Fibonacci Numbers

(Dover Books on Mathematics)

Nikolai Nikolaevich Vorob'ev

Dover Publications |  2011 | 80 páginas | rar - epub | 1,4 Mb

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Fibonacci numbers date back to an 800-year-old problem concerning the number of offspring born in a single year to a pair of rabbits. This book offers the solution and explores the occurrence of Fibonacci numbers in number theory, continued fractions, and geometry. A discussion of the "golden section" rectangle, in which the lengths of the sides can be expressed as a ration of two successive Fibonacci numbers, draws upon attempts by ancient and medieval thinkers to base aesthetic and philosophical principles on the beauty of these figures. Recreational readers as well as students and teachers will appreciate this light and entertaining treatment of a classic puzzle.

CONTENTS
Foreword
Introduction
I.    The simplest properties of Fibonacci numbers
II.   Number-theoretic properties of Fibonacci numbers
III.  Fibonacci numbers and continued fractions
IV.  Fibonacci numbers and geometry

V.   Conclusion

Problems in Solid Geometry

(Science for Everyone)
I. F. Sharygin

MIR | 1986 | pdf

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From the Preface:

This book contains 340 problems in solid geometry and is a natural continuation of Problems in Plane Geometry, Nauka, Moscow, 1982. It is therefore possible to confine myself here to those points where this book differs from the first. The problems in this collection are grouped into (1) computational problems and (2) problems on proof.

The simplest problems in Section 1 only have answers, others, have brief hints, and the most difficult, have detailed hints and worked solutions. There are two reservations. Firstly, in most cases only the general outline of the solution is given, a number of details being suggested for the reader to consider. Secondly, although the suggested solutions are valid, they are not patterns (models) to be used in examinations. Sections 2-4 contain various geometric facts and theorems, problems on maximum and minimum (some of the problems in this part could have been put in Section 1), and problems on loci. Some questions pertaining to the geometry of tetrahedron, spherical geometry, and so forth are also considered here.

As to the techniques for solving all these problems, I have to state that I prefer analytical computational methods to those associated with plane geometry. Some of the difficult problems in solid geometry will require a high level of concentration from the reader, and an ability to carry out some rather complicated work.

Índice:
Preface . 6
Section 1. Computational Problems. 7
Section 2. Problems on Proof . 37
Section 3. Problems on Extrema. Geometric Inequalities. 47
Section 4. 
Loci of Points. 54
An Arbitrary Tetrahedron 59. 
An Equi-faced Tetrahedron 61. 
An Orthocentric Tetrahedron 64. ´
An Arbitrary Polyhedron. The Sphere 65. 
An Outlet into Space 68.
Answers, Hints, Solutions . 6

Fascinating Fractions

 N. M. Beskin


Little Mathematics Library 


1986 | MIR | 89 páginas | DJVU  | 827 kb  | 

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This booklet is intended for high-school students interested
in mathematics. It is concerned with approximating real
numbers by rational ones, which is one of the most captivating
topics in arithmetic.

…
Continued fractions represent one of the most perfect
creations of 17-18th century mathematicians: Huygens, Euler,
Lagrange, and Legendre. The properties of these fractions
are really striking.

The following should be borne in mind when reading this
booklet. Topics easily understandable are presented in normal print,
while those more difficult are given in small print. Proofs of some theorems given in small print may be omitted safely. These theorems will necessarily be taken for granted. However, mathematics is not just reading for entertainment. A future mathematician as well as a physicist or an engineer has to acquire skill in dealing with  mathematical constructions and proofs. So take a pencil and a sheet of  paper and study carefully the topics given in small print. You may succeed in simplifying some proofs or finding better ones.

The book was translated from the Russian by V. I. Kisin and was first published by Mir Publishers in 1986.

