1.1] Hermann Grassmann, "A New Branch of Mathematics: The Ausdehnungslehre of 1844 and Other Works", Open Court Pub Co., ISBN: 0812692764, 1995. [1.2] John Browne, "Grassmann Algebra", Barnard Publishing, ISBN: 978-1479197637, 2012.

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[1.3] William E. Baylis (Ed.), Clifford (Geometric) Algebraic, Birkhäuser, ISBN: 3-7643-3868-7, 1996. [1.4] Courant, R. (2010). Differential and Integral Calculus. Ishi Press, New York. [1.5] Apostol, T. M. (1967), Calculus, Vols 1 and 2, J. Wiley, New York. [1.6] David Hilbert, Paris International Congress of Mathematicians (ICM), 1900. [1.7] https://edsource.org/2022/high-calculus-failure-rates-thwart-stude nts-across-csu/664771 [1.8] Michael Harris, “Mathematics Without Apologies”, Princeton University Press, ISBN 978-0-691-1-17583-6, 2017. [1.9] T. S. Kuhn, Structure of Scientific Revolutions, 1962. [1.10] A. Bruce Carlson, “Communication Systems”. McGraw Hill Kogakusha, Ltd., 1968. APPLICATION REFERENCES [7.1] Peter W. Shor . "Algorithms for quantum computation: discrete logarithms and factoring". Proceedings 35th Annual Symposium on Foundations of Computer Science. IEEE Comput. Soc. Press: 124–134, 1994. [7.2] Ed Gerck, Jason A. C. Gallas, and Augusto B. d'Oliveira. “Solution of the Schrödinger equation for bound states in closed 39

form”. Physical Review A, Atomic, molecular, and optical physics 26:1(1). June 1982. [7.3] Ed Gerck, A. B. d'Oliveira, and Jason A. C. Gallas. “New Approach to Calculate Bound State Eigenvalues”. Revista Brasileira de Ensino de Física 13(1):183-300. January 1983. [7.4] Jason A. C. Gallas, Ed Gerck, Robert F. O'Connell. “Scaling Laws for Rydberg Atoms in Magnetic Fields”. Physical Review Letters 50(5):324-327. January 1983. [7.5] Ed Gerck, Augusto Brandão d'Oliveira. “Continued fraction calculation of the eigenvalues of tridiagonal matrices arising from the Schroedinger equation”. Journal of Computational and Applied Mathematics 6(1):81-82. March 1980. [7.6] Ed Gerck, Augusto Brandão d'Oliveira. “O Problema de Três Corpos Não Relativístico com Potencial da Forma K1.r^n + K2/r + C”. Brazilian Journal of Physics 10(3):405. January 1980. [7.7] A. B. d’Oliveira, H. F. de Carvalho, Ed Gerck. “Heavy baryons as bound states of three quarks”. Lettere al Nuovo Cimento 38(1):27-32. September 1983. [7.8] Ed Gerck, Luiz Miranda. “Quantum well lasers tunable by long wavelength radiation”. Applied Physics Letters 44(9):837 839. June 1984. [7.9] Ed Gerck. “On The Physical Representation Of Quantum Systems”. Computational Nanotechnology 8(3):13-18. October 2021.

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[7.10] Ed Gerck. “Tri-State+ Communication Symmetry Using the Algebraic Approach”. Computational Nanotechnology 8(3):29-35. October 2021. [7.11] Dirk Bouwmeester, Arthur Ekert, and Anton Zeilinger, (Eds.). “The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation”. Springer Publishing Company. 2010. [7.12] Leon Brillouin. Science and Information Theory. Academic Press, N. Y., 1956. [7.13] G. Mussardo, “The Quantum Mechanical Potential for the Prime Numbers.” Arxiv; https://arxiv.org/abs/cond-mat/9712010, 1997. [7.14] https://www.researchgate.net/publication/352830765/

[7.15] https://www.researchgate.net/publication/339988557/

Please use the space below to enter your references and notes.

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Chapter 2: Number Systems One means objective values, not subjective symbols, when one talks here about numbers. The natural numbers (the set N), as well as any dependent number system, such as Z and Q, show 3 quantum properties (QP). This leads to a revisitation of calculus, and an evolution of many Cauchy ideas. This discovery is based on a QP < -- >Number Theory "wormhole". This follows the seminal development of QC by Peter Shor in 1994 [7.1], using the same set N. Each member of the set N is recognized as showing 3 quantum properties (QP): discrete, rigorous, and isolated. Each member of the set N is: 1. Discrete: digital, to use a 21st century term, being separated from each other by exactly 1; 2. Rigorous: showing absolute accuracy with width 0; and 3. Isolated: surrounded by "nothingness", where even the word "nothingness" may be too much. One understands that numbers are not digits, as we can use different digits to represent the same number. But numbers can be thought of as a 1:1 mapping between a symbol and a value. Digits

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become a “name”, a reference, and it is clear that one can use different “names” (even vocally, in different languages, such as “one”, “um”, and “Eins”) for the same value. This leads us to the set of irrational numbers, yet undefined in mathematics. Irrational numbers continue unnamed, neither proved nor disproved. That is, they are binarily independent of the Field Axioms of the mathematical real-numbers --- or, mathematically undecidable in the language of Kurt Gödel. Other books by Planalto Research will consider this, also in Laplace and Fourier transforms, and in QM, and as a revisitation of the Heisenberg Principle. In this book we provide evidence that there is no mathematical continuity. No accumulated errors. Numbers module a finite set of integers can be exact, because integers are exact, and mathematically decidable in the language of Kurt Gödel. They can work like Lego. The sets Z, and Q, are constructed using natural numbers, images of N. The sets N, Z, and Q, have all the properties described for those sets in [1.5], and are themselves mathematically decidable in the language of Kurt Gödel. Mathematics, with calculus, can become “click-mathematics”, like Lego. Anything constructed can be taken apart again, and the pieces reused to make new things. The sets N, Z, and Q, are here called “natural” numbers. Every “natural” number system inherits the same 3 QP of N in their image, albeit with a different separation.

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1__________________________________________________ A basis of general logic is, contrary to common belief, that some things are impossible. Transitions involve change. This is impossible to be continuous, or there is no change. Hence, one must be able to differentiate discontinuous functions, contrary to conventional theory in [1.4-6], but according to all experimental findings [7.1-10]. One finds evidence of the difficulty in how this has progressed, in the work of T. S. Kuhn [1.9] -- where technical changes happen in jumps, called paradigm shifts. This book is such a jump, and its understanding may face difficulties. However, this is also natural, and expected. Our motivation for this book, nonetheless, is that the goal is meritorious for society, and timely for QM, QC, GR, physics, cryptography, biology, and other fields. It is impossible to have a half hole, for example, in nature. Using nature, this book shows that it is impossible that mathematical real-numbers can exist as conventionally thought [1.4-5], as if they would validate some basic “truth” of microscopic continuity, or “faith” (as unreasoned belief) in infinitesimals. But one can use mathematical real-numbers coherently as a human-made “scaffolding” over irrationals, as an interpolation. The idea of infinitesimals, however, is not only against the old concept of a microscopic nature in numbers, giving rise to continuity, but against their rigorous use in calculus, and will be abandoned in this book.

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Mathematics is right in a 17/18th century casual way -- things can look approximately continuous using mathematical real-numbers, infinitesimals, macroscopically and microscopically to any scale one wants, using older equipment/experiments. However, using rigor, the microscopic domain imposes itself as discrete, and this has become clear in the digital 21st century, while offering new applications “for the initiated”. By following this book, you gain access to understand a new “tongue” due to a paradigm shift [1.9], albeit hidden in today's language. Measurements in physics can be exact, in the 21st century, as exemplified in [7.1], notwithstanding the old formulation of the Heisenberg principle. Not only rational numbers are the only numbers measurable, but they are also the only numbers produced. No production is continuous. Nature appears digital in its most basic aspect -numbers. Universality (Chapter 4) defines a macroscopic quality that, although not existing microscopically, emerges at a large scale by collective effect -- a similar effect that produces the apparently continuous, but artificial (and with discrete members, such as N, Z, and Q) mathematical real-numbers, and waves. 2__________________________________________________ Theorem 2.1 -- The 3 QP followed by the set N (the set of natural numbers -- 1, 2, 3, 4, ….) , are induced to every function of N, or to any set that contains N, Z, or Q. Please write down your proof.

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3__________________________________________________ Verify here, and in the next Frame. 4__________________________________________________ A function (arithmetic or algebraic) must be univocal, by definition of a function in Chapter 3, Frame 6. Every value of N is rigorous (a point of dimension 0), and digital (each point in N is separated by 1 from the next, starting with 1), and also isolated -- there is a "margin of nothingness" around every point. These are the 3 QP of interest, and the image of a function is formed likewise, by definition of a function. The separation can be scaled from 1 by the function, but is always ≠ 0. The 3 QP happen in any image function of the elements of the set N, even if it seems to be a blob or continuous. The sets N, Z, and Q, are included. QED. The sets R, of the mathematical real-numbers, and C, of the mathematical decimal complex numbers, are not included, because “natural” numbers cannot map 1:1 to mathematical decimal complex numbers or mathematical real-numbers [1.4-5]. For example, the mathematical real-numbers include the irrational numbers, without any existing correspondence between R and Q. 5___________________________________________________ Corollary 2.1.1 -- Every number system inherited from N has at least 3 QP (rigorous, digital, isolated), albeit with a different separation between numbers. Prove the corollary, below. 46

6__________________________________________________ Try with your calculator, or read Frame 2 again. 7__________________________________________________ For the curious: other QP are possible in a ternary pattern, and can be treated mathematically, e g., in QC [7.9-10]. 8__________________________________________________ But how can one have equidistant points along a curve element expressed in mathematical real-numbers? This is not trivial, since the set R only provides functionality to evaluate the curve based on their internal parametrization (which is supposed to be continuous), and not based on “natural” number coordinates (such as N, Z, and Q), which must be discontinuous. Basically, one has to move along the curve using a fixed step size in the curve parameter space -- also called "natural length of a curve", or "the natural parametrization of a curve". Equal distances in the curve's natural length are transformed to non-equal distances in R coordinates, especially when moving along sharp bends in it. Determining points at equidistant positions along the curve, measured along the curve in R coordinates instead of the curve in natural length coordinates, basically requires integration, which we see in Chapter 6.

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One would need to evaluate the curve step by step in increments, close enough to represent the observed variation of the curve, and measure the sum of distances between the evaluation points until one reaches the desired distance, then add a new marker point at that position. 9__________________________________________________ In euclidean space, this is done by the formulas in the cartesian coordinate construction.

A familiar diagram showing the relations in a right-angle triangle is given above by the Pythagorean Theorem, and is discussed online at https://www.learnalberta.ca/content/memg/division03/pythagorean %20theorem/index.html The cartesian formula for the length c in 2D space, gives the square of the length (also called the square of the “absolute value” 48

of the “norm”) as the square of an inner product, with c2 = a2 + b2 [5.1], and figure above. Similar formulas, not used in this book, apply to the length in a 3D flat space (d2 = c2 + a2 + b2 [5.1]), in the Minkowski 4D spacetime, and in Einstein’s GR flat space with 4D. Thus, the square of the separation c (the ‘length”) between two points in 2D space without axes, A and B, is given by a and b in each coordinate, orthogonal, axis in 1D, as: c2 = a2 + b2.

(2.1)

We can use this expression to interpolate between points in 2D space, using Eq.(5.1), and reproducing Fermat’s Last Theorem. We can use Eq.(2.1) measuring c in the set Q, while measuring with the same set Q in each axis, a and b. Eq.(2.1) means that the Pythagorean theorem is satisfied in 2D euclidean spaces, for Q. Note: The set R, for mathematical real-numbers, will no longer play a key role. No physical production is continuous. Nature appears physically digital. Not only rational numbers are the only numbers measurable physically, but they are also the only numbers produced physically. Each process must be finite. 10__________________________________________________ One can also measure anywhere in-between the points A and B, in numbers in 2D space, using the cartesian construction from 1D axes, over Q.

