Table of Contents

Table of Contents

“The Book of Mathematics: Volume 1”

INTRODUCTION

FIRST PART: ELEMENTARY MATHEMATICS

ELEMENTARY MATHEMATICAL LOGIC

ELEMENTARY ARITHMETIC OPERATIONS

SET THEORY

LITERAL CALCULATION

PLANE EUCLIDEAN GEOMETRY

SOLID EUCLIDEAN GEOMETRY

ALGEBRAIC EQUATIONS AND INEQUATIONS

ELEMENTARY ANALYTICAL GEOMETRY

GONIOMETRIC FUNCTIONS AND TRIGONOMETRY

EXPONENTIAL, LOGARITHMIC AND HYPERBOLIC FUNCTIONS

FUNCTION THEORY

COMPLEX NUMBERS

SECOND PART:  MATHEMATICAL ANALYSIS, FUNCTIONAL ANALYSIS AND ADVANCED GEOMETRY

GENERAL TOPOLOGY

LIMITS

CONTINUOUS FUNCTIONS

DIFFERENTIAL CALCULATION

INTEGRAL CALCULATION

STUDY OF REAL VARIABLE FUNCTIONS

SUCCESSION AND NUMERICAL SERIES

SUCCESSION AND SERIES OF FUNCTIONS

POWER, TAYLOR AND FOURIER SERIES

VECTORS AND VECTOR MATHEMATICS

MATRICES AND MATRIX MATHEMATICS

ADVANCED ANALYTICAL GEOMETRY

NON EUCLIDEAN GEOMETRY

“The Book of Mathematics: Volume 1”

“The Book of Mathematics: Volume 1”

SIMONE MALACRIDA

Most of mathematics is presented in this book, starting from the basic and elementary concepts to probing the more complex and advanced areas.

Mathematics is approached both from a theoretical point of view, expounding theorems and definitions of each particular type, and on a practical level, going on to solve more than 1,000 exercises.

The approach to mathematics is given by progressive knowledge, exposing the various chapters in a logical order so that the reader can build a continuous path in the study of that science.

The entire book is divided into three distinct sections: elementary mathematics, the advanced mathematics given by analysis and geometry, and finally the part concerning statistics, algebra and logic.

The writing stands as an all-inclusive work concerning mathematics, leaving out no aspect of the many facets it can take on.

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

ANALYTICAL INDEX

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INTRODUCTION

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FIRST PART: ELEMENTATY MATHEMATICS

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1 – ELEMENTARY MATHEMATICAL LOGIC

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2 – ELEMENTARY ARITHMETIC OPERATIONS

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3 – SET THEORY

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4 – LITERAL CALCULATION

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5 – PLANE EUCLIDEAN GEOMETRY

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6 – SOLID EUCLIDEAN GEOMETRY

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7 – ALGEBRAIC EQUATIONS AND INEQUATIONS

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8 – ELEMENTARY ANALYTICAL GEOMETRY

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9 – GONIOMETRIC FUNCTIONS AND TRIGONOMETRY

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10 – EXPONENTIAL, LOGARITHMICAL AND HYPERBOLIC FUNCTIONS

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11 – FUNCTION THEORY

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12 – COMPLEX NUMBERS

SECOND PART: MATHEMATICAL ANALYSIS, FUNCTIONAL ANALYSIS AND ADVANCED GEOMETRY

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13 – GENERAL TOPOLOGY

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14 - LIMITS

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15 – CONTINUOUS FUNCTIONS

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16 – DIFFERENTIAL CALCULATION

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17- INTEGRAL CALCULATION

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18 – STUDY OF REAL VARIABLE FUNCTIONS

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19 – SUCCESSION AND NUMERICAL SERIES

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20 – SUCCESSION AND SERIES OF FUNCTIONS

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21 – POWER, TAYLOR AND FOURIER SERIES

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22 – VECTORS AND VECTOR MATHEMATICS

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23 – MATRICES AND MATRIX MATHEMATICS

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24 – ADVANCED ANALYTICAL GEOMETRY

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25 – NON EUCLIDEAN GEOMETRY

INTRODUCTION

INTRODUCTION

In today's society, mathematics is the basis of most scientific and technical disciplines such as physics, chemistry, engineering of all sectors, astronomy, economics, medicine, architecture.

Furthermore, mathematical models govern everyday life, for example in the transport sector, in energy management and distribution, in telephone and television communications, in weather forecasting, in the planning of agricultural production and in waste management, in definition of monetary flows, in the codification of industrial plans and so on, since the practical applications are almost infinite.

Therefore mathematics is one of the fundamental foundations for the formation of a contemporary culture of every single individual and it is clear both from the school programs that introduce, from the earliest years, the teaching of mathematics and from the close relationship between the profitable learning of mathematics and the social and economic development of a society.

This trend is not new, as it is a direct consequence of that revolution which took place at the beginning of the seventeenth century which introduced the scientific method as the main tool for describing Nature and whose starting point was precisely given by the consideration that mathematics could be the keystone to understand what surrounds us.

The great "strength" of mathematics lies in at least three distinct points.

First of all, thanks to it it is possible to describe reality in scientific terms, that is by foreseeing some results even before having the real experience.

Predicting results also means predicting the uncertainties, errors and statistics that necessarily arise when the ideal of theory is brought into the most extreme practice.

Second, mathematics is a language that has unique properties.

It is artificial, as built by human beings.

There are other artificial languages, such as the Morse alphabet; but the great difference of mathematics is that it is an artificial language that describes Nature and its physical, chemical and biological properties.

This makes it superior to any other possible language, as we speak the same language as the Universe and its laws. At this juncture, each of us can bring in our own ideologies or beliefs, whether secular or religious.

Many thinkers have highlighted how God is a great mathematician and how mathematics is the preferred language to communicate with this superior entity.

The last property of mathematics is that it is a universal language. In mathematical terms, the Tower of Babel could not exist.

Every human being who has some rudiments of mathematics knows very well what is meant by some specific symbols, while translators and dictionaries are needed to understand each other with written words or oral speeches.

We know very well that language is the basis of all knowledge.

The human being learns, in the first years of life, a series of basic information for the development of intelligence, precisely through language.

The human brain is distinguished precisely by this specific peculiarity of articulating a series of complex languages and this has given us all the well-known advantages over any other species of the animal kingdom.

Language is also one of the presuppositions of philosophical, speculative and scientific knowledge and Gadamer has highlighted this, unequivocally and definitively.

But there is a third property of mathematics which is far more important.

In addition to being an artificial and universal language that describes Nature, mathematics is properly problem solving , therefore it is concreteness made science, as man has always aimed at solving problems that afflict him.

To remove the last doubts on the matter, it is advisable to report some concrete examples referring to millennia ago.

The discovery of irrational numbers made by Pythagoras, above all pi and the square root, was not a mere theoretical speculation.

At the basis of that mathematical symbolism there was the resolution of two very concrete problems.

On the one hand, since the houses had a square plan, the internal diagonal had to be calculated exactly in order to minimize the material wasted in the construction of the walls, on the other, pi was the mathematical link between straight and curvilinear distances, such as the radius of a wheel and its circumference.