Contents

Preface 7
Chapter 1. Two Historical Puzzles
1.1. Archimedes’ Puzzle 9
1.1.1. Archimedes’ Number 9
1.1.2. Approximation 10
1.1.3. Error of Approximation 12
1.1.4. Quality of Approximation 13
1.2. The Puzzle of Pope Gregory XIII 15
1.2.1. The Mathematical Problem of the Calendar … 15
1.2.2. Julian and Gregorian Calendars 17

Chapter 2. Formation of Continued Fractions
2.1. Expansion of a Real Number into a Continued Fraction 19
2.1.1. Algorithm of Expansion into a Continued Fraction 19
2.1.2. Notation for Continued Fractions 21
2.1.3. Expansion of Negative Numbers into Continued Fractions 21
2.1.4. Examples of Nonterminating Expansion 22
2.2. Euclid’s Algorithm 24
2.2.1. Euclid’s Algorithm 24
2.2.2. Examples of Application of Euclid’s Algorithm 26
2.2.3. Summary 27

Chapter 3. Convergents
3.1. The Concept of Convergents 29
3.1.1. Preliminary Definition of Convergents 29
3.1.2. How to Generate Convergents 30
3.1.3. The Final Definition of Convergents 33
3.1.4. Evaluation of Convergents 34
3.1.5. Complete Quotients 34
3.2. The Properties of Convergents 36
3.2.1. The Difference Between Two Neighbouring Convergents 36
3.2.2. Comparison of Neighbouring Convergents  37
3.2.3. Irreducibility of Convergents 39

Chapter 4. Nonterminating Continued Fractions
4.1. Real Numbers 40
4.1.1. The Gulf Between the Finite and the Infinite . . 40
4.1.2. Principle of Nested Segments 41
4.1.3. The Set of Rational Numbers 44
4.1.4. The Existence of Nonrational Points on the  Number Line 45
4.1.5. Nonterminating Decimal Fractions 46
4.1.6. Irrational Numbers 48
4.1.7. Real Numbers 49
4.1.8. Representing Real Numbers on the Number Line 50
4.1.9. The Condition of Rationality of Nonterminating Decimals 52
4.2. Nonterminating Continued Fractions 52
4.2.1. Numerical Value of a Nonterminating Continued Fraction 52
4.2.2. Representation of Irrationals by Nonterminating Continued Fractions 54
4.2.3. The Single-Valuedness of the Representation of a Real Number by a Continued  Fraction 55
4.3. The Nature of Numbers Given by Continued Fractions 58
4.3.1. Classification of Irrationals 58
4.3.2. Quadratic Irrationals 60
4.3.3. Euler’s Theorem 66
4.3.4. Lagrange Theorem 69

Chapter 5. Approximation of Real Numbers
5.1. Approximation by Convergents 72
5.1.1. High-Quality Approximation 72
5.1.2. The Main Property of Convergents 72
5.1.3. Convergents Have the Highest Quality 76

Chapter 6. Solutions
6.1. The Mystery of Archimedes’ Number 81
6.1.1. The Key to All Puzzles 81
6.1.2. The Secret of Archimedes’ Number 81
6.2. The Solution to the Calendar Problem 83
6.2.1. The Use of Continued Fractions 83
6.2.2. How to Choose a Calendar 84
6.2.3. The Secret of Pope Gregory XIII 86
Bibliography 88

Outros livros em inglês, do mesmo autor, disponíveis no blog:

Images of Geometric Solids (1985)

Álgebra Extraordinaria

I. M. Yaglom

Editorial Mir Moscu | 1983 | 88 páginas | PDF | 8 Mb

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Lecciones populares de matematicas

En Álgebra Extraordinaria se encuentra una muy lúcida exposición del Álgebra de Boole y sus conexiones con el Álgebra de Conjunto y de Proposiciones, así como aplicaciones a Circuitos de Contacto. Este libro refleja la maestría de los Matemáticos Rusos para exponer cuestiones que van más allá de los planes de estudio en la Enseñanza Media, pero de mucho provecho para Profesores y Alumnos de este grado de la enseñanza.

