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Quickest Calculus:

Class Use

First Edition, v6.1

Ann Vogel Gerck, B.A. Ed Gerck, Ph.D.

Founder CEO & Founder

Planalto Research

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

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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

researchplanalto@gmail.com

Gerck, Ann V. and Gerck, Ed.

Quickest Calculus: A Self-Study Guide With New Applications.

2

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 researchplanalto@gmail.com

3

PREFACE

For advanced students, the mathematical theory has been

published, and is available without cost at

https://www.mdpi.com/2227-7390/11/1/68

This is an edition for class use, mainly with mathematical subjects.

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

4

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

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.

Mathematics becomes what we call “click-mathematics”. It works

like Lego, just assembling parts that fit exactly. There is no longer

any error term. Here, one can treat dy and dx separately. This

formalism allows this, but baffles conventional authors, such as

Courant and Apostol. You will learn Integral calculus just by

observation, not numerical calculation. Discontinuous functions

can now be differentiated. New applications appear.

There is no need for physical motivation, or examples. Physics

becomes a consequence.

But technical language deceives us.

5

The process of “steam cleaning” has no steam, just hot water. In

mathematics, to think about calculus is to take the position that

numbers, of whatever type, are used as a way to describe

observations of 'nature'. Whether that means numbers, of

whatever type, "exist in nature", depends on what is meant by

"exist in nature". This is seen as a solipsistic position, not seeking

objectivity, not being even intersubjective.

Measurement theory, in mathematics, is also not what it means in

physics. Both measurement and nature have a different “natural”

definition in physics, which this book follows, guided by

experiments.

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.

The value e^(-π/2) is found in physics, which is the same value for

i^i, which validates ias having a physical existence, thus

objective.

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.

6

Vectors, multivectors, and complex numbers are not used in this

book. This book uses numbers as scalar quantities only. This can

be easily extended to multivectors and complex numbers 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

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.

7

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

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

8

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."

9

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.

10

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.

11

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.

12

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].

14

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

15

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

16

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.

17

● 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.

18

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. Put this study on your calendar.

2. Read the preface, or another text, to inspire you.

3. Write your questions and goals.

4. Visualize reaching your goals.

5. 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.

19

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.

20

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.

21

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”.

22

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

24

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.

25

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

26

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.

27

More will be available over time, email to

researchplanalto@gmail.com

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.

30

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.

31

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 2nnatively. 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.

33

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.

34

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.

36

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.

38

[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.

40

[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.

41

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

42

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.

43

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.

44

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.

45

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.

47

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 aand bin

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 cin the set Q, while measuring

with the same set Q in each axis, aand 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.

49

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 qand nin 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.

50

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:

51

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.

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

√2, which is an irrational number -- which now we know, in this

book, to be exact and simple, as well as √-1.)

52

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.

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

54

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 Sdenote the set of points xfor which a limit

sequence converges.

The function fdefined on Sis called the limit function of the

sequence fnand one [1.5] says that fnhas “pointwise

convergence” to fon set S, although the width > O, which is

shown in Table 1.

55

By the definition of a limit sequence [1.5], this means that for each

xin Sand for each ε > 0 there is an integer N, which may depend

on both xand 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.

56

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.

58

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.

59

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.

60

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.

63

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.

Aset 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.

64

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

65

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.

66

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.

Afunction 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 xis one of the elements in set A, then the element in set B that

the function fassociates with xis denoted by the symbol f (x).

[This symbol f (x) is the value of fat x. It is usually read as "fof

x."]

67

The set A is called the domain of the function. The set Bof all

possible values of f(x) as xvaries 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 fat

g(x). It is usually read as "fof g of x."]

Write below, the expression fog(x) where g(x) = x2and 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.

68

10__________________________________________________

Recall that the product of two negative numbers is positive. Thus

for any integer value of x, positive or negative, x2is positive and a

square. When xis 0, x2is also 0. Therefore, the range of f(x) =

x2is 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.