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The interpolation in each axis is given by following Eq.(2.1) and Eq.(5.1). One can use the set Q in 2D space, by describing measurements in each axis, using the same set Q. This gives a dense covering, such that between any two numbers A and B in Q, there is an infinite number of rational numbers (such as (A+B)q/n, q < n, n > 0, with both q and n in the set Q, where n can be as large as desired). 11__________________________________________________ However, when using mathematical real-numbers in conventional calculus, the covering would have to be modified in type theory (TT) from the set Q to the set R, by means of a metric function that must map Q → R and R → Q. NOTE: One wishes to map Q to R and R to Q, but there is no dependence from Q to R, or vice versa, that one could use exactly. Therefore, the sets Q (coming from natural numbers) and the mathematical R (coming from humans) simply do not share a common mapping Q → R, or R → Q. This problem has spilled lots of chalk, and irritated many students, since the 17/18th century [1.7]. The problem means mapping not only the obvious transformation from Q -> R, as 1 -> 1.000…, 2 -> 2.000…, ⅓ -> 0.3333…, ⅔ -> 0.666…, etc, but also any points in-between, where we need to also find a mapping in reverse, from (for example) an irrational number in the mathematical set R, which is a number … that is not to be found in the set Q, by mathematical definition.

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Such a mapping is mathematically impossible, but is provided in conventional calculus, approximately, by interpolation, as a “scaffolding”, using a metric function. The euclidean metric function [5.1-2] is such a solution, and could be used in the cartesian construction, without further ado. It can be taken, in a flat space, as a mapping from c in Q, to c’ in R, and offering a path in reverse, albeit with an error |c-c’|. This is not rigorous, but has been acceptable in practice, as “unmeasurable”. This step is quite arbitrary, impossible to be exact, and different metric functions can be used. Different error types could also be minimized (e.g., least-square error, mini-max error, least-ripple error, etc.), providing different views. Instead, this book uses the rational number set (the set Q). This creates a question if A, B, or both, are irrational numbers -unreachable by members of the set Q. 12__________________________________________________ In those cases, first consider a continued fraction or an infinite series (AAIS), defining an approximating member of the set Q, such that the irrational member is included. For example, we can apply the Hurwitz Theorem [2.1]. The decimal expansion of an irrational number gives a familiar sequence of rational approximations to that number, using only natural numbers. For example since π = 3.14159... the rational numbers are:

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r0 = 3, r1 = 3.1 = 31/10, r2 = 3.14 = 314/100, r3= 3.141 = 3141/1000, ... This gives a sequence of better and better approximations to π, using natural numbers, providing a physical representation of π, measurable by members of the set Q as an arbitrary-length rational number, not as an “irrational”. We can measure the quality of these approximations by applying the Hurwitz Theorem [2.1], which converges fast the finer the separation: Hurwitz Theorem: Every irrational number has infinitely many rational approximations p/q, where the approximation p/q has error less than 1/(√5 ⋅ q^2). Thus, |π - rk| < 1/(√5 ⋅ 10k) Similarly √2 = 1.41421... can be approximated by the sequence of rational numbers: r0 = 1, r1 = 1.4 = 14/10 r2 = 1.41 = 141/100, r3= 1.414 = 1414/1000, … with the same accuracy as the approximations to π, providing a physical representation of √2 (Which baffled the Greeks and, more recently, in the UK, Edward Titchmarsh. He is well-known to have observed, in his opinion, that √-1 is a much simpler concept than

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√2, which is an irrational number -- which now we know, in this book, to be exact and simple, as well as √-1.) Any curve can be measured by the set Q in 2D (or in higher dimensions), using the cartesian construction from 1D axes in the set Q, as described in the previous Frame, yet as an approximation. That was the first approach. But the error (irrespective of irrational numbers) is now 0 if one considers that any measurement must be a rational number. This is the second approach. Thi is a consequence of 3 QP in N, as further explored in this book: that every path begins and ends in a natural number, but can go through a path in the integer numbers, rational numbers, mathematical real-numbers, mathematical decimal complex numbers, irrational numbers, surreal-numbers, and any other number system. Anything artificial, made by humans, such as the mathematical real-number system and mathematical decimal complex numbers, must be harmonized. But the set Q, being “natural” , is already harmonized, and error-free in the measurement in Q. Again, the error (irrespective of any irrational numbers) is 0 if one considers that any measurement must be a rational number. This offers a wider scope, new solutions, and absolute accuracy in measuring Q from any axis to the 2D space, and can be extended easily to 3D and higher-D. This will be used in Chapter 5. Write your notes below. 53

13__________________________________________________ The number 0 is not in the natural numbers, but there is no mystery. The subtraction of two equal natural numbers (each natural number is always positive) is always 0. Integers (with positive and negative signs) can come from simple subtractions of natural numbers. Irrational numbers cannot be written as rational numbers, but can be approximated by rational numbers (Hurwitz’s Theorem [2.1]). However, a mathematical real-number or a mathematical decimal complex number are human inventions, artificial, and we do not need them. This book mentions mathematical real-numbers only for compatibility purposes, not for rigor or speed of computation. Computers (with or without coprocessors) also do not need them, and yet can calculate anything. And not even the mathematical real-numbers or mathematical decimal complex numbers are actually continuous, but remain grained, since they include N, Z, and Q [1.4-5] -- themselves “grainy”. The illusion of “pointwise convergence”, aka continuity, does not seem to mathematically exist, even where one attempts to make it.

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So, the idea that “The sequence 1, 1/2, 1/3, 1/4, 1/5 … 1/n, … converges “exactly” to 0 as n increases without bound”' has a logical problem that is not new, and has been well-studied since the Zeno paradox. Even if a computer could squeeze infinitely many computational steps into a finite span of time, still the last step is short of the goal, which is the 0-dimensional number 0. One is never at 0 in the sequence, and so one cannot reach it, no matter how close one gets. The difference between 0 and the sequence cannot be ignored in this view, just because it becomes vanishingly small. When that “vanishingly small” is compounded in calculations, it can grow without bound. The reasoning applies to any finite target, not just to the number 0. Thus, one can never have “absolute accuracy”, which influences applications, as discussed next, in terms of two meanings of the term exact. Table 1: Partial exactness and absolute exactness Type

Accuracy

Name

Partial

Width > 0

“pointwise convergence”

absolute

Width = 0

0-D point convergence

Formally, let S denote the set of points x for which a limit sequence converges. The function f defined on S is called the limit function of the sequence fn and one [1.5] says that fn has “pointwise

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convergence” to f on set S, although the width > O, which is shown in Table 1. By the definition of a limit sequence [1.5], this means that for each x in S and for each ε > 0 there is an integer N, which may depend on both x and ε, such that |fn(x) - f(x)| < ε whenever n>= N. So, this so-called pointwise convergence rule in mathematics [1.5] is just an abuse of technical language and cannot apply when one wants to be absolutely exact, to be rigorous. The width ε > 0 in a neighborhood in the image of any function must be always definable for n>= N [1.5]. So, we can revisit calculus using the set Q, basically using natural numbers. The result is a discrete, isolated, and rigorous number system, showing 3 QP, and complete. This works without visibly changing the mathematical real-number equations that have been proved qua rational numbers in experiments, and are visibly seen as continuous. This expands to new results using Q, hoping to reach wider application conditions and faster computation. New applications motivated us to calculate with absolute accuracy [7.9-10], using the set Q in QC. We realize that the mathematical real-numbers or mathematical decimal complex numbers are interpolations over unknown numbers, and not rigorous. Therefore, they are not used in this book.

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__________________________________________________ REFERENCES [2.1] Hurwitz, A. Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche. Math. Ann. 39, 279–284 (1891). https://doi.org/10.1007/BF0120665 YOUR REFERENCES:

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Chapter 3: Set Theory, Functions, and Calculator As Tom M. Apostol [1.5] says, the branch of mathematics known as integral and differential calculus (also called analysis) serves as a natural and powerful tool for approaching a variety of problems that arise in experimental sciences -- physics, astronomy, engineering, chemistry, geology, biology, and other fields including, rather recently, some of the social sciences. There, calculus must apply to physical measurements. The language choice must be based on experimental science. Experimental sciences have allowed this book to stand on the shoulders of giants, preparing the student for the future. Therefore, infinity is not needed here. This book considers that not only rational numbers are the only numbers measurable, but they are also the only numbers produced. No production is continuous. Nature appears digital. The idea that a measurement could go to infinity is not in experimental sciences, neither in places nor in value. Think of the best tape measure in the universe; its graduations can only be rational.

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Mathematics does not have to follow, but applications, qua experimental science, do. Therefore, this book considers that calculus follows the experimental science rule: not only rational numbers are the only numbers measurable, but they are also the only numbers produced. Infinity will be used here, as well as the symbol ∞, meaning an unknown number in algebra, as high as wanted. The number is not predetermined, or fixed (i.e., as a number would), but is finite and reachable. Albeit, it is a result that cannot be counted. The original symbol ∞, is not a number. Similar considerations, mutatis mutandis, apply to -∞. Other books by Planalto Research will consider this in Laplace and Fourier transforms, also in QM, and as a revisitation of the Heisenberg Principle. 1___________________________________________________ Cauchy epsilon-deltas and microscopic continuity are useful, but lead to unseen interpolations [Foreword, 1.4], [1.5-6] and difficulties [1.7-8]. They are not used in this book. This book presents what one can call “a natural view of calculus”, which is taken as the quickest and best way to learn calculus. Exactly, more intuitive -- and yet refreshingly rigorous, ready for computers and the 21st century. Comfortingly, it also includes mathematical real-numbers. It shows using mathematical real-numbers how one can profit from the interpolations leading to a false continuity, but smooth graphs. One can just accept Cauchy epsilon-deltas, etc., instead of trying to "justify" them with a false microscopic continuity -because they can be justified “enough”, elsewhere.

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We use Universality, explained in Chapter 4. It is based on experiments -- that we can now see exactly and clearly in both scales, macroscopically and microscopically, in the 21st century. We can also see smooth graphs, using the rational numbers. If this is clear to you, in a first reading you can Skip to Frame 3. Otherwise, Go to Frame 2. 2___________________________________________________ A similar thing happened, soon after Galileo and other European astronomers developed the first telescopes at the start of the 17th century. They observed dark spots speckling the Sun’s surface. The appearance of dark spots on the Sun, as an experimental fact, had potential consequences in physics, mathematics, and even theology, that impacts daily life today —- with satellite missions to explore the possibility of extraterrestrial life. But … what if we do find a civilization way more advanced than us? Can theology accept that? Mathematics? How? On Earth we are finding that even invertebrates and fish, without digits, can do simple additions and subtractions. We knew that about birds already, but birds have and can see their digits. The invertebrates and fish must use different ideas of numbers, maybe not using digits, but at least their simple arithmetic is equivalent to ours. We are not the pinnacle of evolution. Other species might use more advanced mathematics, and without digits.

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However, until today, mathematics has believed in a number of older paradigms that were typical of the 17th/18th century, but that have been revealed to be lacking in the physical reality revealed by more rigorous measurements, in the 21st century. One can abandon interpolations, to reach absolute accuracy and make mathematics more useful to work with other sciences, including computer science. This is not only according to the HP, but also a necessity in our daily, integrated, life and is exemplified in new applications (see Chapter 7). We are then bound to encounter new realities, new applications -and mathematics should be better able to help, more than other sciences or humanities, when one no longer needs interpolations. Physics, for example, is about what exists, albeit mathematics is about what may exist. Contemplating what may exist, is useful to forecast consequences, for example. In contrast, even our school children know in the 21st century that quantum mechanics (QM) exists, the Internet is available 24/7 worldwide, computers can be easily networked, and one can use cell phones, lasers, and DVDs. To a contrarian, it may seem like we are losing things, but they were not even worthwhile. With this book, we suggest what may seem like a long-road for such contrarians, who can still see continuity in older mirages that no longer exist in the imagination, and cannot exist in nature. Continuity has now become a PTSD, a verifiable pathological condition from a ghostly past. Interpolations may be used, but are 61

limited to the large scale, and to an illusion, albeit sometimes useful, but always imprecise. For the new ideas, they are a complete paradigm shift [1.9]. This is common in science, and potentially brings turmoil. These paradigm shifts create, however, the shortest road into calculus and beyond, using absolute accuracy, as we explore in this book. Natural numbers are isolated and have width 0. They have no error, and induce a digital system in Z and Q (albeit with a separation different from 1, and different from 0, for Q), Mathematics can finally become a 21st century subject, 1.8], [7.1], rigorous and holographic in behavior. However, the material in this book is offered with zero physics, for the benefit of a simpler use. Other books by Planalto Research will consider this in Laplace and Fourier transforms, also in QM, and a revisitation of the Heisenberg Principle. Mathematics can bring together all sciences as innovative and deductive sciences, based on reason, that continues to evolve. Humanities, Law, and Political Sciences can also profit, in a HP, which is used today to physically protect our credit cards. Some problems in this book require the use of a scientific calculator – we recommend a free version for Android and iPhone phones, with ads, or a few US dollars version, called HiPER Calc PRO, shown in the next page. The suggested calculator provides trigonometric functions, logarithms, complex functions, special functions, algebra, 62

derivative and integrals, graphs, and more. It uses hardware only in natural numbers, that uses only addition and encoding, yet performs all operations. The calculator has up to 100 digits of significand and 9 digits of exponent. It detects repeating decimals and numbers can be also entered as fractions or converted to fractions. You can compare with your answers, and learn by observation.