Spanish, Russian (translation)

Problemas elementales de máximo y mínimo, Suma de cantidades infinitamente pequeñas

I. P. Natanson

Ed MIR | 1977 | 112 páginas | PDF | 16,6 Mb

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Se exponen algunos procedimientos elementales para solución, de problemas de máximo y mínimo. La obra está destinada a los alumnos de los grados superiores de la escuela secundaria que deseen adquirir algunas nociones respecto al carácter de los problemas que se examinan en las matemáticas superiores. El material que se expone puede utilizarse en el trabajo de los círculos matemáticos escolares

Mathematical Handbook Elementary Mathematics

M. Vygodsky 

Mir Publishers | 1984 | 422 Páginas | PDF | 9 Mb

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However, it is well to bear in mind that neither handbook, nor textbook alone suffices  to give the reader a knowledge of the subject: he must use pencil and paper and work through the examples and problems for himself.

This is how the the preface of the Mathematical Handbook – Elementary Mathematics by M. Vygodsky ends. The book has about 420 pages and was first published in 1979 by Mir Publishers.

The book has following sections

Tables, Arithmetic, Algebra, Geometry (Plane and Solid), Trigonometry, Functions and Graphs

Fun with Maths and Physics: Brain Teasers Tricks Illusions

Ya. I. Perelman

MIR Publishers | 1988 | 380 páginas | PDF | 12 Mb

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In 1913, in Russian bookshops, appeared a book by the outstanding educationist Yakov Isidorovich Perelman entitled Physics for Entertainment. It struck the fancy of the young who found in it the answers to many of the questions that interested them. Physics for Entertainment not only had an interesting layout, it was also immensely instructive.
In the preface to the 11th Russian edition Perelman wrote: "The main objective of Physics for Entertainment is to arouse the activity of scientific imagination, to teach the reader to think in the spirit of the science of physics and to create in his mind a wide variety of associations of physical knowledge with the widely differing facts of life, with all that he normally comes in contact with." Physics for Entertainment was a best seller.

Probability Theory (First Steps)

E.S. Wentzel

MIR | 1982 | 87 páginas | PDF | 2 Mb

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The author of this booklet describes in popular language how probability theory was developed and found wide application in all fields of modern science. This book can be considered as an introduction towards a more thorough study of probability theory and is intended for a wide circle of readers.

Contents
Probability Theory and Its Problems 6
Probability and Frequency 24
Basic Rules of Probability Theory 40
Random Variables 57
Literature 87

Images of Geometric Solids / Representación de figuras espaciales

N. M. Beskin


Mir Publishers, Moscow | 1985 | 78 páginas | djvu | 663 kb

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Drawing a plane figure is not geometrically difficult because the image drawn is either an exact copy of the original or a similar figure, e.g. the drawing of a circle looks like the original circle. Drawing geometric solids is quite a different matter. Unfortunately, there are no “spatial pencils” which can trace an object in the air. Such a pencil would “draw” a cube by tracing along its edges. Hence, we have to sketch a cube on paper with an ordinary pencil. A plane image will never be an exact copy of a solid and, therefore, a certain routine ought to be followed in drawing a solid that would create an image of the original in the best way.

What is the book about. Descriptive geometry embraces so
many methods that even a brief account would make up a rather thick volume. Therefore, we shall discuss just one of these methods, so as to enable the reader to make stereometric drawings and solve the respective problems…

This book presents a geometric theory of constructing
stereometric drawings. Having mastered this theory, a reader will be able to make the drawings himself rather than have to stick to the few sample ones.

The first chapter presents the theory, the second one is devoted
to its applications (drawing of a cube, a cone, a cylinder, etc.),
and the third one describes a method of plotting the points of an
image if their coordinates are known.

It is these strategies and routine that this book discusses. Though many things are possible with modern computer programs, but the logic may not be known to people who are using them.