Faced with concrete problems, the human intellect has invented this mathematical language whose property is precisely that of solving problems by describing Nature.

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The first part of this book has the express purpose of providing the rudiments of elementary mathematics, that is, of all that part of mathematics prior to the introduction of mathematical analysis.

The notions and concepts exposed in this part were, in part, already known in antiquity (at the time of the Greeks for example), especially as regards the part of elementary logic, together with elementary operations and geometric relationships.

The remaining chapters of the first part describe the knowledge acquired by humanity over the centuries, in particular after the great explosion of thought that occurred in the Renaissance, up to the end of the seventeenth century.

This limit is considered as a demarcation between elementary and advanced mathematics, precisely because mathematical analysis, introduced at the end of the seventeenth century by Newton and Leibnitz, allowed the qualitative leap towards new horizons and towards the real description of Nature in mathematical terms.

Precisely for this reason, although each paragraph constitutes a complete topic in itself, the exposition of the topics follows a logical order, allowing the continuous progression of knowledge based on what has been learned previously.

The first part of the book coincides, more or less, with what was taught until the end of high school (only for scientific high schools, with the end of the fourth year and not the fifth).

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The second part of the book provides all the foundations of advanced mathematics, encompassing in it both the great discipline of mathematical analysis and all the disparate fields that have arisen over the last two centuries, including, to mention but a few of them, the differential and fractal geometry, non-Euclidean geometries, algebraic topology and functional analysis.

Almost all of these notions were developed after the introduction of the formalism of mathematical analysis at the end of the seventeenth century and, since then, the path of mathematics has always continued in parallel between this sector and all the other possible sub-disciplines that gradually side by side and have taken independent paths.

It remains to understand why mathematical analysis has introduced that watershed between elementary and advanced mathematics.

There are two areas that complement each other in this discourse.

On the one hand, only with the introduction of mathematical analysis has it been possible to describe, with a suitable formalism, the equations that govern natural phenomena, be they physical, chemical or of other extraction, for example social or economic.

In other words, mathematical analysis is the main tool for building those mechanisms that allow us to predict results, to design technologies and to think about new improvements to introduce.

On the other hand, mathematical analysis possesses, within its very nature, a specific peculiarity which clearly distinguishes it from the previous elementary mathematics.

It provides for local considerations, not exclusively punctual.

Just the passage from punctuality to locality will allow to build a discourse of globality, going far beyond the previous knowable.

This part presents concepts usually addressed at the university level in various analysis and geometry courses.

In the third part of the book topics of general interest that can be separated from mathematical analysis will be exposed, such as advanced algebra, statistics and advanced logic.

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Each single chapter of the book can be considered as a complete field of mathematics in itself, but only by analyzing all the topics it will be possible to touch the vastness of mathematics and it is for this reason that the order of the chapters reflects a continuous succession of knowledge to progress.

In fact, mathematics has an almost unlimited breadth of sectors and applications.

There is no science that can do without mathematical concepts and there is no application that has not borrowed mathematical notions and made them evolve with particular languages.

This is how many disciplines and many theories not presented in this book were born, citing just a few examples we can include game theory and financial mathematics in the economic field, the applications of group theory and advanced algebra for theoretical physics and elementary particles, the evolution of tensor calculus for problems in cosmology and astrophysics.

For this reason, this book, although very vast, is certainly not complete and all-encompassing.

There are over 1,000 exercises done, but the number of possible problems and exercises is almost unlimited.

Furthermore, in the whole book there are no proofs of theorems that would have further burdened the bulkiness and understanding.

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The evolution of mathematics applied to individual disciplines and technologies has led to extreme ramifications and continuous evolution that continues even today.

This has an important consequence: mathematics is a "living", contemporary and future science and is not relegated to a historical role.

What has been said does not apply only to the countless applications, but also to "pure" mathematics, i.e. to the mathematical problems presented in this manual.

Making a historicism about the notions and the results expressed, one could clearly see how some assumptions and some demonstrations are very recent (an example above all is the demonstration of the Poincaré conjecture) that is, they took place in the Twenty-First Century.

It is no coincidence that there are prizes for solving problems that are still open and that are both historic, such as Hilbert's famous questions from the early twentieth century, and very modern in relation to computational calculation, logic, complexity and chaos theory, as well as geometric and algebraic concepts.

Being a living science, just like a universal language, mathematics is continuously enriched with new words and new constructs and that is why what is presented in this book is only a stepping stone towards even more advanced and specific knowledge.

Taking up the challenge of writing a new chapter or a single chapter in this compelling story of the only universal artificial language that describes Nature is part of the evolution of our species and that is why each of us is called to participate in it.

FIRST PART: ELEMENTARY MATHEMATICS

FIRST PART: ELEMENTARY MATHEMATICS

1

ELEMENTARY MATHEMATICAL LOGIC

ELEMENTARY MATHEMATICAL LOGIC

Introduction

Mathematical logic deals with the coding, in mathematical terms, of intuitive concepts related to human reasoning.

It is the starting point for any mathematical learning process and, therefore, it makes complete sense to expose the elementary rules of this logic at the beginning of the whole discourse.

We define an axiom as a statement assumed to be true because it is considered self-evident or because it is the starting point of a theory.

Logical axioms are satisfied by any logical structure and are divided into tautologies (true statements by definition devoid of new informative value) or axioms considered true regardless, unable to demonstrate their universal validity.

Non-logical axioms are never tautologies and are called postulates.

Both axioms and postulates are unprovable.

Generally, the axioms that found and start a theory are called principles.

A theorem, on the other hand, is a proposition which, starting from initial conditions (called hypotheses) reaches conclusions (called theses) through a logical procedure called demonstration.

Theorems are, therefore, provable by definition.

Other provable statements are the lemmas which usually precede and give the basis of a theorem and the corollaries which, instead, are consequent to the demonstration of a given theorem.

A conjecture, on the other hand, is a proposition believed to be true thanks to general considerations, intuition and common sense, but not yet demonstrated in the form of a theorem.

Symbology

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Mathematical logic causes symbols to intervene which will then return in all the individual fields of mathematics. These symbols are varied and belong to different categories.

The equality between two mathematical elements is indicated with the symbol of , if instead these elements are different from each other the symbol of inequality is given by .

In the geometric field it is also useful to introduce the concept of congruence, indicated in this way and of similarity .

In mathematics, proportionality can also be defined, denoted by .

In many cases mathematical and geometric concepts must be defined, the definition symbol is this .

Finally, the negation is given by a bar above the logical concept.

Then there are quantitative logical symbols which correspond to linguistic concepts. The existence of an element is indicated thus , the uniqueness of the element thus , while the phrase "for each element" is transcribed thus .

Other symbols refer to ordering logics, i.e. to the possibility of listing the individual elements according to quantitative criteria, introducing information far beyond the concept of inequality.

If one element is larger than another, it is indicated with the greater than symbol >, if it is smaller with that of less <.

Similarly, for sets the inclusion symbol applies to denote a smaller quantity .

These symbols can be combined with equality to generate extensions including the concepts of "greater than or equal" and "less than or equal" .