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3___________________________________________________ Set Theory ____________________________________________________ The definition of a function makes use of the idea of a set, which we will use also for relationships in logic, change and area -- such as the differential and integral, in Chapters 5 and 6. Thus, it is very important in this book. Do you know what a set is? If so, go to 10. If not, read on.

A set is a collection of objects – not necessarily material objects – described in such a way that we have no doubt as to whether a particular object either does or does not belong to it, creating a LEM --- where a 3rd state is mathematically undecidable in the language of Kurt Gõdel. This may have avoided confusion, having an internal law for success (the LEM), but may act as a “Procrustean bed”, creating unresolvable indeterminacy, FUD. This does not happen with 3-or-more-states logic.

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There is one clear answer (Yes), another clear answer (No), and room for indeterminacy (Maybe), something in-between. This can be represented in a digital circuit by Intel and other manufacturers, using three-state logic [7.10], to achieve better performance and scalability. 4___________________________________________________ The conventional set theory, however, uses Boolean, binary logic. This was supposed also by Shannon’s information theory to represent the logic of switching circuits, where there is by force no MAYBE – the LEM is valid, always. A familiar diagram showing the relations in a binary set theory is called the Venn diagram, shown above and is discussed online at https://www.onlinemathlearning.com/shading-venn-diagrams.html A set may be described by listing its elements. Example: the set of some natural numbers, 23, 7, 5, 10. Another example: the set of components of matter, as atoms, molecules, and ions. We can also describe a set by a rule, for example, all the odd natural numbers, or all the mathematical real-numbers (these sets contain an infinite number of objects). Another set defined by a rule is the set of all objects physically bound in stable orbits around our solar system (large, unknown, but finite). A particularly useful set is the infinite set of all natural numbers, which includes all numbers such as 1, 2, 3, 4, etc. The set of natural numbers is easy because it involves isolated values we can name, and does not include values that are not

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exact -- such as with decimal points, irrationals, p-adic numbers, roots, surds, complex numbers, transcendental numbers, etc. The difficulty of including an infinite number of members is solved by working with just p elements, where p is a prime or a power of a prime, in Z/Zp or a FIF (where different p in Z/Zp exist, and that can always be included in the same set). The set N not only includes values that are exact, and isolated, but they are all separated by 1. This is seen as being described by 3 QP, and we also see it in nature. This was described in Chapter 2, before Frame 1. The mathematical use of the word "set" is similar to the use of the same word in ordinary conversation, as "a set of cards", where you can use your 21st century knowledge as "a set of emojis". In the blank below, list the elements of the set which consists of all the odd natural numbers between 5 and 10.

5___________________________________________________ Here are the elements of the set of all the odd natural numbers between 5 and 10: 5, 7, 9.

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6___________________________________________________ Functions ____________________________________________________ Now we are ready to talk about functions, using set theory. We did refer to functions intuitively in Chapter 1 as a "wormhole" connecting different universes. Here is a more formal definition. A function is a rule that assigns to each element in a set A, the domain, one and only one element in a set B, the image. It is like a hole made from an injection needle. We say it is a 1:1 mapping if it is also reciprocal – we call it a one-to-one function; basically denoting the reciprocal mapping of two sets. More formally: A function f is one-to-one if every element in the image of f corresponds to exactly one element in the domain of f. We can picture this, as when there is only one injection needle that can make holes. The function’s rule can be specified by discrete values, but also by a mathematical formula supposing continuity in Universality, such as y=x2, or by tables of discreetly associated numbers. If x is one of the elements in set A, then the element in set B that the function f associates with x is denoted by the symbol f (x). [This symbol f (x) is the value of f at x. It is usually read as "f of x."]

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The set A is called the domain of the function. The set B of all possible values of f(x) as x varies over the domain is called the range or image of the function. 7___________________________________________________ In general, A or B need not be restricted to the sets of “natural” numbers (N, Z, Q). It is any formula between any A and B, and can also be composite; as when one function is evaluated after another, written as fog(x). [The symbol fog(x) is the value of f at g(x). It is usually read as "f of g of x."] Write below, the expression fog(x) where g(x) = x2 and f(x) is √x , for any real x. Why is fog(x) different from gof(x)? 8___________________________________________________ The first answer is |x|, the module of x, always a non-negative value. In general, fog(x) is different from gof(x), because f is different from g. For another example, for the function f (x) = x2, with the domain being all integer numbers, the range is:

________________________________________

9___________________________________________________ The answer is: any square natural numbers, adding zero. For an explanation, go to Frame 10. Otherwise, Skip to Frame 11.

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10__________________________________________________ Recall that the product of two negative numbers is positive. Thus for any integer value of x, positive or negative, x2 is positive and a square. When x is 0, x2 is also 0. Therefore, the range of f(x) = x2 is all squares of natural numbers, plus 0. One could also say all non-negative square integers, as they include 0. On the other hand, square positive integers do not include 0. In-between the values in the images of natural numbers, there is a nothingness (reflecting what happens in the domain); but seen from afar, in Universality, one can fictionalize a “continuity” uniting the points, or imagine any figure that passes through the points, crossing “nothingness”, or misses some points, or even all. The decimal expansion of an irrational number seems to give a familiar sequence of rational number approximations to that number, using only natural numbers. However, this is fictionalized, crosses “nothingness”, or misses some points, or even all -- it does not exist in a domain with only natural numbers, but this is not seen from afar. We call this Universality; look for Universality in the next Chapter, write your notes below using priming to learn, and return to the next Frame.

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11__________________________________________________ Our chief interest will be in rules for evaluating functions defined by formulas, presenting results that seem continuous. We realized in Chapter 2, Frame 13, that not even the mathematical real-numbers or the mathematical decimal complex numbers are actually continuous, but fine-grained when observed. If the domain is not specified, it will be understood that the domain is the set of any natural numbers for which the formula produces any value, and for which it makes sense. For instance, (a) f(x) = √x

Range =

_________________________

(b) f(x) = 1/x

Range =

_________________________ Check your answers in Frame 12. 12__________________________________________________ f(x) is an yet unspecified number for x a natural number; so the answer to (a) is all square roots of natural numbers, which can be seen as “continuous” when seen far enough, in Universality. One can use the familiar decimal expressions to illustrate such “continuity”, such as √2 = 1.4142…, where one can truncate at any point, or represent by a fraction, such as 1414/1000 or 239/169 (see Chapter 2, Frame 12).

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An irrational number can never be expressed by any single rational number, but can be well-approximated as seen in Chapter 2, Frame 12. Someone may argue here for R, the set of mathematical real-numbers, in conventional mathematics [1.4-5]. However, a mathematical real-number cannot be physically infinite in digits or value, and must always be written truncated and/or approximated, in any basis or notation, in implementation. But mathematics can also work in terms of observation, without any physical limitations -- as one may argue. Yes. One can use an infinite series, use any mathematical real-number, or envision an infinite process -- all as an observation, in one’s own mind, or in written short notation; even though one’s implementation must be finite -- paper is finite, computer memory is finite, human memory is finite, time is finite, cost is finite, we also live in a finite universe. One can indeed separate observation from implementation, both of which can be represented mathematically, as belonging to different dimensions. The higher dimension is observation, where implementation must exist in a lower dimension. Connecting both, as in a “wormhole”, must involve discontinuities in the lower dimension. We call this Topological Relationship, and results from a well-known theorem in topology [1.10]. This is pictorially represented by projecting a 3D helix onto a 2D surface -- one loses continuity and chirality information. One cannot tell anymore if the 3D helix is right-handed or left-handed, by its 2D projection.

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Approximating rational numbers can be put into a 1:1 mapping, using a "wormhole", with the set N, but the mathematical real-numbers R (as representing continuity) cannot come from a 1:1 mapping with the set N, as Cantor is well-known to have shown. This is seen by mapping not only the obvious transformation from Q -> R, as 1 -> 1.000…, 2 -> 2.000…, ⅓ -> 0.3333…, ⅔ -> 0.666…, etc, but also any points in-between, where we would need to find a mapping in reverse, from (for example) an irrational number in the set R (such as √2), to a number … that is not to be found in the set Q (√2 is well-known not to be rational). One also may require to “fit” the mathematical real-numbers in a certain interval, as one can do with the physical diagonal of a physical unit square, as √2, exactly. By Archimedes' axiom, between any two distinct irrational numbers, we have a rational number, in fact, an infinite number of them. One says that the Q is dense in R, so there is no fear of a "hole" in the mathematical real-numbers, even if there is an irrational point one wants to include (see Chapter 2, Frame 12). 1/x is defined for all values of x a natural number (this excludes zero); so the range in (b) is all inverses of natural numbers, which can be seen as “continuous” in Universality, for large x, or expressed as a continued fraction using natural numbers, or using AAIS in an infinite series.

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13__________________________________________________ With a function defined by a formula, such as f(x) = ax3 + b, the x is called the independent variable, and f(x) is called y or the dependent variable. The function is considered valid for any set that satisfies the formula. One advantage of this notation is that the value of the dependent variable, say for x = 3, can be indicated by y = f(3). Often, a single letter is used to represent the dependent variable, as in: y = f(x) Here x is the independent variable and y is the dependent variable. 14__________________________________________________ In mathematics the symbol x frequently represents an independent variable, f often represents the function, and y = f(x) usually denotes the dependent variable. However, any other symbols may be used for the function, the independent variable, and the dependent variable. For example, we might have z = H(u) which reads as "z equals H of u". Here u is the independent variable, z is the dependent variable, and H is the function. The formula s = W(t) is valid, where t is the independent variable, s is the dependent variable, and W is the function. 73

15__________________________________________________ Trigonometric functions are very important in life, to understand, for example, shadows, the division of areas, in physics, and in other sciences. Even in Humanities in understanding equivalence, as when a metaphor can be fitted with trigonometric rules for equivalence – avoiding mixed metaphors. But trigonometry can be introduced with easy absolute accuracy after one studies differential equations, and this is motivated in Chapter 5, Frame 45 and ff. Meanwhile, please explore your calculator, by calculating sin(30 degrees), and cos(-60 degrees). 16__________________________________________________ The answer is 0.5, in both cases. The sine and cosine functions are exactly the same, just shifted horizontally by 90 degrees. The graph below, from the calculator, can be just shifted by 90 degrees, to show either sin(x) or cos(x). In written form, one may write sin(x) or cos(x). Note the periodicity, important for QC [7.1].

In this case, 90 - 60 = 30. This fact will simplify many calculations.

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Please explore further the trigonometric functions in your calculator. Check with your calculator that sin(Θ)2 + cos(Θ)2 =1, always, for any angle Θ, with peaks exactly compensating valleys, in any interval. Use the space below to write other trigonometric identities that you find useful.