The book was translated from the Russian by Valery Barvashov and was first published by Mir in 1985. 

Contents

Chapter 1. Theory
1. The subject matter of the Theory of images .7
2. Requirements of an image. 7
3. What is the book about .8
4. The method of parallel projection …8
5. A comment on notation .10
6. Properties of parallel projections . 11
7. Free images .13
8. Constructing the images of plane figures ..14
9. Some examples of representing polygons .15
10. The image of a circle .16
11. Another viewpoint of constructing the images of plane figures  .17
12. Pohlke-Schwartz theorem.20
13. Representing geometric solids…26
14. Reversibility of an image .28
15. Specified images .30
Chapter 2. Practical Exercises
16. Cross sections of polyhedrons .32
17. Metric problems  ..35
18. Solids of revolution…37
19. The image of a plane..42
20. Inscribed and circumscribed solids.…43
21. Some drawing conventions..46
22. Drawing obvious images…47
Chapter 3. A computation method
23. Theory… 50
24. Application of the computation method …53

Appendix 1. Expression of the Coordinates of the Image Points Using the Coordinates of the Original Points
25. A characteristic property of a linear homogeneous function .62
26. Formulas for the coordinates of the points of an image .64

Appendix 2. The Ellipse

27. Uniform compression  .67

28. The definition of an ellipse ..70

29. Some properties of ellipse …70

30. The ellipse as the projection of a circle .73

31. The cross sections of a circular cylinder …75

3;2. Some constructions connected with the ellipse .76

Representación de figuras espaciales

88 páginas | PDF | 15,7 Mb

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Al representar una figura plana no aparece ninguna dificultad geométrica. El dibujo resulta una copia exacta del original o bien representa una figura semejante al mismo. Cuando consideramos el dibujo de un círculo, nuestra percepción visual es la misma que al considerar el círculo original. La situación es otra por completo cuando se trata de la representación de figuras espaciales. Desgraciadamente, no existe un "lápiz espacial" cuya punta deje huella en el aire. Tal lápiz permitiría "dibujar" un cubo auténtico trazando sus aristas. Pero como no existe, nos vemos obligados a dibujar el cubo desplazando sobre el papel la punta de un lápiz corriente. La imagen plana no puede ser copia exacta de una figura espacial. Esta discordancia plantea el problema: ¿a qué normas atenerse para construir la representación a fin de que ésta dé la imagen más adecuada del original?.

Problems in Geometry

A. Kutepov, A. Rubanov

Mir Publishers, Moscow | 1978| 208 páginas | PDF | 21,1 Mb

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1.ª edição 1975, 2.ª edição 1978.

The book contains a collection of 1351 problems (with answers) in plane and solid geometry for technical schools and colleges. The problems are of varied content, involving calculations, proof, construction of diagrams, and determination of the spatial location of geometrical points. It gives sufficient problems to meet the needs of students for practical work in geometry, and the requirements of the teacher for varied material for tests, etc

Contents

CHAPTER I. REVIEW PROBLEMS

1. The Ratio and Proportionality of Line Segments,
Similarity of Triangles 7
2. Metric Relationships in a Right-Angled Triangle 10
3. Regular Polygons, the Length of the Circumference
and the Arc 15
4. Areas of Plane Figures 17

CHAPTER II. SOLVING TRIANGLES

5. Solving Right-Angled Triangles 22
6. Solving Oblique Triangles 29
Law of Cosines 29
Law of Sines 31
Areas of Triangles, Parallelograms and
Quadrilaterals 32
Basic Cases of Solving Oblique Triangles 34
Particular Cases of Solving Oblique Triangles 34
Heron’s Formula 35
Radii r and R of Inscribed and Circumscribed Circles and the Area S of a Triangle 36
Miscellaneous Problems 37