Obviously one can also have the negation of the inclusion given by .

Another category of logical symbols brings into play the concept of belonging.

If an element belongs to some other logical structure it is indicated with , if it does not belong with .

Some logical symbols transcribe what normally takes place in the logical processes of verbal construction.

The implication given by a hypothetical subordinate clause (the classic “if...then”) is coded like this , while the logical co-implication (“if and only if”) like this .

The linguistic construct "such that" is summarized in the use of the colon:

Finally, there are logical symbols that encode the expressions "and/or" (inclusive disjunction), "and" (logical conjunction), "or" (exclusive disjunction).

In the first two cases, a correspondent can be found in the union between several elements, indicated with , and in the intersection between several elements .

All these symbols are called logical connectors.

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Principles

There are four logical principles that are absolutely valid in the elementary logic scheme (but not in some advanced logic schemes).

These principles are tautologies and were already known in ancient Greek philosophy, being part of Aristotle's logical system.

1) Principle of identity: each element is equal to itself.

2) Principle of bivalence: a proposition is either true or false.

3) Principle of non-contradiction: if an element is true, its negation is false and vice versa. From this it necessarily follows that this proposition cannot be true

4) Principle of excluded middle: it is not possible that two contradictory propositions are both false. This property generalizes the previous one, since the non-contradiction property does not exclude that both propositions are false.

Property

Furthermore, for a generic logical operation the following properties can be defined in a generic logical structure G (it is not said that all these properties are valid for each operation and for each logical structure, it will depend from case to case).

reflective property :

For each element belonging to the logical structure, the logical operation performed on the same element refers internally to the logical structure.

Idempotence property :

For each element belonging to the logical structure, the logical operation performed on the same element results in the same element.

Neutral element existence property :

For each element belonging to the logical structure, there is another element such that the logical operation performed on it always returns the starting element.

Inverse element existence property :

For each element belonging to the logical structure, there is another element such that the logical operation performed on them always returns the neutral element.

Commutative property :

Given two elements belonging to the logical structure, the result of the logical operation performed on them does not change if the order of the elements is changed.

transitive property :

Given three elements belonging to the logical structure, the logical operation performed on the chain of elements depends only on the first and last.

Associative property :

Given three elements belonging to the logical structure, the result of the logical operation made of them does not change according to the order in which the operations are performed.

Distributive property :

Given three elements belonging to the logical structure, the logical operation performed on a group of two of them and on the other is equivalent to the logical operation performed on groups of two.

The concepts of equality, congruence, similarity, proportionality and belonging possess all these properties just listed.

Ordering symbols satisfy only the transitive and reflexive properties.

In this case, the idempotence property is satisfied only by also including the ordering with equality, while the other properties are not well defined.

The logical implication satisfies the reflexive, idempotence and transitive properties, while it does not satisfy the commutative, associative and distributive ones.

On the other hand co-implication satisfies all of them as do logical connectors such as logical conjunction and inclusive disjunction.

An operation in which the reflexive, commutative, and transitive properties hold simultaneously is called an equivalence relation .

In general, De Morgan's two dual theorems hold :

These theorems involve the definitions of logical connectors and the distributive property.

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Boolean logic

For logical connectors it is possible to define, with the formalism of the so-called Boolean logic, truth tables based on the "true" or "false" values attributable to the individual propositions.

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DENIAL

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The negation is true if the proposition is false and vice versa.

LOGICAL CONJUNCTION

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The logical conjunction is true only when both propositions are true.

INCLUSIVE DISJUNCTION

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The inclusive disjunction is false only when both propositions are false.

EXCLUSIVE DISJUNCTION

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Exclusive disjunction is false if both propositions are false (or true).

LOGICAL IMPLICATION

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The logical implication is false only if the cause is true and the consequence is false.

LOGICAL CO-IMPLICATION

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Logical co-implication is true if both propositions are true (or false).

In case the logical implication is true, A is called a sufficient condition for B, while B is called a necessary condition for A.

The logical implication is the main way to prove theorems, considering that A represents the hypotheses, B the theses, while the logical implication procedure is the proof of the theorem.

Logical co-implication is an equivalence relation.

In this case A and B are logically equivalent concepts and are both necessary and sufficient conditions for each other.

Recalling the exposed properties, the logical co-implication can also be expressed as:

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Applications of logic: proof of theorems

The mathematical proof of a theorem can be based on two large logical categories.

On the one hand there is the deduction which, starting from hypotheses considered true (or already demonstrated previously), determines the validity of a thesis by virtue of the formal and logical coherence of the demonstrative reasoning alone. Generally , following this pattern, a mechanism is applied that reaches from the universal to the particular.

On the other hand, we have the induction which, starting from particular cases, abstracts a general law. As repeatedly highlighted throughout the history of logic, every induction is actually a conjecture and therefore, if we want to use the inductive logical method, these propositions are to be considered axioms.

In modern logic, which we will not go into in this paragraph as it deals with advanced concepts far beyond the scope of these simple elementary bases, the inductive method is not accepted as the correct logical reasoning to prove theses mathematically.

The deductive method is therefore the main method of mathematical proof.

It is distinguished in the direct method, in which the thesis is actually demonstrated starting from the hypotheses, and in the indirect method, in which the thesis is assumed to be true and the logical path is reconstructed backwards to reach the hypotheses.

The indirect method can, in turn, make use of the proof by contradiction which, by denying the thesis, leads to a logical contradiction and therefore the thesis remains proved for the principle of the excluded middle.

The method by contradiction therefore consists not in proving that it is true, but that it is false.

Sometimes, one can resort to the proof of the so-called contranominal to arrive at the proof of the theorem.

This originates from the following logical relationship.

If it is true , then it is necessarily true too .

In some particular sectors of mathematics, for example in geometry, particular demonstrative constructs such as those of similarity and equivalence can be used.

Logical demonstration procedures are constructive and iterative, in the sense that previous results can be used to demonstrate new theses (this is the case of lemmas and corollaries for example) or the same logical procedures can be used a sufficient number of times to reach the proof of the thesis.

Finally, we point out that mathematical theorems, precisely because they have to be proved, are neither true nor false in absolute terms; it is the hypotheses that determine the veracity or otherwise of the theses.

Precisely for this reason, a general extension of mathematical knowledge is given by the mechanism of the weakening of hypotheses.

Given a general thesis proved under suitable hypotheses, which of the latter can be "relaxed" to obtain the same thesis?

If, on the other hand, other hypotheses are changed, what new theses can be deduced?

These are the main questions that lead to overcoming previous knowledge in both logic and mathematics.

Applications of Boolean logic: electronic calculators

Boolean logic, also called Boolean algebra, is the basis of modern electronic calculators.

Indeed, a computer memory, or a processor of the same or of a smartphone, is based on single units that are connected through logical operations.

In electronic calculators, every single command is encoded by high-level languages (for example operating systems) which in turn are based on medium-level programming codes.

These codes are mediated by other programs which act directly on the physical part of the machine.

The heart of every electronic calculator is given by a logic unit capable of coding and executing a large number of logical operations per second.