17__________________________________________________ Exponentials and logarithms will be introduced next, and viewed again, using differential equations, in Chapter 5, Frame 45 and ff. The exponential is defined by the formula: z = ax where z, x, a ε R or Q. Using Universality, one can fictionalize the mathematical real-numbers as an interpolation between points, or use rational numbers. The logarithm is the inverse function of the exponential function, exactly. log a x = y ; ay = x The number a is often called the base. When a is the special 2.71828… mathematical real-number (see [1.4-5]), it is called the Euler number and is written as e; in that case, the logarithm is called the natural logarithm and is written as “ln” instead of “log”. The formula ln(e) = 1 defines e, exactly. Sometimes, in that case, the inverse function is called the natural exponential. When using bits, the base is 2.

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Please explore further the exponential and logarithmic functions in your calculator. Check that ln(e) = 1, and log10 = 1. In logarithms, one can simplify equations by noting the identities: Product-Sum rule: log(u ⋅ v) = log(u) + log(v) Division-Subtraction rule: log(u/v) = log(u) - log(v) where u, v are real-numbers, or functions. These identities were very useful before hand-held 21/st century calculators, even before slide rules, and are useful today. Now that we know what a function means, and the main functions, write your notes below. Let's move along to a discussion of cartesian graphs.

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18__________________________________________________ Graphs ____________________________________________________ If you know how to plot graphs of functions, you can Skip to Frame 24. Otherwise, your calculator can do it! If you are comfortable with that, you can also Skip to Frame 24, in a first reading. 19__________________________________________________ A convenient way to represent a function defined by y = f(x) is to plot a graph. We start by constructing coordinate axes, usually in a rectangular Cartesian coordinate system. First we construct a pair of mutually perpendicular intersecting lines, one horizontal, the other vertical. The horizontal line is called the x-axis, and the vertical line the y-axis. One can also add a vertical axis, called z. This book will only use 2D relationships and graphs, with the x-axis and the y-axis. The point of intersection is the origin, and the axes together are called the rectangular coordinate axes. This can be done in 2D, or 3D in three mutually perpendicular directions. In 3D, obeying the right-hand rule of chirality, one can hold the z-axis with the right-hand, pointing the thumb up, curling

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the fingers from the x-axis to the y-axis, as shown on the previous page. Chirality, in chemistry, means 'mirror-image, non-superimposable molecules', and to say that a molecule is chiral is to say that its mirror image (it must have one) is not the same as itself. Whether a molecule is chiral or achiral depends upon a certain set of overlapping conditions. 20__________________________________________________ This cannot be visualized completely today in higher dimensions, larger than 3D. This will not be used in this book. 21__________________________________________________ Next, we select a convenient unit of length and, starting from the origin, mark off a number scale on the x-axis, positive to the right and negative to the left. This can be done with numbers in the set Q. In the same way, we mark off a scale along the y-axis with positive numbers going upward and negative downward, in the next page.

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In 2D, one has what is shown in the picture above. The scale of the y- axis does not need to be the same as that for the x-axis (as in the drawing). In fact, y and x can have different units, such as voltage and time. 22__________________________________________________ We can represent any pair of values (x,y), and achieve a cartesian construction in 2D space, from values in 1D axes. This was explained in Chapter2, Frames 9-12, and is used in Chapter 5, to define a derivative. This point is important in terms of using priming to learn.

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Use the axes above, to mark the points A = (3,5) and B = (5,3). Check in the image above. Let a represent some other particular value for the independent variable x, and let b indicate the corresponding value of y = f(x), as the dependent value. Thus b = f(a), point A, or shown as (3,2) for x=3 and y=2, in the figure, using the previous page. The four quadrants are noted above. We now draw a line parallel to the y-axis at distance A from that axis, and another line parallel to the x-axis at distance B = 2. The point P at which these two lines intersect is designated by the pair of values (A,B) for x and y respectively. The number 2 is called the x-coordinate of the point marked as A, and the number 3 is called the y-coordinate of the same point. 80

(Sometimes the x- coordinate is called the abscissa, and the y-coordinate is called the ordinate.) In the designation of a typical point by the notation (a,b) we will always designate the x-coordinate first and the y-coordinate second. As a review of this terminology, encircle the correct answers below. For the point (-4.5, 5): x-coordinate [ -4.5 | -3 | 3 | 5 ] y-coordinate [ -4.5 | -3 | 3 | 5 ] (Remember that answers are ordinarily given next, but in this case it is already marked in the graph above in Quadrant II. You can always check also with your calculator before continuing.) 23__________________________________________________ The most direct way to plot the graph of a function y = f(x) is to make a table of reasonably spaced values of x and of the corresponding values of y = f(x). Then each pair of values (x,y) can be represented by a point as in the previous frame. A graph of the function is obtained by using Universality -- by visualizing the points, as if the points are connected with a smooth curve, such as a straight-line. For the mathematical connection, we have to introduce a metric function -- usually, the euclidean metric. Of course, the points on the “continuous” resulting curve are always only approximate, using relative accuracy. Even if we want an accurate plot, and we are very careful, use a 0.5 mm diameter mechanical pencil, account for that diameter 81

when drawing, be careful with the scaling, and use many points, we can only get a graph that uses relative accuracy. Absolute accuracy can be achieved, easily however, with a CAD, where just one point of 0-dimensions is the intersection of two non-parallel lines. This can be very useful in tracing optical ray lines, showing a rigorous result, separating observation from implementation. In drafting, however, one usually considers three lines, to estimate the intersection better (absolute accuracy is often not needed, and unreachable in mechanical drafting, try as we may). 24__________________________________________________ As an example, the next page shows a plot of the function y = 3x2, done by the calculator we recommended. A table of values of x and y is not shown but some points could be indicated on the graph (but it is usually not necessary if using a calculator). To test yourself, encircle the pair of coordinates that corresponds to a point in the figure, as: (2,12). Check your answer, or use your calculator.

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If incorrect, study Frames 23 and 24 once again. Afterward, go to Frame 25. 25__________________________________________________ Definition 2.1: For any given function f(xk) → yk, on its independent discrete, isolated variable xk, one has the dependent variable yk also necessarily discrete, isolated, as one and only one value, yk = f(xk), with xk and k ∈ N, and one has a sequence of related points in xk and yk, denoted as (xk, yk). The 3 QP of N are transmitted from domain to image, by the function f. The above definition follows from the definition of a function. Far enough, in Universality, the distance between the points in (xk,yk) may not be seen, and one can have the impression, in approximation, and as a visual interpolation, that the function is continuous, and can write y= f(x), or calculate (x,y).

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If that is clear, Skip to Frame 26. If not, proceed anyway and we trust that usage may make it clear. 26__________________________________________________ The graph in this page below, can be used to show these relationships.The student can magnify any section (easier to see in curved sections) and see the individual points (xk,yk) that make up the image. The curve uses points in the set Q, set to be visible to the naked eye.

Far enough, one seems to "see" a "continuous" curve y = f(x), which continuity is fictional, when looked closer. It is created by a collective effect, called Universality.

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Universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display Universality in a scaling limit, when a large number of parts presenting collective effects come together. We detail this in the next Chapter. Analytically, this book uses Universality to obtain a smooth enough behavior in the scaling limit, resembling continuity as well as one can measure, whereas the underlying process is inevitably isolated, discrete, and pixelated, originally representable by the set of natural numbers, N, separated each number by 1. This induces a discrete behavior in the image of any function, with absolute accuracy, even when invented to look continuous like the mathematical real-numbers -- and even it may indeed look continuous when seen closer … but we are always using the set Q in a rigorous measurement. Henceforth, the relative accuracy is represented by the mathematical real-numbers, whereas the absolute accuracy is represented by the set of natural numbers, N, or a derived set, such as Z, and Q. Please go to Chapter 4. __________________________________________________ REFERENCES See Chapter 1

Please use the space below to enter your references and notes.

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Chapter 4: Universality Continuity is defined in the classical calculus textbooks by Courant [1.4] and Apostol [1.5]. If you understand Universality, you can Skip this Chapter on first reading. Otherwise, please go to Frame 1. 1___________________________________________________ Numbers, qua values, could not be invented without some aspect of reality. Even if natural numbers are a model only, they should be a fundamental archetype, useful when considering different species, and they should offer ontic functions. In ontology, ontic is physical, real, or factual existence. In more nuance, it means that which concerns particular, individualized beings rather than their modes of being; the present, actual thing in relation to the virtual -- a generalized dimension which makes a thing what it "is". In other words, there needs to be something about reality in the value of numbers, most of it ageless and widespread, and such need we consider to be satisfied by natural numbers. In the evolution scale, billions of years far from us, even invertebrates and fish, without digits, have a notion that one can associate with natural numbers -- and, experimentally, they are able to (somehow, in their own system) do simple additions and 86

subtractions, when mapped to our words -- where everything begins and ends exactly in what we call a natural number. When we use decimal complex numbers, likewise, the path goes into complex space in our minds, that one can be trained to invent, but everything begins and ends exactly in what we call a natural number. Computers can also do any calculations in mathematical real-numbers and mathematical decimal complex numbers, exactly, but by calculating only in natural numbers in hardware. When we use mathematical real-numbers, likewise, the path goes into mathematical real-number space in our minds, that one can be trained to invent, but everything begins and ends exactly in what we call a natural number. All four operations of arithmetic (+-×÷) can be done in a computer only by addition, and encoding, using natural numbers in hardware. 2___________________________________________________ The reason there is no FUD about incompleteness, uncertainty, imaginary, and even abstract, when using natural numbers, is that the path always goes through the natural numbers -- that have 3 QP. They are known objectively, ontically, and with absolute accuracy. Each natural number, or derived set, has at least 3 QP, being discrete, rigorous, and isolated, as explained in Chapter 2. By the mathematical definition of a function, any “natural” number system is then made to depend on the natural numbers, even

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using composite functions. This also induces the same 3 QP in the final image, in any “natural” number system. 3___________________________________________________ Try any expression with your calculator. This is also your laboratory -- where all calculations are done with natural numbers in hardware, including mathematical real-numbers, mathematical decimal complex numbers, and when using coprocessors. If you agree, you can go to Frame 4. Otherwise, write your question below, proceed and expect your question to be answered in time. Come back to write the answer!

4___________________________________________________ Two QP of natural numbers are that they are isolated and rigorous -- a mathematical point surrounded by a region of nothingness -where even the word nothingness may be too much. And each natural number has a third QP, called digital, as each one is separated by 1. This induces discrete functionally into any “natural” number system, using a particular separation, with 3 QP. This microscopic reality is induced mathematically when we make every “natural” number system map (showing 3 QP) by means of a function to the natural numbers. The natural numbers, Z, and 88

the set Q, become ontic. This does away with any FUD. Everything becomes absolutely accurate, in a science that can be both inventive and deductive. This is “click-mathematics”, and works as a Lego. 5___________________________________________________ Where is continuity? Macroscopically, one has the set R of mathematical real-numbers, and the set C of mathematical decimal complex numbers, created by humans, and they do not allow 3 QP to be induced, in a macroscopic continuity that has no microscopic continuity origin. They are not exact, by definition. Microscopically, though, one has innumerous discrete, rigorous, and isolated, natural numbers in set N, integer numbers in set Z, and rational numbers in set Q -- all showing 3 QP. They are all exact, by definition. Use the space below to draw these relationships. And, answer: how to pursue objectivity? Should one use what one sees, as macroscopic, or what one infers, with instruments, as microscopic? Move to the next Frame.