CHAPTER III. STRAIGHT LINES AND PLANES IN SPACE

7. Basic Concepts and Axioms. Two Straight Lines in Space 43
8. Straight Lines Perpendicular and Inclined to a Plane 46
9. Angles Formed by a Straight Line and a Plane  52
10. Parallelism of a Straight Line and a Plane 55
11. Parallel Planes  59
12. Dihedral Angles. Perpendicular Planes 63
13. Areas of Projections of Plane Figures 67
14 Polyhedral Angles 69

CHAPTER IV. POLYHEDRONS AND ROUND SOLIDS

15. Prisms and Parallelepipeds 71
16. The Pyramid 77
17. The Truncated Pyramid 81
18. Regular Polyhedrons 84
19. The Right Circular Cylinder 86
20. The Right Circular Cone 89
21. The Truncated Cone 93

CHAPTER V. AREAS OF POLYHEDRONS AND ROUND SOLIDS

22. Areas of Parallelepipeds and Prisms 97
23. Areas of Pyramids 102
24. Areas of Truncated Pyramids 105
25. Areas of Cylinders 108
26. Areas of Cones Ill
27. Areas of Truncated Cones 115

CHAPTER VI. VOLUMES OF POLYHEDRONS AND ROUND SOLIDS

28. Volumes of Parallelepipeds 118
29. Volumes of Prisms 122
30. Volumes of Pyramids 127
31. Volumes of Truncated Pyramids 133
32. Volumes of Cylinders 137
33. Volumes of Cones 141
34. Volumes of Truncated Cones 145

CHAPTER VII. THE SPHERE

35. Spheres 149
36. Areas of Spheres and Their Parts 152
37. Volumes of Spheres and Their Parts 155
38. Inscribed and Circumscribed Spheres 159

CHAPTER VIII. APPLYING TRIGONOMETRY TO SOLVING GEOMETRIC PROBLEMS

39. Polyhedrons 164
40. Round Solids 168
41. Areas and Volumes of Prisms 172
42. Areas and Volumes of Pyramids  176
43. Areas and Volumes of Round Solids . 181
Answers 187

Combinatorial Mathematics for Recreation

N. Vilenkin

MIR Publishers | 1972 | 207 páginas | Djvu | 3,76 Mb

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Division de Un Segmento en La Razao Dada

N. M. Beskin

MIR | 1976 | PDF | 14 Mb

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O objetivo deste livro é levar o leitor à profundidade. Análise de um problema mais básico ("dividir o segmento em uma razão dada adquirir novas habilidades que os conhecimentos complementares matemáticos podem ser ampliados para o comprimento ea largura do sujeito do conhecimento Ampliar significa estudar novos ramos da matemática;.. Aprofundar significa uma análise aprofundada questões que fazem parte do currículo. Não há matemático que a pretensão de saber todo o caminho deste ou daquele ramo de lamateria, como no estudo de cada problema, é o mais elementar, sempre revelar ligações inesperadas com outros problemas, Portanto, este processo de aprofundamento é interminável.

Fracciones Maravillosas

Beskin

Mir | 1987 | 98 páginas | PDF | 11 Mb

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Este libro está destinado a escolares quienes se interesan por las matemáticas. Está dedicado a uno de los apartados más cautivadores de la aritmética, la aproximación de los números reales mediante los racionales. La teoría de las fracciones continuas es bastante amplia. En el presente libro se exponen únicamente los datos fundamentales. Sin embargo, en el mismo está presente todo lo que debe conocer cada uno quien se interesa por las matemáticas. Los especialistas deben conocer mucho más.