In electronics, logical operations are defined as follows:

- the negation is called NOT

- the logical conjunction is called AND

- the inclusive disjunction is called OR

- the exclusive disjunction is called XOR.

Furthermore, the negations of the previous ones are called NAND, NOR and XNOR.

Electronic calculators are made up of billions of elementary logic cells, each of which encodes one of these logical operations.

The binary numbering system, which has only two digits 0 and 1, is very well suited to interpreting Boolean logic. The digit 0 corresponds to the status of "false", to the digit 1 the status of "true."

In computer science, these digits are called bits.

Physically, the false state is made up of an unbiased circuit (that is, without the application of an electric voltage), while the true state is constituted by a polarized circuit.

Thus applying a direct reference voltage (for many years it was 5 volts continuously, but today there is a tendency to decrease this value from 3.3 volts to 2.1 up to 1.8 or 1.3 or 0.9 volts ), it is possible to identify the different logical states and build the physical foundations of an electronic calculator.

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Insight: syllogism and mathematical logic

The syllogism develops around this reasoning divided into three statements:

First proposition: all men are mortal.

Second proposition: Socrates is a man.

Third statement: Socrates is mortal.

Translated with the symbology of mathematical logic it becomes (called A the set of all men, b the identifying element of Socrates and C the fact of being mortal):

It is clear that, logically, this reasoning is flawless.

The real problem lies precisely in the first statement.

To say "all men are mortal" is in itself already knowing that Socrates, as a man, is mortal. In other words, the first statement derives from an induction already known a priori and, as such, it is a conjecture that cannot be demonstrated, but taken as true (common sense tells us that this is the case).

As such, the syllogism, being a reasoning based on a first inductive statement, does not generate real knowledge.

At the end of the third sentence we know that Socrates is mortal, but in reality we already knew it at the beginning, since, in order to be able to affirm that all men are mortal, we had to necessarily have already included Socrates himself.

Modern logic disregards the use of the syllogism to enrich knowledge, relying on other logical constructs, based on the deduction and demonstration of theorems.

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Exercises

Exercise 1

Prove De Morgan's first theorem using logical properties.

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De Morgan's first theorem states that:

Applying the distributive property of negation with respect to logical conjunction we arrive at the result of De Morgan's theorem.

Similarly, the second theorem is proved.

An alternative method of proof is to use truth tables.

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Exercise 2

Construct the truth table for the following logical construct.

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By applying the inclusive disjunction to the two tables it is clear that the logical construct is always true.

It is therefore a tautology.

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Exercise 3

Justify, through Boolean logic, the veracity of the method of demonstrating the contranominal.

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The method of demonstrating the contranominal is based on denying the thesis and demonstrating that this denial implies the denial of the hypotheses.

In logical terms it means admitting that if it is true , then it is necessarily true too .

From Boolean logic we know that the logical implication is false only if the cause is true and the consequence is false.

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As can be seen, the two truth tables coincide.

2

ELEMENTARY ARITHMETIC OPERATIONS

ELEMENTARY ARITHMETIC OPERATIONS

Introduction

In addition to logic, the mathematical alphabet relies on numbers which are conceptual abstractions to encode the different quantities of a given element.

Almost all numerical alphabets, such as ours, are based on characters given by numbers; in our decimal number system, the digits are ten, including zero which indicates zero quantity.

A number is given by the composition of several digits; starting from the right, the last digit represents the units, the penultimate the tens, the antepenultimate the hundreds, the fourth last the thousands.

We can define elementary operations related to each numerical alphabet, for ease of use we only consider the decimal system that we use extensively.

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Addition and subtraction

Addition takes into account the increase of one quantity by another (or others).

The individual quantities are called addends, while the result of the addition is called the sum.

For addition, the commutative and associative properties are valid, furthermore the neutral element is given by zero.

Addition also satisfies an ordering property since the sum is always greater than the single addends and, conversely, each addend is always less than the sum.

The mathematical symbol for addition is +.

Subtraction, on the other hand, takes into account the reduction of one quantity by another (or others).

The quantity to be subtracted is called the minuend, the quantity to be subtracted is called the subtrahend, while the result is called the difference.

For subtraction the associative property holds, the neutral element is always given by zero and an ordering property is satisfied being the difference always smaller than the minuend and, vice versa, the minuend always greater than the difference.

The mathematical symbol of subtraction is minus –.

A special case of subtraction occurs when the subtrahend is greater than the minuend.

In this case, the difference is negative, i.e. less than zero.

Negative numbers are exactly the same in shape as positive numbers except with the prefix - .

In doing so we see that the subtraction does not satisfy the commutative property, but another one called anti-commutative:

This formulation allows us to unify the concepts of addition and subtraction.

We can associate the + and – signs with individual numbers and not with the operation.

Therefore subtraction is an addition between a positive and a negative number, applying the well-known rule of signs according to which an even number of agreeing signs (two plus or two minus) returns a positive sign, while an even number of discordant signs (a plus and a minus) results in a negative sign.

It happens vice versa if there are odd numbers of agreeing and discordant signs.

In this unifying view, the commutative property is always valid since subtraction falls into addition. Furthermore, each number has an inverse with respect to the addition/subtraction operation given by its negative counterpart.

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Multiplication and division

Multiplication is an operation that summarizes the iterated addition of equal numbers.

The numbers to be multiplied are called factors, while the result is called product.

The multiplication symbol is given by , even if in mathematics the dot is more often used or the multiplication symbol is totally omitted (and this happens in the vast majority of cases).

For multiplication the commutative and associative properties hold, furthermore the distributive property holds with respect to addition and subtraction:

The neutral element is given by unity (every number multiplied by 1 always gives itself), while the rule of the signs previously exposed is always valid.

Furthermore, for multiplication there is also a zero element, given precisely by zero (every number multiplied by zero always gives zero).

Division is the reverse of multiplication.

The number to be divided is called the dividend, the number that divides is called the divisor and the result is called the quotient.

The division symbol is given by , sometimes the slash / is also used.

The properties listed for multiplication do not apply to division.

The neutral element is given by unity (every number divided by 1 always gives itself) and the rule of signs previously exposed is always valid.

However, the division by zero operation is not defined.

If the dividend is greater than the divisor, the quotient is greater than 1, and if the dividend is less than the divisor, the quotient is less than 1.

The division operation brings up non-integer numbers, i.e. numbers that can only be defined using digits smaller than the unit.

For such figures, the convention of placing a comma between the upper part and the lower part is used.

The digits after the decimal point respectively express tenths, hundredths, thousandths and so on.

When the dividend is a multiple of the divisor, the quotient is an integer and is called quota and, in this case, the dividend is divisible by the divisor, the remainder being zero.

A number is always divisible by itself (giving the value 1) and by 1 (giving the value itself).

Numbers that are divisible only by themselves and 1 are called prime numbers.

Numbers that are divisible by 2 are called even, those that are not divisible by 2 are called odd.

A quotient can always be expressed as the sum of a quota and a remainder.

Non-integer quotients can have a limited number of decimal digits or an infinite number of such digits.

In the latter case, we speak of periodic numbers as the decimal digits (all or part of them) are always repeated in the same sequence.