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6___________________________________________________ The answer to Frame 5 is not Boolean, which would be either macroscopic or microscopic. This book uses both, breaking the LEM. The LEM, and Boolean logic, can be broken in different situations. Microscopic reality has macroscopic effects, like the laser, albeit the microscopic reality of the laser cannot be denied even on a large scale. Without denying the "graininess" of microscopic reality, one can use a smooth macroscopic interpolation to replace infinitesimals, as done in this book with the set Q. There is also no LEM. Likewise, universality allows the microscopic "graininess" of reality to be ignored in macroscopic formulas in the well-known Maxwell equation, while providing smooth waves in the large scale, when microscopic effects are interpolated. But, the Maxwell equations are well-known to fail in the microscopic regime, and cannot macroscopically explain diamagnetism, the laser, particles, the electromagnetic spectrum, and other phenomena. A computer program can show discrete points with 0-dimensions, but zoom out to a smooth line, in CAD. This abstracts implementation (must be discrete) from observation (continuous), common in CS, and we see that in “continuous” computer graphs in our high-resolution cell phones, in the 21st century. Mathematics has believed in a number of older mirages that were typical of the 17th/18th century, where the imagined physical reality does not turn out to exist in the small scale, but the imagined reality forms in the large scale, and can be used as an 90

interpolation while psychologically retaining the small-scale imagined reality as a form of PTSD, potentially pathogenic. This book stands for a possible cure. The "graininess" of 3 QP is induced to every image of N, i.e, to every “natural” number system. Then, they must show 3 QP. The mathematical real-numbers are a number system that humans invented and interpolates on the natural numbers, trying to avoid the “graininess” of 3 QP -- yet showing it, as it includes the sets N, Z, and Q. 7___________________________________________________ One can work in Universality at a large scale. There, the mathematical real-numbers in the macroscopic domain can be used well -- even though they are interpolations. Cauchy epsilon-deltas and accumulation points cannot be used well, and introduce ghostly contradictions as they move to the microscopic domain. Now, we have problems that do NOT accept that interpolation treatment to solve. For instance, it is not necessary to assume continuity to have a derivative, but standard references, using continuity, affirm so in [1.4-6]. The p-adic numbers are not “natural”. This is also contra sensical, and “hides” solutions, as explained in Chapter 1. Current models blur from discreteness to superposition as one approaches uncertainty limits. No one seems to be able to say

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with certainty that there is “no microscopic continuity" under such conditions of relative accuracy. Likewise, a mechanical drafting cannot define precisely a mathematical point as the intersection point of two non parallel lines, and it is a recommended practice to use three lines, to find a more trustworthy point. The relative cannot define the absolute, with rigor. Albeit, this is not necessary with a CAD in the 21st century, nor using Euclid’s results from 300 BC -- showing that one needs to pursue rigor, achieving absolute accuracy that stays the same at all ages. Now, as Chapter 2 shows, with rigor provided by the natural numbers and derived number systems, such as Z, and Q, one can work with any number system that is determined by the natural number system. This has the "graininess", isolation, and exactness of 3 QP -- hence all “natural” number systems have 3 QP, albeit the isolation changes value. Thus, there is no "eternal" contradiction between continuity and discreteness. It is a matter of scale, in a non-Boolean logic, and even though the scale is quite arbitrary where it ends or starts, exact accuracy can always be obtained with the 3 QP of N, Z, and Q, as in the CAD and Euclid examples above. If you agree, you can Skip to Frame 9. Otherwise, please write below your objections, and advance to Frame 8.

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8__________________________________________________ Google can help you find more examples by searching for Universality. Please write below what you find.

9__________________________________________________ The discrete aspects of 3 QP also invalidate the important mean value theorem, when not taking into account Universality. We can, however, use the mean value theorem, and mathematical real-numbers, in an interpolation within a scale of Universality. The conventional treatment of calculus is hereby objectively affirmed in many cases -- but only in Universality. We can move in two regimes, from relative accuracy and subjectivity with Universality, with the sets of R and C, to objective, absolute accuracy with the sets N, Z, or Q. In this book we will be doing both, and validating calculus with rigor. 10__________________________________________________ Thus, continuity cannot be produced in the large scale, because the large scale must bear an image of the set N, necessarily discontinuous, and following discrete rules, with 3 QP. But, the large scale can be provided with continuity built-in, or with differences too small for the naked eye or an error term to resolve,

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when using relative accuracy. Thus, the large scale can also provide visual continuity and rigor, with the set Q. It is absurd to pretend that 0 can be reached by decreasing something proportionally to it, by fractional decrements of it, as one is always not at 0. One can also see a line, retraced many times with slight offsets, and imagine a continuous line, for example. Or a blob, without being able to resolve any discrete point in it. The large scale is discrete by definition of the set N, in the microscopic scale. This discrete nature is induced macroscopically, even if we cannot see it microscopically. But, humans wanted to use mathematical real-numbers and mathematical decimal complex numbers -- systems that do not follow a "wormhole" from the set N, and can offer what is not naturally provided: continuity, in the illusion that more precision is thereby to be attained. However, cryptography found out that absolute precision could not be reached with such artificial continuity, but was provided by a finite set of integers. Cryptography can be exact and complete because it does not use mathematical real-numbers, it uses modular arithmetic over FIF. The ancient Mayans and the Greeks used integer fields in astronomical calculations over millennia, with no errors. They did not use mathematical real-numbers or mathematical decimal complex numbers. The principle here seems to be, what we call the HP: that all creation (Sciences no matter where discovered, other species no 94

matter where they live, including humans), have to be holographic with nature, any small part reflecting the whole. There is no bottom-up or top-down model -- there is a HP. 11__________________________________________________ Some things were invented in conventional mathematics, such as Cauchy microscopic continuity, Cauchy accumulation points, and continuity in general, before the discrete nature of objects was historically known. However, they were not invented willy-nilly -- they were invented to the best of knowledge according to 17th/18th century life. 12__________________________________________________ This book considers macroscopic properties in a large class of systems that are quite independent of the microscopic details of the system. This is a motivation to accept mathematical real-numbers and mathematical decimal complex numbers as “they are”, because they work as macroscopic systems, but without any need for microscopic “justification” as attempted in [1.4 Foreword], to the despair of students [1.7] as well as teachers [1.8]. Continuity in the macroscopic scale can be advantageous as a simplified macroscopic model, for example in reading using interpolation, as in Frame 14. One can expect the same, by the HP, in mathematics, and any science, or even in humanities, or poetry.

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Thus, students of this book can learn Cauchy accumulation points, etc., and use them logically as interpolations, instead of using rote/group work, to memorize or “justify” a "rule" that one cannot see or confirm, or is counter-intuitive. This can avoid suffering [1.7] and PTSD. The continuous mathematical real-numbers are just the Universality view of countless, underlying, discrete, separate oscillations, yet unresolvable macroscopically. Albeit, a finer microscopic resolution is the reality in a finer scale of microscopic oscillations, where even matter disappears and is replaced by exact oscillations of energy, according to the formula that everyone seems to know: E = mc2, in the 21st century. But, such things are unresolvable at a large scale, where they macroscopically build a "continuity" model within a very small error, even immeasurable, although they reveal themselves to be important for reaching new results in QC [7.10]. Thus, mathematics has been trying to teach us to treat as continuous what is, actually, discrete, by accepting a small error, as if it would be negligible. This is not the result of a “conspiracy theory”, “fake news”, or malice, but results from a factor that was “unknown to be unknown” -- Universality, the matter of scale, in a Dunning-Kruger effect of the first kind. This book presents a solution to this by using the set Q. This “unknown to be unknown” factor may feel, referred to today, as “out of left field" in American slang -- meaning "completely unexpected", "unusual" or "very surprising". 96

The phrase came from baseball terminology, referring to a play in which the ball is thrown from the area covered by the left fielder to either home plate or first base, surprising the runner. So, it was ignored by experiments in the 17/18th century, and not detected using faulty Boolean logic of the 19th century. This book also may seem to come “out of the left field”. In the 21st century, anyone who sees a "continuous" graph under high magnification in a cell phone or display, can appreciate its underlying, barely hidden, graininess. Albeit, the discrete aspect is manifest in the graininess of nature itself. This is “out of the left field” in the macroscopic view of Universality. 13__________________________________________________ We use continuity in this book, as an interpolation. This will allow students to interact with those practicing previous mathematics. But, we first show a basic, 3 QP description of differential calculus in Chapter 5, that anyone can use, easily, but based on natural and rational numbers, albeit arriving at the same formulas with new results, using what we called the algebraic approach. In the blank below, describe what happens when an apparently continuous function on a display is seen under 3x, 10x, and 100x magnification. You can use your cell phone to take pictures of your display, and affix them. You can get to 100x or even higher magnifications easily, by taking repeated pictures and amplifying from a lower scale, as feasible.

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14__________________________________________________

Any apparently "continuous" curve is seen as grainy, with spaces between the “dots” (shown as rectangles or circles), that a naked eye could not see. An example is given above. And even the “dots” are grainy. 15__________________________________________________ Universality allows you to read the words "PCM" in frame 14, at a distance, by interpolation. Further magnification will probably make it harder, if not impossible. Even the “dots” are grainy. Going closer does not increase readability, because it makes interpolation more difficult. Interpolating is a macroscopic property. Can you name a microscopic property in frame 14? (Hint: Each pixel is microscopic).

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Answer in the space below, with your text. Skip to Frame 17, if Yes. 16__________________________________________________ Try with your calculator, and experiment. You can also try any TV image on a flat-screen. This may not work so well in older TV screens, not made in the 21st century, and showing a raster image, with indistinguishable pixels. However, a photomultiplier tube could “see” the individual electrons making the image grains that the naked eye cannot see. Today, you should be able to distinguish the individual pixels, as they make the image. Write your experience, below.

17__________________________________________________ Any matter is, actually, made mostly of empty space -- with grains moving around. These grains are made out of various atoms, molecules, and ions (charged atoms or molecules). Matter shows these objects as grains, under very high magnification. The table you see as solid, is actually made mostly of empty space -- with those grains moving around. Even under higher magnification, would you see any continuous matter? Please draw your answer below, and answer YES, NO, or MAYBE. If NO, Skip to Frame 19. If YES/MAYBE, go to Frame 18.

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18__________________________________________________ There is no wave microscopically, just a collective motion as particles. No wave-particle conundrum exists. Think of pure water; it seems continuous. But, pure water is made with a union of 3 atoms: 2 of Hydrogen and 1 of Oxygen. They create a chemically covalent bond, with an acute angle. This gives pure water a polar bond that creates the appearance of a volumetric continuity, by attraction between different molecules, forming a strong spatial grid, much like a link chain, allowing a wave to appear macroscopically. Try watching pure water on a high magnification microscope, though, and the molecules become separable -- if you don't have access to one, try online videos. Write about your experience.

19__________________________________________________ In the 21st century, mathematics must follow the underlying nature. This includes Universality -- and makes this “completely unexpected” factor a matter of scale, although not a Boolean variable, as the two realities coexist. Thus, no one can postulate microscopic continuity in the 21st century, or risk being called insane -- not attuned to reality.

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How would you classify today the notion that continuity is possible? Please describe your answer, as if in front of your peers.

20__________________________________________________ Continuity is possible as a collective effect, with errors one cannot resolve, hiding the “grainy” nature of matter, and all empty spaces, including in numbers. The underlying reality is always grainy, though, like the screen seen in Frame 14. The “completely unexpected” factor is a matter of scale, and non-Boolean, a Dunning-Kuger effect of the first type. Far away, we see a wave in pure water, a continuity we can feel and experience on a beach, but under magnification we see the grainy molecules that make up the water. Recognizing that, allows one to break up the molecules into atoms, and even use different molecules. Absolute accuracy down to one atom or molecule is today considered certainly measurable, it is considered a certitude even in our scale of physical reality, for example, with an atomic force microscope. A larger size necessarily means more atoms or molecules, and can provide only relative accuracy, albeit one can use interpolation. The “completely unexpected” factor, so “out of the 101

left field” in the 17/18th century, became a matter of scale, in the 21st century. 21__________________________________________________ There is an empty space between natural numbers. They are represented in N -- the natural numbers -- separated by exactly 1 unit. Each natural number is then isolated from the next by a unity, in a clear and smallest possible separation. However, due to "wormholes" (as functions) we can invent, we can "warp" the natural numbers, increasing or decreasing their separation -- albeit not to 0, or it would not be a function. We can also invent new number systems altogether. One example is the square-root function, with √2 as an example. This can create square-roots out of natural numbers, with a smaller space between numbers. So, if we create a unity square, we can fit a diagonal that has a measure of exactly √2. But, √2 is exact, as √4 is exact. If you agree, Skip to Frame 23. If in doubt for any reason, go to Frame 22. 22__________________________________________________ To see that √2 is a number that we can measure exactly, and with absolute accuracy, note that the natural number 2 has 3 QP -- and we can use these properties mathematically, afterward, in the image of the well-formed function √ in the positive branch -- still with 3 QP after the function (i.e., after the "wormhole"). Write below your own diagram to this.

