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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