El método de coordenadas / The Method of Coordinates

I. Gelfand E. Glagolieva A. Kirillow.

Mir Moscú | 1968 | 95 páginas | PDF | 15 Mb

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En este opúsculo se expone, en forma muy didáctica, el sistema de coordenadas de un punto en una recta, en un plano o en el espacio como una preparación para lo medular de él: introducir el espacio de dimensiones y una serie de aplicaciones. Sólo se requieren los conocimientos a nivel de Enseñanza Media y una fuerte disposición para estudiarlo con atención

The Method of Coordinates 

Birkhäuser Boston | 1990 | 84 páginas | PDF |  2,6 Mb

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The need for improved mathematics education at the high school and college levels has never been more apparent than in the 1990s. As early as the 1960s, I. M. Gel'fand and his colleagues in the USSR thought hard about this same question and developed a style for presenting basic mathematics in a clear and simple form that engaged the curiosity and intellectual interest of thousands of high school and college students. These same ideas, this same content, unchanged by over thirty years of experience and mathematical development, are available in the present books to any student who is willing to read, to be stimulated and to learn. "The Method of Coordinates" is a way of transferring geometric images into formulas, a method for describing pictures by numbers and letters denoting constants and variables. It is fundamental to the study of calculus and other mathematical topics. Teachers of mathematics will find here a fresh understanding of the subject and a valuable path to the training of students in mathematical concepts and skills.

Sistemas de desigualdades lineales / Systems of Linear Inequalities

A. S. Solodovnikov

Lecciones populares de matemáticas

Editorial Mir - Moscu | PDF | 16 Mb

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El libro que ofrecemos al lector tarta sobre la relación entre sistemas de desigualdades lineales y poliedros convexos, contiene la descripción de los conjuntos de todas las soluciones para desigualdades lineales; estudia las cuestiones de compatibilidad e incompatibilidad; en el se da una introducción elemental a la programación lineal que, de hecho, es uno de los capítulos de la teoría de desigualdades lineales. En el último párrafo se expone el método de solución del problema del transporte en la programación lineal 

Algunos hechos de la geometria analitica. Sentido geometrico de un sistema de desigualdades lineales con 2 ó 3 incognitas.Capsula convexa de un sistema de puntos. Region de soluciones...

Systems of Linear Inequalities

MIR Publishers |1979 | 123 páginas | PDF | 3,9 Mb

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The book tells about the relation of systems of linear inequalies to convex polyhedra, gives a description of the set of all solutions of a system of linear inequalities, analyses the questions of compatibility and incompatibility; finally, it gives an insight into linear programming as one of the topics in the theory of systems of linear inequalities. The last section but one gives a proof of the duality theorem of linear programming. The book is intended for senior pupils and all amateur mathematicians. 

CONTENTS
Preface 7
1. Some Facts from Analytic Geometry 8
2. Visualization of Systems of Linear Inequalities In Two or Three Unknowns 17
3. The Convex Hull of a System of Points 22
4. A Convex Polyhedral Cone 25
5. The Feasible Region of a System of Linear Inequalities in Two Unknowns 31
6. The Feasible Region of a System in Three Unknowns 44
7. Systems of Linear Inequalities in Any Number of Unknowns 52
8. The Solution of a System of Linear Inequalities ‘by Successive Reduction of the Number of Unknowns 57
9. Incompatible Systems 64
10. A Homogeneous System of Linear Inequalities. The Fundamental Set of Solutions 69
11. The Solution of a Nonhomogeneous System of Inequalities 81
12. A Linear Programming Problem 84
13. The Simplex Method 91
14. The Duality Theorem in Linear Programming 101
1.5. Transportation Problem 107 

Acerca de la geometria de Lobachevski / Lobachevskian Geometry

A. S. Smogorzhevski

Editorial Mir Moscu | 1978 | 80 páginas | PDF | 13 Mb

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Se da a conocer al lector los fundamentos principales de la geometría no euclidiana de Lobachevski. Las geometrías de Euclides y Lobachevski tienen mucho de común; en ellas sólo con diferentes las definiciones, los teoremas y las fórmulas ligadas al axioma del paralelismo. Para comprender qué es lo que suscitó esta diferencia se debe examinar cómo surgieron y desarrollaron las nociones geométricas fundamentales.