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23__________________________________________________ Another number is eiπ , a natural number that one can obtain in absolute accuracy as -1, from two irrational numbers and an imaginary number. Magic? Explain below why not. 24__________________________________________________ You can verify with your calculator. The “imaginary” exponential is equal to cos(π) which is -1, and i times sin(π) which is 0, both exactly. From that, you can use algebra to calculate (and check with your calculator) that ii = 0.20787957635046 … = e-π/2, which shows the reality of i. 25__________________________________________________ Some people don't think that structures need to exist in nature, as mathematics study, but they think that they may need to use approximate structures that can exist undetected in nature. Undetected until now, there exists an inner compensation mechanism, there is a marvelous realization that can make all answers exact, and “save” calculus. This is seen in 3 QP -- how natural, numeric reality works. This is provided by the natural numbers and derived number systems, Z and Q, with 3 QP. Computers use natural numbers, in hardware exclusively, but humans prefer to imagine mathematical real-numbers and mathematical decimal complex numbers, costing more time … and decreasing precision. Some still think that continuity could exist in mathematics (since there seems to be no natural limit to the subdivision of a spatial 103

object using mathematical real-numbers or mathematical decimal complex numbers, even if its other physical properties are discrete), and are trying to calculate this even to today. Many doctor's theses, careers, and chalk have been lost to that misconception. Please say in your words, how would you answer such objections?

26__________________________________________________ Loss of time. As we can choose the "wormhole" (as a series of functions in an expression), one can imagine, by absurdity, that it would lead to a continuous universe. Then, we could have a mapping from the natural numbers in our universe, to a continuous blot or curve. But we can still define a microscopic reality in our universe, the domain, following the natural numbers. This must induce 3 QP in the supposedly “continuous” image. Then, we can calculate using this separation between points in the domain, even for all the points in the range, in the other universe. One could go to a blot, allowing only observations in partial accuracy, albeit in absolute accuracy the individual points must be preserved in the image, according to 3 QP.

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Thus, absolute accuracy must exist in the supposedly “continuous” universe, in the function image, independently of the function, but it forms from the function domain, if we use the sets N, Z, or Q. Some may think that a “way around” to obtain continuity is to be human-made, artificially made. The mathematical real-numbers, or mathematical decimal complex numbers, for example. But one still finds 3 QP. Every number system is discrete with a clear rule: the domain separation is 1 unit. This induces 3 QP in the image, albeit with a different separation, albeit not 0, due to the definition of a function. 27__________________________________________________ When one includes collective effects, coherence can be investigated further by the student. Coherence can make many particles indistinguishable. We use this effect in superconductivity, or lasers, for example. However, it is not continuity, because the particles exist as particles. We can replace the particles with one another, annihilate them, or create them, but we cannot eliminate their borders. Many coherent particles can behave as one, as in a hologram, but are generated and recorded individually. The interferometer cavity, in a stimulated emission source (a laser), allows the individually generated photons to behave in lock step. This effect can be achieved also without any mirrors or stimulated emission, in superradiance [7.9-10].

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28__________________________________________________ This century has included a shift away from the notion that signal processing on a digital computer was merely an approximation to a mythical analog (continuous) signal in processing techniques. Most now prefer music to play on a DVD, to long-play vinyl records, for the DVD higher quality, lower cost, and smaller reproduction size. Digital has imposed itself as the true and desirable signal, masking as an interpolated analog signal. One recognizes that the discrete signal is the actual cause of the interpolated analog signal, which seems continuous. The analog signal is now mythical, includes measurement errors in the x-axis and in the y-axis, and takes into account the recipient as well as the environment, while the discrete signal is more likely what is produced. Thus, one can prognosticate that the reduction of prejudice against digital (as a quantum/obscure method) is signaling the direction of evolution in many fields, with absolute precision in less time. The Fast Fourier Transform (FFT), for example, used in a DVD player, has reduced the needed computation time by orders of magnitude. The FFT uses the natural numbers efficiently, achieving continuity in interpolation as a collective effect, and concludes our presentation on Universality. You can search online for further

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references. This ends Chapter 4. Review as needed. Go to Chapter 5.

REFERENCES See Chapter 1.

Please use the space below to enter your references and notes.

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Chapter 5: Differential Calculus This is the central Chapter in this book. A basic fact is that the set of rational numbers (Q) is closed under subtraction. This means that the subtraction of two numbers in Q, will always yield a number in Q. The set Q has all the properties described in Chapter 2 and [1.5]. 1___________________________________________________ This Chapter follows a familiar process, where a solution is easier to find when an equation is seen through a connection as shown below, taken from [5.2], page 934.

Fig.(5.1) Method for an easy solution of difficult problems.

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2___________________________________________________ Consider two numbers in the set Q (or in the set R, as mathematical real-numbers), A and B, in a flat 2D space. A and B must differ if there is a change. Problem: How to measure the change? According to Fig.(5.1), we seek to transform the problem, to find an easy solution. To demand continuity before one is able to measure change is contradictory. To “measure” is always intended to mean “of some physical, real-world quantity”, as stated in Chapter 1. This is not what one could calculate using infinite sets, as explained in Chapter 1. One cannot also demand continuity before one starts to measure what must represent … a lack of continuity, a change. Our purpose is to be able to measure the change between A and B, i.e., the lack of continuity. We use the cartesian construction to transform a 2D problem, hard to solve, into 2 simpler 1D problems in the set Q, simpler to solve, using the concept of Fig.(5.1). The euclidean distance between those points in the set Q, A and B, represents the most intuitive concept of linear distance on the line [5.1]. The closer the linear distance between A and B, the smaller their euclidean norm. This is natural, and trustworthy. 3___________________________________________________ To measure that distance, we set a cartesian coordinate system, with orthogonal axes as rulers in x and y; the length is to be expressed between A and B, in 2D space using rational numbers (all that one can measure with numbers, see Chapter 2). This is to be measured by cartesian construction from the two 1D axes using Q, as we saw in Chapters 2 and 3. 109

We are able to choose the orientation and scales of the x and y rulers, without changing A or B. We measure A = (ax , ay ) and B = (bx , by) as the coordinates, making sure that ax - bx ≠ 0. What is the change in each axis? Write your response below. 4___________________________________________________ The change in the x-axis is ax - bx , and the change in the y-axis is ay - by . This makes the change in both axes to be the ratio of two rational numbers (ay - by)/(ax - bx), with ax - bx ≠ 0. The error (irrespective of any irrational numbers) is 0 as one considers that any measurement must be a rational number. Therefore, one does not need to use the set R to measure the change from A to B -- and doing so would reduce rigor. Notes: 1. The set of mathematical real-numbers are fictively continuous. It mixes numbers in the set Q with the irrational numbers; but no mapping between them is possible, by definition. 2. The x-y axes are orthogonal, and are freely oriented and positioned as one needs, to make physical sense. Changing the x-y axes position and right-left orientation (chirality) does not change the difference between A and B, but may simplify or even allow calculations. We make sure that the denominator is valid, with ax - bx ≠ 0.

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5___________________________________________________ Because A = (ax , ay) and B = (bx , by) are in cartesian coordinates, the change between A and B in the x-axis is ax - bx , and the change in the y-axis is ay - by, making the change in both axes to be expressed also in Q, as (ax - bx , ay - by). This can be measured as a ratio of those two numbers, taken as (ay - by)/(ax - bx). They are always in Q, thus their ratio is also always in the set Q, provided (as stated before), that ax - bx ≠ 0. We are also free to modify the units of measurement in each axis, as finely as desired, provided that ax - bx ≠ 0. 6___________________________________________________ The change from A to B is very important in Science and Engineering, and can reflect in other areas of knowledge by the HP -- and this is always intended to mean “of some physical, real-world quantity”, as stated in Chapter 1. A and B may be of different sizes or units on the axes. This may just make the change be comparatively large or small in each axis. We define the total change in 2D as the differential, and it is in Q, as we can measure only rational numbers, “of some physical, real-world quantity”. Definition 5.1, Differential: The differential, total differential, derivative, or slope in 2D, between A and B, is defined as the rational number (all that we can measure): = y’ = dy/dx = (ay - by)/(ax - bx)

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The numbers A and B, belong to the set Q, while ax, ay, bx, and by are also in the set Q, provided that ax - bx ≠ 0. The mapping is from Q to Q. This mapping always works. The derivative can be measured as finely as desired using the set Q, and can seem visually continuous. Points A or B may be from the irrational numbers (which are not in the set Q). We are to use the Hurwitz Theorem [2.1] to also have them represented in the set Q, which will be discussed in Frame 13, or considered not produced in finite time. We often write the differential as y’ (pronounced y prime, no connection to primes), dy/dx, or use y with a “dot” when referring to time differentiation. Note that we define this as dy divided by dx, and that dx ≠ 0. On their own, dy and dx have an exact and well-defined meaning, contrary to [1.4-5]: the change in each axis. We can take the expression also as a symbol dy/dx on its own, that may always be split up exactly into those 2 parts, using “click-mathematics”, just like Lego. Anything constructed can be taken apart again, and the pieces reused to make new things. The quantity dx does not have to be “small” -- the relationship to dy is not even assumed to be linear, and can be discontinuous. The symbol d/dx can be considered as an operator. You can apply this operator to a discontinuous function f. One gets a new function f’ = df/dx. This is also contrary to [1.4-5]. So if f' is a function, it makes sense to "apply" the differential operator again to f', and write f'', and in succession. We can also write f'' as d2f/dx2, and so on.

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If one writes y’ = f(x), then this is the first derivative, the second derivative uses the same concept of a differential in regard to a change of a change in x, and in succession. Is there any inconsistency or error term in definition 5.1, Yes or No/Maybe? Circle your answer and Skip to Frame 8. If No/Maybe read Frame 7. 7___________________________________________________ No errors. Definition 5.1 is exact. The set Q is closed under subtraction, as finely as desired, provided that ax - bx ≠ 0. Thus, the numerator and denominator are always well-defined in the set Q. They can be split up if we want, into dy and dx. We avoided the case of ax = bx, by simply rotating the chosen x-y axes. This does not change the difference between the points A and B, and just reassesses. Rotation of the chosen axes, however, can change the differential. 8___________________________________________________ The square of the length of separation on each axis is: a2 = (ax - bx)2 and b2 = (ay - by)2. The cartesian formula for the length c in the 2D space gives the square of the length (also called the square of the “absolute value” of the “norm”) as the square of an inner product, which is always non-negative. It is c2 = a2 + b2 [5.1], measuring in each of the two 1D coordinate axes, and leading to one 2D value.