Periodicity is indicated with a sign above the periodic digit or digits.

For example, the quotient obtained from the division between 1 and 3 is given by a number having infinite decimal digits all equal to 3 and is indicated as follows .

Another way of expressing division is to use the concept of fraction.

In this case, the dividend and the divisor are called numerator and denominator, respectively.

Since division by zero is not defined, the denominator of a fraction can never be equal to zero.

A fraction is indicated with the symbol of fraction ––, the numerator goes in the upper part, the denominator in the lower part.

A fraction is said to be reduced to its lowest terms, or irreducible, if the numerator and denominator are prime numbers, i.e. they are no longer divisible by each other, resulting in a fraction.

If the numerator is greater than the denominator, the fraction is greater than 1 and is said to be improper.

Conversely, it is less than 1 and is called a proper one.

Finally, the fraction is apparent if the numerator is a multiple of the denominator (because, in this case, the fraction is actually an integer).

We define the reciprocal of a number as that number which, multiplied by the first, always gives 1.

In other words, the reciprocal of a number is its inverse element with respect to multiplication.

With this definition and with the notation of fractions, we can unify the concept of division with that of multiplication.

A division is nothing but a multiplication between the dividend and the reciprocal of the divisor;.

For example:

We define least common multiple (abbreviated lcm) of two or more integers, the smallest positive integer multiple of all the numbers considered.

If any of these numbers is zero, then that lcm is zero.

We define the greatest common divisor (abbreviated GCD) of two or more integers that are not all equal to zero, the largest positive integer by which all numbers can be divided.

In the case of two numbers, if one of them is zero, then GCD is equal to the other number.

Two prime numbers to each other have a GCD equal to 1.

Exponentiation and root extraction

Exponentiation is an operation that sums up the iterated multiplication of equal numbers.

The number multiplied several times is called the base, the number of iterated multiplications is called the exponent.

The exponent and the base can be integers or decimals, both positive and negative.

The symbol for exponentiation is given by the base with an upper superscript where the exponent is positioned, for example

and reads "two raised to three" or "two to three".

If the exponent is equal to 2 it is called squaring, if it is equal to 3 it is called cubing.

Every exponentiation of a zero base always gives zero, while if the base is one, the result is always one.

One and zero are therefore the two neutral elements of exponentiation.

A negative base will give a negative power if the exponent is odd, positive if the exponent is even, while a positive base will always give a positive power.

With the same base, the negative exponent is equivalent to taking the reciprocal of the number: for example

Therefore there is a link between the operations of multiplication, division and exponentiation which also extends to the concepts of neutral and inverse elements.

In light of exponentiation, we can review the concepts of units, tens, hundreds and thousands.

Units are those digits that multiply the value of , tens of , hundreds of and thousands of .

That's why our system of calculation is called decimal, since it follows powers of base 10.

The inverse operation to exponentiation is called root extraction and is indicated with the symbol where the exponent of the root must appear at the top left.

The number from which to extract the root is called the radical while the result is called the radical.

If we take the example above, we have

where 8 is the radical, 3 is the exponent, and the radical is 2.

If the exponent is equal to 2 it is called a square root, if it is equal to 3 it is called a cube root.

If the radicals are integers, the respective radicals are called perfect (perfect squares in the case of square root, perfect cubes in the case of cube root).

All other radicals are decimal numbers, but have infinitely many non-repeating digits after the decimal point.

If the exponent of the root is even, the radical must necessarily be greater than or equal to zero, if it is odd the radical can be either positive or negative.

Finally, root extraction of one and zero are always equal to one and zero, respectively, whatever the exponent. One and zero are therefore the two neutral elements of root extraction.

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Numerical expressions and number systems

Mathematical expressions contain numbers (integers, decimals, in the form of fractions, powers or roots) and mathematical operations, such as those shown so far.

Multiplication, division, exponentiation and root extraction have priority over addition and subtraction in the sense that, in an expression like this 2x3+4, the multiplication between 2 and 3 takes place first and then the sum of this result with 4.

If you want to give different priorities, you need to introduce brackets: the braces have higher priority than the square ones and the latter over the round ones.

For example, if the previous expression had been written as 2x(3+4), one would first have to add 3+4 and then multiply this result by 2.

The system we use is the decimal one, but there are many others, each of which is characterized by a different number of digits.

From the property of powers, we can understand how a number system other than the decimal one has a different base.

In particular, the binary systems (whose digits are only 0 and 1) and the hexadecimal systems (in addition to the ten digits of our system, there are also the letters A, B,C,D,E,F).

Finally, for geometric systems it makes sense to define, within the decimal system, a method of sexagesimal positional numbering, i.e. which divides the geometric "numbers", called degrees, not into hundredths, but into fractions of 60.

This system is the same we use for measuring time in minutes and seconds.

3

SET THEORY

SET THEORY

Introduction

We define the primitive and intuitive concept of mathematical set as a collection of objects, called elements, indicated with lowercase letters, while sets with uppercase letters.

If an element belongs to a given set, it is indicated with the logical symbol of belonging.

Two sets coincide if and only if they have the same elements.

A set is said to be finite if it has a finite number of elements, conversely it is said to be infinite.

The number of elements of a finite set is called cardinality and is denoted by card(A).

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Operations

On sets it is possible to perform the logical operations, already described in the first chapter, of union, intersection and negation.

Union corresponds to inclusive disjunction, while intersection to logical conjunction.

We can also define the difference between set B and set A in the same way as we define the difference of two numbers.

We define the Cartesian product as the set of all possible ordered pairs (a,b) with a belonging to set A and b to set B.

The Cartesian product looks like this:

Two sets are said to be disjoint if they have no elements in common

Where the empty set is present on the second member.

A set contained in another is called a proper subset , if equality is also valid it is called improper.

The empty set is a subset of any existing set.

Instead, the set of parts is indicated as that set which is formed by the elements composed of the subsets of the starting set.

Calling A the starting set, the set of parts is P(A) and this relation always holds:

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Numerical sets

We can construct numerical sets, that is, sets whose elements are numbers.

The set of natural numbers, denoted by N, is the set of positive integers.

The set of relative numbers, denoted by Z, is the set of both positive and negative integers.

The set of rational numbers, denoted by Q, is the set of numbers obtainable as ratios between two whole numbers, both positive and negative.

The set of irrational numbers, denoted by I, is the set of non-repeating decimal numbers that cannot be expressed as a ratio between two integers.

The set of real numbers, denoted by R, is the union of the set of irrational numbers with those of rational numbers.

Therefore, the following properties hold between these sets:

In the set of natural numbers the addition and multiplication between two natural numbers are defined, furthermore the associative, commutative, distributive properties and the existence of the neutral element (zero for addition and one for multiplication) are valid ).

Such a set is closed in the sense that the sum and product of natural numbers are also natural numbers.

The set of natural numbers can be obtained axiomatically from the Peano axioms which are respectively:

1) There exists a natural number corresponding to the null quantity called zero.