Lobachevskian Geometry (Little Mathematics Library) 

Mir |1982 | 73 páginas | djvu | 633 kb

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The aim of this book is to acquaint the reader with the fundamentals of Lobachevsky's non-Euclidean geometry.
The famous Russian mathematician N. I. Lobachevsky was an outstanding thinker, to whom is credited one of the greatest mathematical discoveries, the construction of an original geometric system distinct from Euclid's geometry. The reader will find brief biography of N. I. Lobachevsky in Sec. I.

Euclidean and Lobachevskian geometries have much in common, differing only in their definitions, theorems and formulas as regards the parallel-postulate. To clarify the reasons for these differences we must consider how the basic geometric concepts originated and developed, which is done in Sec. 2.

Apart from a knowledge of school plane geometry and trigonometry reading our pamphlet calls for a knowledge of the transformation known as inversion, the most important features of which are reviewed in Sec. 3.

We hope that the reader will be able to grasp its principles with profit to himself and without great difficulty, since it, and Sec. 10, play very important, though ancillary, role in our exposition.  

Contents are as under:

Author’s Note 7
1. A Brief Essay on the Life and Work of N. I. Lobachevsky 9
2. On the Origin of Axioms and Their Role in Geometry  11
3. Inversion 21
4. Map of a Lobachevskian Plane  29
5. A Circle in a Lobachevskian Plane 42
6. Equidistant Curve 46
7. Horocycle 47
8. Selected Theorems of Lobachevskian Geometry 49
9. Supplementary Remarks 52
10. On Natural Logarithms and Hyperbolic Functions 53
11. Measurement of Segments of Hyperbolic Straight Lines 57
12 Basic Formulas of Hyperbolic Trigonometry 60
13. The Lengths of Certain Plane Curves in Lobachevskian Geometry 64
Conclusion 68

Curvas maravillosas, Números complejos y representaciones conformes, Funciones maravillosas / Remarkable Curves

Curvas maravillosas, Números complejos y representaciones conformes, Funciones maravillosas

A. I. Markushevich

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Este libro esta destinado principalmente para los escolares y todos los que esténinteresados en ampliar sus conocimientos matemáticos adquiridos en la escuela. Se basa en una conferencia que dicto el autor a un grupo de alumnos moscovitas de séptimo y octavo grados.

Al preparar las publicación de la conferencia, el autor la ha ampliado un poco tratando de conservar el estilo accesible de la exposición. El complemento mas esencial es el punto 13 en el que se trata de la elipse, la hipérbola y la parábolaen tanto que secciones de una superficie cónica

Remarkable Curves 

MIR Publishers | 1980 | djvu | 1,08 Mb

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This book has been written mainly for high school students, but it will also be helpful to anyone studying on their own whose mathematical education is confined to high school mathematics. The book is based on a lecture I gave to Moscow schoolchildren of grades 7 and 8 (13 and 14 years old).
In preparing the lecture for publication I expanded the material, while at the same time trying not to make the treatment any less accessible. The most substantial addition is Section 13 on the ellipse, hyperbola and parabola viewed as conic sections.
For the sake of brevity most of the results on curves are given without proof, although in many cases their proofs could have been given in a form that readers could understand.
The third Russian edition is enlarged by including the results on Pascal's and Brianchon's theorems (on inscribed and circumscribed hexagons), the spiral of Archimedes, the catenary, the logarithmic spiral and the involute of a circle. 