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9___________________________________________________ The potential existence of irrational numbers at A, at B, or even at both points, create an interference of values measured in each axis, so that best variations in c cannot be neatly separated for each axis. This has no influence here, because we are using rational numbers throughout, but will be handled by partial differentiation in Frame 50, including mathematical real-numbers. With mathematical real-numbers, there is no connection possible between R → Q and Q → R. This would be impossible -- i.e. one cannot map the irrational numbers to Q, which are not in Q. We want to be able to adjust the measuring device for best measurement. We can zoom in or out in Q, without changing A or B. This can correspond with a factor k, k ≠ 0, in the a and b axes. It can be represented by a zooming of the axes, leading to a resultant zooming of the points in c, measuring better the separation of the points within A and B. Or, a meaning using different units in each axis. The measurement is done in the set Q and the object is assumed to be in the set Q. Possibly, however, the reader may argue, the object may be hypothesized to be in the mathematical real-numbers. All measurements are in the set Q, what to do? 10__________________________________________________ Zooming, harmonizes, as possible, the numbers in the different sets. If any irrational number needs to be included in A, B, or at

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both points, we can use the Hurwitz Theorem [2.1], or not consider -- due to finite time to produce (see Frame 6). As the coordinates x-y zoom in or zoom out with factor z, z ≠ 0, in each axis equally, we have, by cartesian coordinate calculation on the orthogonal axes, the resulting length of c in 2D (see above) is calculated according to the expression, with z, a, b, c in Q: (cz)2 = (az)2 + (bz)2

(5.1)

11__________________________________________________ Theorem 5.1. The main property is called the Linearity Property. This corresponds to the measurement of the separation between A and B, as defined by the zoom factor z, z ≠ 0. This is achieved by reflecting the zooming in each coordinate axis, each one measured with the set Q, as finely as desired in each axis, where we make the final z, as the common zoom factor. This does not change the points A and B themselves, including points that can be irrational numbers. We just expand or contract, zooming in or zooming out, the rational numbers that fit in-between the points A and B, through Eq.(5.1). One can fit-in finer and finer members (az, bz) of the set Q. Geometrically, this means that the slope (the numerator divided by the denominator, in the differential) does not change at all when each axis zooms by the same common factor z, where z > 0 zooms in and z < 0 zooms out. It is written:

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d(z⋅f(x))/d(z⋅x) = d(f(x)/dx, with z ≠ 0,

(5.2)

The zoom factor z cancels exactly in the differential, and the measurement is given by cz, Eq.(5.1), in the set Q. If you agree, skip to Frame 13. Otherwise, please read on. 12__________________________________________________ Please, try with your calculator. Use this space to write your notes. 13__________________________________________________ Frame 8 explains what happens to the measurement of the separation within the points A and B, in Q. In case A, B or both points are irrational numbers, they are not in the set Q. See Frames 2 and 6. We say, however, that Q is dense in R [1.5], so this can be approximated by the Hurwitz’ theorem [2.1], or not be considered -- due to finite time to produce (see Frames 6, 10). For example, 22/7 is a well-known rational approximation to the irrational number π. The error in the approximation is 0.00126. Another rational approximation to π is 355/113; this time the error is 0.000000266. To be exact is, indeed, impossible by definition: the irrational numbers are not in Q. But, the error is 0 if one considers that any measurement must be a rational number. An exact result can be obtained, as the object is in Q, and the measurement is also in Q. This agrees with TT. If you agree, Skip to Frame 15. Otherwise, go to Frame 14. 14__________________________________________________ 116

Write your comments below. Please try with your calculator. See Chapter 4, Frame 1. 15__________________________________________________ Theorem 5.1 is the main result of this Chapter. We use this to show how the total derivative is constructed coordinate-wise from discrete rational numbers in two 1D axes, to discrete rational numbers in 2D, which uses rigor. There is no error in the formalism, everything matches in a “click-mathematics”, like a Lego. If we had used the euclidean metric in a 2D flat space, we could be looking for macroscopic continuity in mathematical real-numbers. In that case, as [1.4-6] propose, the rational numbers of the set Q (microscopically discrete) would be used with interpolation on both axes as the cause of mathematical real-numbers (macroscopically continuous). This can only be approximate. Theorem 5.1: Because we will be using only rational numbers both for the object and for the measurement, all derivative formulas will be the same as in [1.4-5], and yet have absolute accuracy. If you agree, Skip to Frame 17. Otherwise, read on. 16__________________________________________________ Please, try with your calculator. See Chapter 4, Frame 1.

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17__________________________________________________ Corollary 5.1.1: We can only measure rationals. We can also only produce rationals. Both processes must be finite. 18__________________________________________________ There is no absolute precision one can measure using mathematical real-numbers -- even with infinite digits. This is the lesson we learned with AES in cryptography, with modular arithmetic and FIF. The ancient Mayans (with modular arithmetic) and the Greeks (with gears, in the Antikythera Mechanism) used integer numbers, in exact astronomical predictions over millennia. 19__________________________________________________ In this book, one does not need to use mathematical real-numbers to do calculus, which would decrease rigor. Use the space below to write your notes.

20__________________________________________________ Use Theorem 5.1 to show Theorem 5.2: Theorem 5.2: d(c)/dx = 0, where c in Q,is a constant (does not change with x or y). The derivative of a constant (no change in both axes) is zero. Theorem 5.2 will be very useful for integration, in Chapter 6. 118

If you agree, Skip to Frame 22. Otherwise, write your comments, read on. 21__________________________________________________ Using Frame 50, ∂(s(x,y))/∂x = 0 if s(x,y) = cx, a constant in x. Likewise, ∂(s(x,y))/∂y = 0 if s(x,y) = cy, a constant in y. The addition is 0 + 0 = 0. 22__________________________________________________ Use Theorem 5.1 to show that d(x)/dx = 1, where x is in the set Q. The derivative of a line with a 45 degree inclination is 1. If you agree, Skip to Frame 24. Otherwise, use the space below to write your notes, and read on.

23__________________________________________________ Please, try with your calculator. 24__________________________________________________ Use Theorem 5.1 to show that: d(u ⋅ v)/dx = v ⋅ d(u)/dx + u ⋅ d(v)/dx,

(5.3)

where u = u(x), v = v(x), are functions of x, all defined in the rational numbers. If you agree, Skip to Frame 26. Otherwise, read on.

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25__________________________________________________ Please, write any comments. Try with your calculator, or use Theorem 5.1. 26__________________________________________________ Show that d(xn+ c)/dx = n ⋅ xn-1, where n, c, and x are in the set Q. If you agree, Skip to Frame 28. Otherwise, read on.

27__________________________________________________ Please, try with your calculator, or use Theorem 5.1. 28__________________________________________________ Show that d(A ⋅ ea⋅x + b)/dx = A ⋅ a ⋅ ea⋅x + b, where e is the Euler constant and A, a, b and x are in the set Q. If you agree, Skip to Frame 30. Otherwise, read on. 29__________________________________________________ Please, use the space below to try the formula. Try also with your calculator, or use Theorem 5.1.

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30__________________________________________________ Contrary to [1.4-6], one can now differentiate a discontinuous function. This is important for the mathematical formulation, because continuity is not assumed, so we allow new applications as in [7.1-10]. Also, contrary to [1.4-6], the symbol dy/dx means that the derivative is the ratio of two well-defined quantities, dy and dx. This is useful in many Science applications, and allows “click-mathematics” that works as Lego. When one needs to consider a change in voltage or time, for example, one can consider what changes can occur in a dependent variable. The change in dx, however, does not have to be small (small, compared to what?), according to Definition 5.1. 31__________________________________________________ If f(x) = x4 + 5, x in set Q, then the derivative of f(x) can be written in any of the equivalent forms: df(x)/dx = d(x4 + 5)/dx = d(x4)/dx + d(5)/dx. This, according to your study above, is 4 ⋅ x3. If you agree, Skip to Frame 33. Otherwise, use the space below to write your notes, and read on.

32__________________________________________________ Please, try with your calculator, or use Theorem 5.1. 33__________________________________________________ 121

Thus, d( )/dx means “differentiation with respect to x”, including changes in both axes (x and y), according to Eq.(5.3). We can use any function f(x,y) inside the parentheses. We can also use the definition many times operating on the same function. This leads us to second derivatives and multiple derivatives, written as: d2( )/dx2 and dn( )/dxn, mean, respectively,:

34__________________________________________________ The function |x| (module of x) seems to create a problem in conventional mathematics [1.4-6]. Its differential is discontinuous at x=0, and cannot be differentiated twice, where the graph on the left (done with the calculator) shows that the derivative of |x| have to deal with a discontinuity at 0. However, the discontinuity is a change, and creates no problem in this formulation. The function |x| can be differentiated twice. Write below the result of d2(|x|)/dx2. 35__________________________________________________ 122

If you obtain 0, a step-function with value 2 must be added at 0. Skip to Frame 36. Otherwise, try also with your calculator, or use Theorem 5.1. 36__________________________________________________ In the next page, find the graph of a test function y = f(x):

Sketch y’ in the space provided in this page, the derivative.

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37__________________________________________________ Here is the derivative of the test function, using the calculator we recommended. If your sketch is similar, Skip to Frame 38. Otherwise, read on.

To see that the plot of y’ above is reasonable, note that the function is flat in the middle, which is at 0 and has slope 0. The slope increases at each side, but is negative when x is positive, and actually the slope is positive when x is negative; it then reaches 0 on both sides. 38__________________________________________________ Using the calculator, solve d2( -sin(x) )/dx2 and plot the result.

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39__________________________________________________ The second derivative is sin(x). The plot of the result is given in Chapter 3, Frame 16. This can define the sine function: d2(y)/dx2 = -(y - y0). Verify graphically that this differential equation is obeyed both with y = sin(x) or cos(x). Fixating the initial condition of y = 0 when x = 0 (y0,), fixes the solution to only sin(x). 40__________________________________________________ Using the calculator, solve d2( 100x)/dx2. 41__________________________________________________ If you obtain 0, Skip to Frame 43. Otherwise, read on. 42__________________________________________________ If you disagree, or for more practice, repeat from Frame 1. 43__________________________________________________ Calculate the derivative of f(fx) = x⋅sin(x) + cos(x) + c 44__________________________________________________ The result is f’(x) = x⋅cos(x). If you get a different result, use your calculator. Write in Frame 45 your notes so far. This is important in the priming process of learning, as used in this book.

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45__________________________________________________

46__________________________________________________ INTERLUDE ____________________________________________________ We accomplished a lot in this chapter. All has been done using rational numbers, the set Q. By interpolating rational numbers, one can obtain continuously-looking plots without postulating microscopic continuity, or ghostly infinitesimals. No use of “small” errors or “negligible” error terms were assumed in the physical measurements either. Physically, not only rational numbers are the only numbers measurable, but they are also the only numbers produced. The derivative is always mathematically exact in the set Q, as with Legos that fit, providing absolute certainty, and faster execution. The absolute certainty can be verified also if one considers one or more points in the irrational numbers, considered in [1.4-5] to be somewhat “murky”, and that have been unnamed so far, qua some sort of “pariah” among the numbers. No approximation was used in the formalism. The calculator followed along, presenting nice graphs, also using only natural numbers -- that is all that hardware can do! In the next two Frames, you are noted to memorize just a few results, in order to cover most applications of differential calculus,

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next, preparing for Chapter 6, and developing more mathematical intuition for the relationships. 47__________________________________________________ The derivative to x, with y in the set Q as well as in the mathematical real-numbers: … of a constant is zero, … of a linear function with a 45 degree inclination is 1, … of a parabola is a linear function, … of xn is n ⋅ xn-1 … of A ⋅ ea ⋅ x + b is A ⋅ a ⋅ ea ⋅ x + b … of tx is tx ⋅ ln(t) … of log(n) is 1/n … of the sin(x) is cosin(x). … of the sin(ax + b) is a ⋅ cosin(ax + b). … of the cosin(x) is -sin(x). … the derivative (the second derivative) of -sin(x) is sin(x). If you agree with all these statements, keep them as your first shortcuts. They occur so often, that it is useful to remember them. If you disagree with any, calculate! You can use the previous Frames, or your calculator. Write your notes, for better priming. 48__________________________________________________ Derivative measures change. With no change, the derivative is zero. With large change, the derivative is large. To save space, u(x) and v(x) will be represented by u and v. Sum rule: d(u + v)/dx = du/dx + dv/dx 127

Product rule: d(u ⋅ v)/dx = u ⋅ d(v)/dx + v ⋅ du/dx = u ⋅ v’ + v ⋅ u’ Division rule: d(u/v)/dx = (v ⋅ u’ - u ⋅ v’)/v2 Chain rule: d(u(v))/dx = du/dv . dv/dx = du/dx Yes! Simple cancellation was used. If you agree with all these statements, keep them as your second list of shortcuts. If you disagree with any, calculate! You can use the previous Frames, or your calculator. You can also add to the list of shortcuts, according to your use, using the space below. 49__________________________________________________ Maxima and Minima ____________________________________________________ The heart of differential calculus is given by Maxima / Minima problems. To interest students, and to peak one’s curiosity, these problems solve within absolute accuracy what algebra would need trial-and-error, and arrive at relative accuracy. Thus, they represent what algebra cannot calculate. But, with differential calculus, one first looks for one condition: dy/dx = 0. Maxima / Minima problem #1: Find out what figure maximizes area, for a given perimeter. This problem is important, for example, to design a submarine, or buy a tent. The solution is a circle. The student can calculate, or search. Maxima / Minima problem #2: Find out what figure mimizes use of material, in dividing a space. The solution is a hexagon, and bees use it in their hives (honeycomb). The student can calculate, or search.