2) Every natural number a has a successor natural number, denoted as S(a)

3) Zero is not the successor of any natural number

4) Distinct natural numbers have distinct successors

5) If a property P is possessed by zero and by the successor of every natural number which possesses this property, then the property is possessed by all natural numbers

From the foregoing, it can be seen that Peano's last axiom makes use of the principle of induction.

In the set of relative numbers, the same properties mentioned for the set of natural numbers apply, with the addition of the operation of subtraction and the existence of the opposite element (which is the negative of the selected number).

In the set of rational numbers the same properties mentioned for the set of relative numbers apply with the addition of the division operation, always defined except for denominators equal to zero.

In the set of irrational numbers, root extraction is uniquely defined, provided that the radical of an even root is greater than or equal to zero.

In the set of real numbers all the above operations are defined with the two conditions of existence deriving from the set of rational numbers (denominator other than zero) and from the set of irrational numbers (radical of even root greater than or equal to zero).

The set of real numbers is a field with respect to addition and multiplication as the associative, commutative, distributive and existence properties of the neutral and inverse elements with respect to the two operations mentioned are valid.

Furthermore, this set is ordered in a total way since the reflexive, antisymmetric and transitive properties hold for the lowering or increasing ordering relations, in addition to the dichotomy property (given two non-coinciding real numbers, or one is greater than other or vice versa).

To tell the truth, both the properties of total ordering and of being a field are also proper to the set of rational numbers.

The big difference of the real numbers is that the ordering is complete, i.e. every non-empty subset of R has a supremum in R.

This is Dedekind's axiom and it derives directly from having incorporated the irrational numbers into the set of real numbers.

This difference is also found on the cardinality of these numerical sets.

In fact, although they are all infinite sets, they do not have the same cardinality, i.e. there are infinites of different order.

Two sets are said to be equicardinal or equipotent if a one-to-one correspondence can be established between their elements, i.e. if one and only one element of B is associated with each element of A and vice versa.

The property of equicardinality is an equivalence relation and we can divide finite sets into equivalence classes, each of which can be represented by a natural number.

At this point, that class of sets which can be placed in one-to-one correspondence with the set of natural numbers has the same cardinality as the latter and is called the cardinality of the countable (the set is therefore called countable, even if it is infinite) .

In doing so we see that N and Z possess the cardinality of the countable.

Cantor proved that also Q has the cardinality of the countable, that is, it can be placed in a one-to-one relationship with the set of natural numbers under suitable equivalence classes.

The set of real numbers, on the other hand, cannot be placed in one-to-one correspondence with that of the natural numbers due to the presence of irrational numbers which cannot in any way be included in any equivalence class as they have infinite non-periodic decimal digits.

The set of real numbers therefore has no cardinality of the countable, but is said to have cardinality of the continuum.

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Exercises

Exercise 1

Prove that in the set of natural numbers the associative, commutative, distributive properties and the existence of the neutral element (zero for addition and one for multiplication) hold.

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In the set of natural numbers the operations of addition and multiplication are defined.

The commutative property of addition comes from the combination of Peano's axioms number 2 and number 4.

The commutative property of multiplication derives from the latter, remembering that multiplication is nothing but a sum (multiplying one number by another means adding the number itself for a number of times equal to the digit it multiplies).

The associative property of addition comes from axioms 2 and 4 and from the commutative property.

Identically for the case of multiplication.

From these two properties derives the distributive one of addition and multiplication.

The neutral element of addition is Peano's first axiom.

The neutral element of multiplication is the first successor of zero.

4

LITERAL CALCULATION

LITERAL CALCULATION

Operations

In mathematics, literal calculus is widely used, i.e. the replacement of numbers with letters that can take on any numerical value. The choice of letters is completely random and does not affect the general validity of what we are going to explain. This sector of mathematics is called algebra and the operations that we are going to list are called algebraic.

In doing so, the addition and subtraction operations can be written as follows:

And in the same way the properties of these operations can be rewritten.

Identically, for multiplication we have (remembering the various symbols and the possibility of omitting them):

For multiplication we can thus rewrite the distributive property with respect to addition and subtraction:

This property, if read in the opposite direction, ie from right to left, is called common factor grouping and is crucial in the development of expressions containing the literal calculus.

For fractions, the following multiplication and addition/subtraction properties hold:

A literal relation linking gcd and lcm between two numbers is given by:

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Power operations

Exponentiation and root extraction are thus denoted

And they read as "a raised to the nth power" and "nth root of a"-

The properties are as follows:

Therefore the product of powers having the same base is given by the sum of the powers, while the exponentiation of a power is given by the product of the powers.

Recalling that negative exponents lead back to fractional symbols:

We have the following dual property relating to the division between powers having the same base:

That is, the division between powers having the same base is given by the difference of the powers.

If instead the bases change but the exponents are the same, we have:

The two properties just mentioned are called common factor grouping of the power.

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Operations on radicals

Recalling that fractional exponents lead to root extraction:

We have the following properties of radicals:

The first property summarizes the definitions of exponentiation and root extraction and states that they are inverse operations, i.e. the exponentiation to the nth power of an nth root of a number returns the number itself.

The second property states that the nth root of an mth root is given by a root whose index is the product of n by m.

The third property indicates the interchangeability of the root and exponentiation operations.

The last property is called rationalization of the denominator (if read in reverse it is called irrationalization).

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Conditions of existence

All the operations presented in this chapter are defined only under two distinct conditions, which from now on we will call existence conditions.

The first is given by the denominator of a fraction, which must always be different from zero.

The second is given by the root of an even root, which must always be greater than or equal to zero.

In formulas we have:

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Monomial

We define monomial as an algebraic expression, in which neither addition nor subtraction appears, consisting of a numerical coefficient and a literal part.

The degree of a monomial is the sum of the exponents present in it.

Two monomials are said to be similar if they have the same literal part raised to the same exponents.

The multiplication and division of monomials derive from the rules expressed when speaking of powers, for example we have, with K and H any numerical coefficients:

Polynomials

The addition and subtraction of monomials that are not similar to each other gives rise to polynomials.

If the polynomial consists of two monomials it is called a binomial, if instead there are three monomials it is called a trinomial.

The degree of a polynomial is the maximum degree of the individual monomials constituting the polynomial. A polynomial of degree zero is a numerical constant, if it is of degree one it is said linear, of degree two quadratic (or conic), of degree three cubic.

The product of polynomials is given by the sum of the products of each single monomial of the first polynomial by all the other monomials of the second polynomial, applying the well-known rule of the distributive property.

With the introduction of polynomials it becomes natural to generalize all literal expressions.

Such expressions can lead to identities, when literal parts are compared to each other, or to equations.

When a value of a literal part of the polynomial is such that it cancels the whole polynomial, it is called the root of the polynomial.

The search for the roots of a polynomial is crucial for solving mathematical problems and makes use of the properties of decomposition of polynomials into prime factors, ie the inverse of the distributive property mentioned above.

The division between two polynomials leads to the formation of two polynomials, one given by the quotient and the other by the remainder, both with degree lower than the starting polynomial.

If the remainder polynomial is zero, it means that the two starting polynomials are divisible between them and that a prime factorization has been carried out.