69

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) = √xRange =

_________________________

(b) f(x) = 1/xRange =

_________________________

Check your answers in Frame 12.

12__________________________________________________

f(x) is an yet unspecified number for xa 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).

70

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.

71

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/xis defined for all values of xa 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.

72

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 xis the independent variable and yis the dependent

variable.

14__________________________________________________

In mathematics the symbol xfrequently represents an

independent variable, foften 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 "zequals H

of u". Here uis the independent variable, zis the dependent

variable, and H is the function.

The formula s = W(t) is valid, where tis the independent variable,

sis 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.

74

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 ax = y ; ay= x

The number ais often called the base. When ais 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.

75

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.

76

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

77

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.

78

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, yand xcan 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.

79

Use the axes above, to mark the points A = (3,5) and B = (5,3).

Check in the image above.

Let arepresent some other particular value for the independent

variable x, and let bindicate 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 Afrom 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 xand yrespectively.

The number 2is called the x-coordinate of the point marked as A,

and the number 3is 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 xand 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 yis 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.

82

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 ykalso necessarily discrete, isolated, as one and only one

value, yk= f(xk), with xkand k∈ N, and one has a sequence of

related points in xkand 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).

83

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.

84

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.

85

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

87

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.

89

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

91

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.

92

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,

93

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.

95

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.

97

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).

98

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.

99

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.

100

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.

102

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.

104

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].

105

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

106

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.

107

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.

108

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 = (a x, 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.

110

5___________________________________________________

Because A = (a x, ay) and B = (bx, by) are in cartesian coordinates,

the change between A and B in the x-axis is a x- 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)

111

The numbers A and B, belong to the set Q, while a x, 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.

112

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)2and 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.

113

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 ccannot 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

114

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:

115

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.

117

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.

119

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 ⋅ e a⋅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.

120

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( )/dx2and 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.

123

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) )/dx2and plot the result.

124

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.

125

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,

126

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 xnis n ⋅ xn-1

… of A ⋅ e a ⋅ x + b is A ⋅ a ⋅ e a ⋅ x + b

… of txis 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.

128

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/dt2is 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.

132

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.

134

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 fon an interval Iif

the derivative of Pis fin 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 aprimitive, rather than the primitive, because if Pis a

primitive of f, so is every function P + c, where cis a constant.

Conversely, any two primitives Pand Qof 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 (6.1)

a a 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.

138

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 Pof 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.

139

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.

140

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 <= x < b.

18__________________________________________________

Calculate:

1

∫xn⋅dx.

0

19__________________________________________________

Using the calculator or the shortcut in Frame 15, we can calculate:

1

∫xn⋅dx = 1/(n+1), where n ≠ -1.

0

If that is right, go to Frame 21. Otherwise, calculate yourself, using

Definition 5.1.

20__________________________________________________

Using the calculator, calculate:

10

∫√x⋅dx, and plot the area under the curve (the Integral).

0

141

21__________________________________________________

The result is 20⋅√(10)/3. The graph of the area is given above. In

spite of using an irrational number under the Integral sign, this is

an exact result. Other books by Planalto Research will consider

this, also in Laplace and Fourier transforms, and in QM, and in a

revisitation of the Heisenberg Principle.

22__________________________________________________

Integrate ∫ x3⋅cos(x)⋅dx.

Use this space to show your work.

142

23__________________________________________________

Using the calculator, the result is x3sin(x) + 3 x2cos(x) - 6 x sin(x)

- 6 cos(x) + c. You may not omit the constant c.

The graph is below, and is highly oscillatory.

24__________________________________________________

Calculate the following cases:

(1) ∫u⋅sin(u)⋅du

(2) ∫a⋅dw

(3) ∫√5\t ⋅ dt

(4) ∫v3⋅dv

(5) ∫ln x⋅dx

143

Using the calculator or the shortcut in Frame 15, we can calculate

these results. Otherwise, calculate yourself, using Definition 5.1.

When all are done, go to Frame 25 and verify.