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Maxima / Minima problem #3: Find out the angle one should throw a stone, to reach maximum distance. This problem is important in basketball and football. The student can calculate, or search. The solution will be given in the last Frame of this Chapter. 50__________________________________________________ Differential Forms and Partial Derivatives ____________________________________________________ An equation that involves the derivative of a function is called a differential equation. Using Fig (5.1) with Laplace transforms, one can simplify a differential equation down to an algebra problem. Solving the algebra problem, one is led to the inverse Laplace transform, to obtain the solved differential equation. Thus, Laplace transforms and their inverse play a crucial role in solving differential equations, which is very useful to engineering. Then, you will be using “click-mathematics”, like Lego, just assembling parts that fit. Anything constructed can be taken apart again, and the pieces reused to make new things. Here, one treats dy and dx separately. This formalism allows this, but baffles [1.4-5]. The notation y’ does not help. The notation dy/dx leads to a simple rearrangement of the total differential, to reach differential forms, with partial differentials. The first rule is simple and self explanatory: dy = (dy/dx) ⋅ dx With this formula, we can treat dx as an independent variable, something we can control, and we can calculate dy. The quantity dx is usually small, but that is not necessary. 129

One of the important uses in mathematics is in defining exponential, logarithmic, and trigonometry functions. They can be introduced with easy absolute accuracy using differential forms and equations. Another important use is, in the definition of a differential in Frame 11, when mathematical real-numbers are hypothesized, one has to account for mutually-dependent x and y variations. We expect that because one needs to measure the set R. This includes points that belong to irrational numbers, although they do not belong to the set Q. This is an approximation that may not include coordinates described by only one pair (x,y), but two pairs. One can write that difference as a linear combination of first-order differential forms, as d(s(x,y)) = ∂(s(x,y))/∂x⋅dx + ∂(s(x,y))/∂y⋅dy

(5.3)

where ∂( )/∂x is called a “partial derivative of x”. This operation is useful when two or more independent variables are required to define a function, as in s(x,y). Then, we can consider all independent variables fixed, except one. The symbol ∂( )/∂x represents the “partial derivative” in that case, of x.

51__________________________________________________ “Click” Mathematics ____________________________________________________ Important to physics, biology, and engineering, we can use Hooke’s law. If one considers a small enough compression (expansion) of a spring, the restoring force is proportional and opposite, the spring expands (compresses). This can be used to 130

model blood vessels, lungs, pipes, soil, tires, chords for music, and more. We write, in differential form: dF= - k ⋅ dx, where dF is the force differential, k is Hooke’s constant, and dx is the movement. Write Newton’s law in differential form: ________________________________________ (Hint: F = M ⋅ a is the ordinary form) 52__________________________________________________ The result is: dF = M ⋅ d2x/dt2, where dF is the force differential, M is the inertia, and d2x/dt2 is the acceleration as the second derivative of position in regard to time. Now, algebraically, like Lego, calculate the oscillatory movement that results from the use of Hooke’s law and the Newton’s law, as the equation of motion of a mass on a sliding, horizontal support (no friction or gravity), shown in the figure below:

__________________________________ 131

53__________________________________________________ The result is: d2x/dt2 = -(k/M) ⋅ dx where x (the position) is a trigonometric (sine/cosine) oscillating function within well-defined limits. This equation of motion is very important in physics, biology, music, and engineering, and is called a “harmonic oscillator”. The solution of this equation of motion is discussed in Frame 39, helping understand how “click-mathematics” work, like Lego. Calculus becomes like Lego, with differential forms. Anything constructed can be taken apart again, and the pieces reused to make new things. 54__________________________________________________ The solution to the third question in Frame 49, without taking into account air resistance or wind, is 45 degrees. Next, go to Chapter 6.

____________________________________________________ REFERENCES [5.1] G. Birkhoff and S. MacLane. A Survey of Modern Algebra, 5th ed. New York: Macmillan, 1996. [5.2] George B. Arfken. Mathematical Methods For Physicists. Elsevier Academic Press, 2005.

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Chapter 6: Integral Calculus Integration can become easy with the 3 QP realization (Chapter 2). We only have to consider 0-dimensional isolated points, with no profile to approximate, and no error terms. An Integral becomes even easier in this book, and already done in Chapter 5, by considering the Integral to form an antidifferentiation pair (i.e., the inverse function of differentiation). That is why one should start with differentiation in Chapter 5. The pairs derivative/Integral are then easier to define, with no resort to continuity, infinitesimals, or even calculations. This book provides the same known formulas as in [1.4-5], with less work, with no illusions that are unphysical, and with less assumptions. The analog, continuous signal on the left, represents what researchers used to want to measure in the 17/18th century, in calculating area. Now, in the 21st century, the interest is in the discrete signal, also represented above. 133

One recognizes that the discrete signal is the actual cause of the interpolated analog signal, which seems continuous. The analog signal is now mythical, includes measurement errors in the x-axis and in the y-axis, and takes into account the recipient as well as the environment, while the discrete signal is more likely what is produced. The discrete signal represents a digital signal and consists of a sequence of samples, which are integers: 4, 5, 4, 3, 4, 6…, at integer values. They could also be rationals, at rational values. We can only measure rationals. We can also only produce rationals. See Corollary 5.1.1. 1___________________________________________________ The Figure above means that the “mean value theorem” [1.4-5] has a flaw, the "consolation" is that it is true in Universality, when continuity with mathematical real-numbers can be assumed. We will not use the mean value theorem. As we concluded in Corollary 5.1.1 -- there is no absolute precision one can measure in nature, using mathematical real-numbers. One works better by using Q, with more precision and less computational time, less memory, and less assumptions. All expressions fit with one another, in such “click-mathematics”. To measure the area of the digital signal is easy: the height of each signal value is added, for the duration of the signal, as a square function. But there is a much easier route at our disposal, in calculating the areas of figures, which can be easily expanded to volume, hyper-volume, and more.

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2___________________________________________________ The discovery of QM made not only the signals cease to be considered continuous in the 21/st century, but the measurement techniques evolved, from a mythical and ghostly analog to digital. One of the forces driving this evolution were Computers, unknown in the 17/18th century. Computers work with hardware using natural numbers only, including coprocessors for simulating mathematical real-numbers. Many early students (as ourselves) used Computers to make calculations “as precisely as possible”, using double-precision mathematical real-numbers and mathematical decimal complex numbers in their programs. They would use lots and lots of computer time, and see phantom mathematical decimal complex numbers in the output. But they soon realized that Computers were always making their calculations using only natural numbers, and they were wasting time, and memory, emulating double-precision mathematical real-numbers as well as mathematical decimal complex numbers, running into errors even when using coprocessors, while only achieving less precision. Not rigorous and wasteful! The advent of the FFT announced a new era in computation, offering orders of magnitude improvement in speed, and deprecating classical ways to calculate the Fourier Transform, no longer requiring mathematical real-numbers and mathematical decimal complex numbers. The FFT is done using only integers. The AES just re-affirmed this new era of computation, of rigor and speed, as the basis of cryptography, using GF(28) as a FIF. 135

We could see the “algebraic approach”, used in this book, emerge from these early results [7.2]. We have been using this approach since 1975, and many researchers have been involved in this realization. 3___________________________________________________ This book follows this advancement, and already allows differential calculus to be done only with rational numbers, using Computers as they work -- digitally, in “click-mathematics”. Continuity can be obtained macroscopically, using interpolation at the final stage, but is not based on a mythical, ghostly, and artificial microscopic continuity for justification. Integral calculus is obtained in this book, also not by following the same Q path, which is also open to us … but would take more time. We are following in this book the shortest shortcut possible. Instead of “enjoying the journey”, we will reach the goal sooner -- and enjoy the journey more, sooner. The shortcut is: all the formulas derived in Chapter 5, can now be reversed, to find the Integral plus a constant. Chapter 5, establishes that the derivative of a 2D constant is zero. The next Frames will formalize this concept toward the Integral. 4___________________________________________________ 6.1. Definition of antiderivative, primitive. A function is called a primitive P (or, an antiderivative) of a function f on an interval I if the derivative of P is f in the interval I. 136

Which options are true within the [first, second] choices in the next phrase? A [sine, log] function is a primitive of the [cosine, x2] function in every interval because the derivative of the [sine, log] is the [cosine, x2]. Skip to Frame 6 if you choose the first option, proceed if you choose the second option. 5___________________________________________________ The first option is consistent. A primitive of the cosine is a sine function, because the derivative of a sine is the cosine in every interval. 6___________________________________________________ We speak of a primitive, rather than the primitive, because if P is a primitive of f, so is every function P + c, where c is a constant. Conversely, any two primitives P and Q of the same function f, can differ only by a constant. This is because their difference P - Q has the derivative 0. If you agree, skip to Frame 8, proceed otherwise. 7___________________________________________________ Calculate P’ - Q’ = f(x) - f(x) = 0, for every x in the interval I, so by Theorem 5.3, P - Q is a 2D constant in the interval I, with derivative 0 therefore. 137

8___________________________________________________ Every rectangle is a measurable function in the 2D plane; every step function is measurable, and its total area is the sum of areas of its rectangular pieces. The first fundamental theorem of calculus [1.5], says that one can always construct a primitive by Integration of a measurable function, whereas you can observe that continuity is not required. 9___________________________________________________ The properties of the Integral have a geometric interpretation in terms of area, and can use set theory (Chapter 3). The first property is the Additive Property. It means that the sum of two areas is the resulting area. Area is additive (volume is also). This is written as: b

b

b

∫(f(x) + g(x))⋅dx = ∫(f(x) + ∫g(x))⋅dx a

a

(6.1)

a

That is why we can say that the “total area is the sum of areas of its rectangular pieces”. If you agree, Skip to Frame 11. Otherwise, read on. 10__________________________________________________ Eq.(6.1) can be seen as the Additive Property in set theory, and corresponds to the union of A and B as C (C = A U B). This can be shown in a Venn diagram.

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11__________________________________________________ The cartesian construction in 2D also represents C = A U B. Descartes made an important connection between geometry and algebra with the cartesian construction, which is well-known to have been pioneered by Omar Khayyam in Persia, when solving the cubic polynomial. And the Pythagorean Theorem, that satisfies both the cartesian construction in 2D, and area sums (Chapter 2, Frame 9), leads to the second fundamental theorem of calculus [1.5]. The same separation c (the derivative) can be calculated by the Additive Property of the Integral or by Eq.(5.1) or, as commonly stated: x

f(x) = f(a) + ∫df(t)/dt)⋅dt = f(a) + [f(x) - f(a)]

(6.2)

a

The theorem shown in Eq.(6.2) means that the problem of evaluating an Integral is transferred to another problem -- that of finding a primitive P of f, which problem we already know how to solve, by observation of the derivative leading to f. We learn by observation, following the method by Pestalozzi. Integral calculus becomes child play! 12__________________________________________________ If you agree, skip to 13. If not, read Chapter 5 Frame 6, first. Write your notes below, for priming.

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13__________________________________________________ In practice, the second problem is a lot easier to deal with than the first. Every differentiation formula, when read in reverse, gives us, by simple observation, an example of a primitive of a function f, and this, in turn, must lead to an Integration formula for this function, plus a constant. 14__________________________________________________ Try for the differentiation formulas derived in Chapter 5. One can construct the Integration formulas as causes of differentiation, with d[P(x) + c]/dx = f(x), finding P(x) for a given f(x). 15__________________________________________________ Shortcut: Use your calculator to observe that the function P(x) = xn+1/(n+1) + c has the derivative xn, where n is any rational number different from n = -1 (for the existence of P(x)). This means that you know the integral just by observation of the derivative. 16__________________________________________________ Revert some cases in Chapter 5, creating a table of Integrals, and list them next. Do not forget the added constant -- it matters.

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17__________________________________________________ To calculate a definite Integral, we just use the end-points of the interval I. 6.2. Definition of a Definite Integral. b

∫f(x)⋅dx = ∫f(x)⋅dx, where I is defined by [a, b], (a, b), etc. xεI

a

We can use open intervals as (a, b), representing a < x < b, or mixed intervals as [a, b), representing a