This theorem is known as the remainder theorem. A corollary is given by Ruffini's theorem, according to which if a polynomial is divisible by (xa) then a is a root of the polynomial.

From this derives the well-known Ruffini's rule which, once the root of a polynomial has been identified, allows to decompose it and to obtain the quotient polynomial, obviously of lower degree than the starting polynomial.

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Notable products

Some useful results for the decomposition of polynomials are given by the so-called remarkable products:

Square of a binomial:

Square of a trinomial:

Cube of a binomial:

Difference of squares:

Sum and difference between cubes:

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Exercises

Exercise 1

Solve the following literal expression:

Applying the distributive property of multiplication with respect to addition, we have:

Adding and rearranging the terms:

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Exercise 2

Solve the following fractional calculation:

The first two terms condense into the fraction:

Having the same denominator as the third term, we simply have:

Remark: the "cross product" rule used makes it possible to greatly simplify the calculations compared to the normal calculation of the greatest common divisor by the denominator and the multiplication of the remaining factors by the numerators. This is one of the many cases in which mathematics prefers a "smart" way, i.e. an elegant (and fast!) way to solve problems without getting bogged down in useless, lengthy calculations that lead to any type of error.

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Exercise 3

Find the least common multiple and greatest common divisor of the following numbers: 15 and 18.

15= 5 x 3

18= 3 x 3 x 2

The greatest common divisor is obviously 3.

The rule allows us to avoid unnecessary calculations, stating that the least common multiple is simply given by 15 x 18 divided by 3 or 90.

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Exercise 4

Solve the following expressions:

a)

b)

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a) The expression develops in this way, recalling the properties of powers:

Where in the third step the common denominator fractional operation was carried out and in the last step the term that appeared in all the literal expressions in the numerator was collected as a common factor.

The expression is defined only if both b and c are non-zero.

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b) From the properties of powers, we have:

The expression is defined only if both a and b are non-zero.

Exercise 5

Solve the following radicals:

a)

b)

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a) From the properties of the radicals we have:

Where in the last passage the property of “carrying in” was exploited, that is to bring a b under the thirtieth root to remove the negative exponent inside the root.

A faster method to solve the radical would have been by remembering that roots can be identified with fractional exponentials, i.e.:

Through this method we can see how the calculation of radicals is nothing if not an application of the properties of powers.

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b)

In the first passage, the term under the cube root was collected as a common factor, while subsequently the rules of the radicals were applied.

Also in this case we could have proceeded with the normal properties of the powers.

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Exercise 6

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Determine the conditions of existence of the following expressions in the field of real values:

a)

b)

c)

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a) It is a question of imposing non-zero denominators, therefore the existence condition is any value of a belonging to R except 0 and 3.

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b) It is a question of setting the radicals greater than or equal to zero. Both conditions must therefore hold:

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c) Roots can take any value, as roots have odd index. The only condition of existence is that relating to the denominator which must be different from zero.

The expression is therefore defined on the whole set R except for a=1.

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Exercise 7

––––––––

Solve the following operations on monomials:

a)

b)

––––––––

a) It is a question of applying the normal rules of powers:

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b) As in the first exercise:

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Exercise 8

Solve the following polynomials by factoring them into prime factors:

a)

b)

c)

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a) Collecting at the common factor we have:

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b) Carrying out the operations and subsequently collecting the common factor, we have:

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c) The first fraction is simplified as follows:

For the second fraction the polynomial division is carried out, i.e.:

Where Q is the quotient and R is the remainder.

It can be seen that a=1 is a root of the numerator and therefore the remainder will be zero R=0.

This means finding that polynomial which multiplied by (a-1) gives the numerator.

The second fraction therefore remains:

The result is therefore:

Where in the last step the remarkable product given by the difference between squares was applied.

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Exercise 9

––––––––

Solve the following polynomial expressions:

a)

b)

c)

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a) The exercise can be solved in two ways. Either by applying the rule on the squares of a binomial and doing the math (useless effort that we leave to the reader), or by looking at the expression in the face and cleverly applying the mathematical rules.

It is a difference between squares, so we have:

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b) Explaining the accounts we have:

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c) The numerator and denominator of the fraction are differences between squares therefore:

No further simplifications can be made.

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Exercise 10

––––––––

Factor the following polynomial using Ruffini's rule:

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It can be seen that the polynomial has a root corresponding to a=1

Applying the scheme according to Ruffini's rule we have:

1

3

0

-2

-2

1

0

1

4

4

0

1

4

4

2

0

The polynomial is therefore decomposed into:

The polynomial in brackets has a root that is not easily calculable by hand, except by resorting to numerical calculation techniques which are beyond the scope of this discussion.

5

PLANE EUCLIDEAN GEOMETRY

PLANE EUCLIDEAN GEOMETRY

Elementary definitions

Geometry is that branch of mathematics that deals with shapes and figures in a given setting.

Below we give the foundations of elementary geometry, largely developed already in ancient Greece.

The primitive concept of geometry is the point, conceived as a dimensionless and indivisible entity, which characterizes the position and is characterized by it.

An infinite and successive set of points is called a segment, if this set is delimited by two points called extremes.

Two segments are consecutive if they have an end point in common, while they are external if they have no point in common.

Two segments are said to be incident if they have only one point in common, called the point of intersection, which however is not an extreme.

The midpoint of a segment is the point that exactly divides the segment in half.

An infinite and successive set of points is called a straight line if this set is not bounded by any end point, while it is called a semi-line if there is only one end point.

A segment can therefore be seen as part of a straight line.

Two consecutive segments are adjacent if they belong to the same line.

Lines, segments and semi-lines are characterized by a single dimension called length.

The geometric entity characterized by two dimensions, called length and height, is the plane, while the one characterized by three dimensions (in addition to those mentioned there is the width) is called space. Plane geometry deals with the study of the two-dimensional case, solid geometry with the three-dimensional case.

Two straight lines or two segments are said to be coplanar if they lie in the same plane, otherwise they are called skew.

In geometry, points are indicated with capital letters, segments with capital letters of the two extremes barred at the top by a line, while straight lines and semi-lines with small letters.

Furthermore, all geometric dimensions are, by definition, positive.

Two segments, two straight lines or two semi-lines are said to coincide if and only if all the points present in the first geometric element are exactly the same as in the second geometric element.

In plane geometry, in the case of two half-lines having a common end point, the concept of angle can be defined.

In fact we see that the two half-lines divide the plane into two parts.

The angles are denoted either with lowercase letters of the Greek alphabet or with the uppercase letters of the extremes, spaced out from the point of origin of the two half-lines (called vertex) with a circumflex accent above this last letter.

By convention, angles are measured counterclockwise.

In plane geometry, if the two half-lines coincide, the angle contains the whole plane and is called a round angle, the measure of which is, by definition, 360°.

Half of a round angle is called a flat angle and measures 180° and occurs when the two half-lines are adjacent.

An angle is said to be convex if it is less than 180°, concave if it is greater, as in the figure:

Half of a straight angle is called a right angle and measures 90°.

An angle between 0° and 90° is called acute, an angle between 90° and 180° is called obtuse.