25__________________________________________________

The results are:

(1) -u⋅cos(u) + sin(u) + c

(2) a⋅w + c

(3) √5⋅ln(|t|) + c

(4) 0.25⋅v4+ c

(5) x⋅ln(x) + c

26__________________________________________________

An Integration method favored by Richard Feynman, is called

“differentiation under the Integral sign”, also called “the Leibniz

Integral rule”.

This is an operation in calculus used to evaluate certain Integrals.

Under fairly loose conditions on the function being integrated,

differentiation under the Integral sign allows one to interchange

the order of Integration and differentiation. In its simplest form,

called the Leibniz Integral rule, differentiation under the Integral

sign makes the following equation valid under light assumptions

on f. Many Integrals that would otherwise be impossible or require

significantly more complex methods can be solved by this

approach.

b b

d( ) .∫ f(x,t)dt = ∫ ∂( ) f(x,t)dt,

dx a a ∂x

144

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. Then, we can consider all independent variables

fixed, except one. The symbol ∂( )/∂x represents the “partial

derivative” in that case, of x, as seen Chapter 5, Frame 50.

In general, the change is due to all the independent variables,

however.

27__________________________________________________

Compute the definite Integral, and graph the Integral.

1

∫(x3- 1)/ln(x)⋅dx

0

Use the calculator and try the method in Frame 26.

28__________________________________________________

Using the calculator, the result is 0.16989904… The graph of the

Integral is in the next page.

To calculate using the method “differentiation under the Integral

sign” of Frame 26, follow this hint:

The appearance of 1/ln(x) in the denominator of the integrand is

quite unwelcome, and we would like to get rid of it. Thankfully, we

know that dtx/dx = tx⋅ln(t) -- see Chapter 5, so differentiating the

numerator with respect to the exponent seems to be what we'd

like to do.

145

Follow the hint, and calculate.

29__________________________________________________

Following the hint, we define a function:

1

g(x) = ∫(tx- 1)/ln(x)⋅dx

0

In this notation, the Integral we wish to evaluate is g(3). Observe

that the given Integral has been recast as a member of a family of

definite Integrals g(x) indexed by the variable x.

It follows that g(x)=ln ∣x+1∣+c, for some constant c. To determine c,

note that g(0)=0, so 0 = g(0) = ln 1+c = g(0).

146

Hence, g(x)=ln∣x+1∣ for all x. such that the Integral exists. In

particular, g(3)=ln 4 = 2⋅ln 2.

30__________________________________________________

The method “Integration by parts” is a special method of

Integration that is often useful when two functions are multiplied

together, but is also helpful in other ways. It uses the result of

Chapter 5, Frame 43, called “Product Rule”.

Calculate the following Integral: ∫x⋅cos(x)dx

31__________________________________________________

The result, using the calculator or “Integration by parts” is:

x⋅sin(x) + cos(x) + c. We saw this result in Chapter 5.

32__________________________________________________

Using your calculator, or your formulas, calculate the following

Integral:

Check the answer on your calculator.

147

33__________________________________________________

Using your calculator, or your formulas, calculate the following

Integral:

34__________________________________________________

The result, using the calculator or formulas, is: 1. Which way was

easier?

For advanced students, the mathematical theory has been

published, and is available without cost at

https://www.mdpi.com/2227-7390/11/1/68

THE END.

____________________________________________________

REFERENCES

See all Chapters.

Add your own references:

148

ACKNOWLEDGEMENTS

This book benefitted from an earlier version, titled “Mathematics

Without Incidents”, now being recalled, and previous editions.

Many thanks to our early readers, ResearchGate, and LinkedIn,

as well as other online forums, in this time of increased isolation in

a pandemic. A special thanks to “devil’s advocates”, that have

been particularly useful as a litmus test.

Readers can comment further, by letter to our postal address or

by email to the authors, at researchplanalto@gmail.com

Questions may also be answered, in support of self-study. You

can use the next page to write a draft of your questions.

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

ending this book. Other books by Planalto Research will follow,

you are invited to write to researchplanalto@gmail.com

149