Preface to the Third Russian Edition
1. The Path Traced Out by a Moving Point
2. The Straight Line and the Circle
3. The Ellipse
4. The Foci of an Ellipse
5. The Ellipse is a Flattened Circle
6. Ellipses in Everyday Life and in Nature
7. The Parabola
8. The Parabolic Mirror
9. The Flight of a Stone and a Projectile
10. The Hyperbola
11. The Axes and Asymptotes of the Hyperbola
12. The Equilateral Hyperbola
13. Conic Sections
14. Pascal’s Theorem
15. Brianchon’s Theorem
16. The Lemniscate of Bernoulli
17. The Lemniscate with Two Foci
18. The Lemniscate with Arbitrary Number of Foci
19. The Cycloid
20. The Curve of Fastest Descent
21. The Spiral of Archimedes
22. Two Problems of Archimedes
23. The Chain of Galilei
24. The Catenary
25. The Graph of the Exponential Function
26. Choosing the Length of the Chain
27. And What if the Length is Different?
28. All Catenarics are Similar
29. The Logarithmic Spiral
30. The Involute of a Circle
Conclusion 

División Inexacta

A. A. Belski, L. A. Kaluzhnin

MIR | PDF | 10,2 Mb

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En esta pequeña obra se tratan algunos problemas interesantes de la teoría de los números. Se da la demostración del teorema de la unidad de la descomposición en factores primos, se estudian el algoritmo de Euclides, las ecuaciones diofánticas, la aritmética de los números complejos enteros y las clases residuales, la representación de los números en los diversos sistemas posicionales, etc. Este libro está dedicado a los alumnos de las escuelas especiales físico-matemáticas. Será de utilidad a los profesores de matemáticas de los centros de Enseñanza Media y a los alumnos de los últimos grados de dicha enseñanza.

The Decomposition of Figures Into Smaller Parts

 (Popular Lectures in Mathematics)

Vladimir Grigor'evich Boltyanskii, Izrail' Tsudikovich Gohberg, Henry Christoffers, Thomas P. Branso
University Of Chicago Press | 1980 | 80 páginas | djvu | 935 kb

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In contrast to the vast literature on Euclidean geometry as a whole, little has been published on the relatively recent developments in the field of combinatorial geometry. Boltyanskii and Gohberg's book investigates this area, which has undergone particularly rapid growth in the last thirty years. By restricting themselves to two dimensions, the authors make the book uniquely accessible to interested high school students while maintaining a high level of rigor. They discuss a variety of problems on figures of constant width, convex figures, coverings, and illumination. The book offers a thorough exposition of the problem of cutting figures into smaller pieces. The central theorem gives the minimum number of pieces into which a figure can be divided so that all the pieces are of smaller diameter than the original figure. This theorem, which serves as a basis for the rest of the material, is proved for both the Euclidean plane and Minkowski's plane.

Preface 
1. Division of Figures into Pieces of Smaller Diameter 
1.1. The Diameter of a Figure 
1.2. Formulation of the Problem 
1.3. Borsuk's Theorem 
1.4. Convex Figures 
1.5. Figures of Constant Width 
1.6. Embedding in a Figure of Constant Width 
1.7. For Which Figures is a(F) = 3? 
2. Division of Figures in the Minkowski Plane 
2.1. A Graphic Example 
2.2. The Minkowski Plane 
2.3. Borsuk's Problem in Minkowski Planes 
3. The Covering of Convex Figures by Reduced Copies 
3.1. Formulation of the Problem 
3.2. Another Formulation of the Problem 
3.3. Solution of the Covering Problem 
3.4. Proof of Theorem 2.2 
4. The Problem of Illumination 
4.1. Formulation of the Problem 
4.2. Solution of the Problem of Illumination 
4.3. The Equivalence of the Last Two Problems 
4.4. Division and Illumination of Unbounded Convex Figures 
Remarks

Figures for Fun: Stories and Conundrums

Yakov I. Perelman

Central Books Ltd | 1973 | 153 páginas | djvu | 4,72 Mb

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um blog com alguns problemas retirados deste livro: figforfun


To read and enjoy this book it suffice to possess a modest knowledge of mathomatics, i.e., knowledge of arithmetical rules  and elementary geometry. Very few problem' require the ability of forming and solving equations-and the simplest at that.
The subjects range from a mostley collection of conundrums and mathe-matical stunts to useful practical problems on counting