Table of Contents

Table of Contents

"Handbook of Elementary Mathematics"

INTRODUCTION

ELEMENTARY MATHEMATICAL LOGIC

ELEMENTARY OPERATIONS

LITERAL CALCULATION

ELEMENTARY GEOMETRY

SET THEORY AND FUNCTIONS

ELEMENTARY EQUATIONS AND INEQUATIONS

ANALYTIC GEOMETRY

GONIOMETRIC FUNCTIONS AND TRIGONOMETRY

EXPONENTIAL, LOGARITHMIC AND HYPERBOLIC FUNCTIONS

SUCCESSION AND SERIES

COMBINATORY CALCULATION AND ELEMENTARY STATISTICS

COMPLEX NUMBERS

VECTOR AND MATRICIAL MATHEMATICS

ELEMENTARY NUMERICAL CALCULATION

"Handbook of Elementary Mathematics"

"Handbook of Elementary Mathematics"

SIMONE MALACRIDA

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This book lays the foundations of mathematics, starting with logic and elementary operations and moving on to topics such as trigonometry, complex numbers, matrix and vector notations, while addressing plane, solid and analytic geometry, as well as the rudiments of combinatorial and numerical calculus.

Such topics are necessary for the understanding of mathematical analysis and all modern developments, while providing a useful extension of knowledge for an initial mathematical description of the natural phenomena around us.

Simone Malacrida (1977)

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

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ANALYTICAL INDEX

INTRODUCTION

I – ELEMENTARY MATHEMATICAL LOGIC

II – BASIC OPERATIONS

III – LITERAL E CALCULATION

IV – ELEMENTARY GEOMETRY

V – SET THEORY AND FUNCTIONS

VI – ELEMENTARY EQUATIONS AND INEQUATIONS

VII – ANALYTICAL GEOMETRY

VIII – GONIOMETRIC FUNCTIONS AND TRIGONOMETRY

IX – EXPONENTIAL, LOGARITHMIC AND HYPERBOLIC FUNCTIONS

X – SUCCESSION AND SERIES

XI – COMBINATORY CALCULATION AND ELEMENTARY STATISTICS

XII – COMPLEX NUMBERS

XIII – VECTOR AND MATRICIAL MATHEMATICS

XIV – ELEMENTARY NUMERICAL CALCULATION

INTRODUCTION

INTRODUCTION

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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 la Naturaand whose starting point was precisely given by the consideration that mathematics could be the keystone for understanding 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 la Naturaits 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 la Natura, mathematics is properly problem solving , therefore it is concreteness made science, as man has always aimed at solving problems that grip 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 describingla Natura.

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This handbook has the express purpose of providing the rudiments of elementary mathematics, i.e. of all that part of mathematics prior to the introduction of mathematical analysis.

The notions and concepts exposed in this manual 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 remainder of the book describes the knowledge acquired by humanity over the centuries, especially 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.

Without the notions exposed in this manual, however, it is impossible to approach mathematical analysis directly, as the mathematical cognitive process is a slow evolution that bases its results on previous knowledge. 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.

Furthermore, in the description of elementary mathematics, concepts explicitly made well beyond the seventeenth century but which, for logical continuity, complete the picture of the individual sub-disciplines we will be discussing will be brought into play.

These mathematical tools are therefore necessary for the full understanding of mathematical analysis and of all modern evolutions and, at the same time, also provide a useful extension of knowledge for a first description of the natural phenomena that surround us.

I

ELEMENTARY MATHEMATICAL LOGIC

ELEMENTARY MATHEMATICAL LOGIC

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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.

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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 the mathematical field, proportionality can also be defined, indicated with . 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 in this way , the uniqueness of the element in this way and the expression "for each element" is transcribed in this way .

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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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.

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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.

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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).

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reflective property:

Idempotence property:

Neutral element existence property:

Inverse element existence property:

Commutative property:

transitive property:

Associative property:

Distributive property:

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The concepts of equality, congruence, similarity, proportionality and belonging possess all these properties just listed. Ordering symbols satisfy only the transitive and reflexive properties; that of idempotence is satisfied only by also including the ordering with equality, while the other properties are not well defined. 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 just as logical connectors do .

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:

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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LOGICAL CONJUNCTION

Impressum

Verlag: BookRix GmbH & Co. KG

Tag der Veröffentlichung: 19.04.2023
ISBN: 978-3-7554-3936-3

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