Two straight lines or two semi-lines or two segments are said to be perpendicular (or orthogonal) to each other if the angle of incidence is a right angle. It goes without saying that these geometric elements are necessarily accidents.

Two angles are said to be complementary if their sum gives a right angle, supplementary if it gives a straight angle, and complementary if it gives a round angle.

The bisector is defined as the straight line or semi-line or the segment which divides the angle subtended between two segments or two semi-lines or two straight lines into two equal parts.

A peculiarity of the bisector is given by the fact that if a half-line is bisector of an angle, its extension is also the same for the angle complementary to the first.

The straight line or half-line or segment that divides the length of a given segment into two equal parts, i.e. passing through the midpoint of the segment, is defined as median.

Height is defined as a line or ray or segment that is perpendicular to another geometric entity such as a line or ray or segment. The point of intersection between the height and the given line or ray or segment is called the foot.

An axis is defined as a line or ray or segment which is perpendicular to a segment at its midpoint.

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Euclid's postulates

Euclid enunciated five postulates which, if accepted, make geometry fall into the so-called Euclidean geometry. The five postulates are given by:

1) Between any two points it is possible to draw one and only one straight line.

2) A straight line can be extended beyond the two points indefinitely.

3) Given a point and a length, it is possible to describe a circumference.

4) All right angles are equal.

5) If a straight line cuts two other straight lines, determining on the same side internal angles whose sum is less than that of two right angles, extending the two straight lines indefinitely, they will meet on the side where the sum of the two angles is less than two right angles .

Euclid's third postulate introduces the concept of circumference which we have not yet introduced. We define circumference as the locus of points on the plane equidistant from a point called center (the constant distance is called radius).

Some consequences of the first four postulates are as follows:

An infinite number of straight lines pass through each point on the plane.

One and only one straight line passes through two distinct points on the plane.

Infinite planes pass through a straight line in space.

Only one plane passes through three non-aligned points in space.

Only one circle passes through three non-aligned points in the plane.

Obviously the plan and the space take the qualifying adjective of Euclidean.

Three or more points in space are aligned if they are contained in a straight line, four or more points in space are coplanar if they are contained in a plane.

Of all Euclid's postulates, it is the fifth postulate that determines Euclidean geometry.

This postulate is also called the parallel postulate and its non-acceptance gives rise to non-Euclidean geometries, which we will not deal with in this manual which, instead, is entirely centered on Euclidean geometry.

The name of parallel postulate derives from the equivalent, and better known, version given to this postulate:

Given any straight line r and a point P not belonging to it, it is possible to draw for P one and only one straight line parallel to the given straight line r .

Therefore two straight lines are intersecting if they have a point of intersection, while if they do not they are said to be parallel.

In Euclidean geometry, two parallel lines always maintain the same distance between them.

In Euclidean geometry, the minimum distance between two parallel lines, or between a point outside a line and the line itself, is given by the height.

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Other definitions

We define translation as a transformation of the Euclidean plane or space that moves all points a fixed distance in the same direction.

We define rotation as a transformation of the Euclidean plane or space that moves all points by a given angle (called the angle of rotation) while maintaining a fixed point around which the rotation takes place (called the center of rotation). By convention, the direction of rotation is counter-clockwise.

The combination of a translation with a rotation is called a rototranslation.

We define reflection as a transformation of the Euclidean plane or space that mirrors all points with respect to a point, a straight line or a plane (respectively called the center, axis or plane of reflection). The mirrored points are said to be symmetrical to the first ones.

The operations of translation, rotation and reflection are isometries, i.e. transformations which keep the distances constant.

Two geometric objects are congruent when they can be transformed into each other using isometries. The congruence relation is a logical equivalence relation.

Transformations which instead increase or decrease the distances, but keep the angles unchanged are called homotheties.

Given two parallel lines a and b and an incident line, we can define the angles as in the figure:

Angles opposite the vertex are defined as the pairs of angles and (obviously also the respective angles with the apex).

Corresponding angles are defined as all pairs of the type ,

interior conjugate angles and ,

exterior conjugate angles and ,

alternate interior angles and ,

alternate exterior angles and .

In Euclidean geometry we have the following properties:

conjugate angles (interior and exterior) are supplementary,

the angles opposite the vertex, corresponding and alternate (internal and external) are congruent to each other.

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Figures: definitions

Plane Euclidean geometry also deals with plane figures that can be defined in Euclidean geometry.

We define perimeter as the measure of the length of the contour of a flat figure (generally indicated with 2p) and semiperimeter as half of the perimeter.

We also define area as the extent of the extension of the two-dimensional region of the Euclidean plane included within the perimeter of a plane figure.

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Circumference

Let's start by taking into consideration the curved figures, i.e. composed of points not aligned along straight lines.

We define circumference as that geometric locus made up of points equidistant from a fixed point called center.

The distance from the center to any point in the geometric locus is called the radius.

Circles are simple closed curves, which divide the plane into an internal surface, called a circle, and an external surface (the whole Euclidean plane minus the circle).

Concentric circles have the same center but different radii.

The diameter is given by twice the radius and is the segment that joins the opposite points on a circumference, passing through the center, as in the figure:

Said O the center of the circle and P a generic point of the circumference, the geometric locus is expressed by this formula:

The perimeter of the circle, given by the circumference, and the area of the circle are respectively:

A segment whose ends are on the circumference is called a chord: the diameter is the only chord that passes through the centre, the other chords will have a shorter length than the diameter.

The diameter divides the circle into two congruent parts, called semicircles.

Each chord divides the circle into two parts called circular segments.

An angle is said to be at the center if its vertex is the center of the circle, while it is said to be at the circumference if the vertex belongs to the circumference.

The intersection between a central corner and the circle itself identifies a part of the circle called a circular sector, as in the figure:

If the angle in the center is right, the circular sector is called a quadrant, if instead the angle is flat, the circular sector coincides with a semicircle.

The perimeter of a circular sector is equivalent to the sum of the two radii and the length of the circular segment, also called arc.

The angle at which the arc equals the radius is called the radian. Generally, angles are measured in radians, starting from this equivalence: a round angle, equal to 360° is also equal to radians.

In the figure, the relative central angle is said to insist on the arc AB.

An angle at the center which points to an arc is always twice the angle of the circle which points to the same arc.

The area between two concentric circles is called an annulus.

A straight line is said to be external to the circle if it has no points of intersection with it, it is said to be secant if it has two distinct points of intersection, while it is said to be tangent if there is only one point of intersection.

A line tangent to a circle makes a right angle with the radius of the circle at the point of tangency.

Ellipse

The ellipse is the locus of points for which the sum of the distances from two fixed points is a constant.

These points are called foci and, if they coincide in a point, we have the circumference.

Considering this figure:

The locus equation is given by:

The distance between the antipodal points of the ellipse, ie between points symmetrical with respect to its centre, is maximum along the major axis which also contains the foci and is minimum along the minor axis, which is perpendicular to the major one.

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Parable

A non-closed plane curve of particular importance is given by the parabola, defined as the geometric locus of points

Imprint

Publisher: BookRix GmbH & Co. KG

Publication Date: 04-18-2023
ISBN: 978-3-7554-3918-9

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