The textbook was written in accordance with the program for studying mathematics in institutions of NPO and SPO and covers all the main topics: number theory, roots, powers, logarithms, lines and planes, spatial bodies, as well as the basics of trigonometry, analysis, combinatorics and probability theory. For students in institutions of primary and secondary vocational education.
Integers and rational numbers.
What do we know about numbers?
1. Natural numbers. Natural numbers are constructed constructively, starting from one, adding one unit at each step: 1, 1 + 1, 1 + 1 + 1, ...
The writing of natural numbers has a long history. Modern society uses a decimal system in which 10 digits are entered: 1, 2=1 + 1, 3 = 2+1, ..., 9 = 8 + 1 and 0. The number following the number 9 is written as 10. Further, counting in tens, hundreds (10 x 10), thousands, etc., each natural number represent in the form a0 + + a1 10 + ... + ak 10k (ak = 0), where 0< аi < 9, и записываем последовательностью цифр akak-1 ... a0.
In computer science, the binary system plays an important role, using two digits: 0 and 1 - and based on the representation of a number as the sum of powers of the number 2, which in the binary system has the notation 10.
Table of contents
Basic notation 3
Preface 4
Chapter 1. DEVELOPMENT OF THE CONCEPT OF THE NUMBER 5
Lesson 1. Integers and rational numbers 5
Lesson 2. Real numbers 9
Lesson 3. Approximate calculations 13
Lesson 4. Complex numbers 16
Conversation. Numbers and roots of equations 20
Chapter 2
Lesson 1. Review of the past 24
Lesson 2. Root nth degree 27
Lesson 3. Degrees 31
Lesson 4. Logarithms 35
Lesson 5. Demonstrative and logarithmic functions 38
Lesson 6. Exponential and logarithmic equations and inequalities 44
Conversation. Calculating powers and logarithms 47
Chapter 3. LINES AND PLANES IN SPACE 50
Lesson 1. Mutual arrangement of lines and planes 50
Lesson 2. Parallelism of lines and planes 54
Lesson 3. Angles between lines and planes 56
Conversation. Geometry of Euclid 59
Chapter 4. COMBINATORICS 64
Lesson 1. Combinatorial constructions 64
Lesson 2. Rules of combinatorics 67
Lesson 3. Number of orbits 70
Conversation. From the history of combinatorics 75
Chapter 5. COORDINATES AND VECTORS 77
Lesson 1 Review 77
Lesson 2. Coordinates and vectors in space 81
Lesson 3. Dot product 83
Lesson 4. Perpendicularity of lines and planes 86
Conversation. Vector space 88
Chapter 6. BASICS OF TRIGONOMETRY 91
Lesson 1. Angles and rotary motion 91
Lesson 2. Trigonometric operations 96
Lesson 3. Transformation trigonometric expressions 101
Lesson 5. Trigonometric equations 112
Conversation. Historical information 118
Chapter 7. FUNCTIONS AND GRAPHS
Session 1 Review general concepts 120
Lesson 2: The Function Study Outline 123
Lesson 3. Function transformations and actions on them 129
Lesson 4. Symmetry of functions and transformation of their graphs 133
Lesson 5: Function Continuity 137
Conversation. Development of the concept of 13S function
Chapter 8. Polyhedra and round bodies 141
Lesson 1. Dictionary of geometry
Lesson 2. Parallelepipeds and prisms
Lesson 3. Pyramids
Lesson 4. Round bodies
Lesson 5. Regular polyhedra
Conversation. Platonic solids 15
Chapter 9. BEGINNINGS OF MATHEMATICAL ANALYSIS 15
Lesson 1. Process and its modeling 15
Lesson 2. Sequences 16
Lesson 3. The concept of a derivative
Lesson 4. Differentiation formulas
Lesson 5. Derivatives elementary functions 171
Lesson 6. Applying the derivative to the study of functions 18
Lesson 7. Applications 18
Lesson 8. Antiderivative 19
Conversation. Taylor Formula 19
Chapter 10. THE INTEGRAL AND ITS APPLICATIONS 19
Lesson 1. Squares flat figures 19
Lesson 2. Newton's theorem - Leibniz 19
Lesson 3. Spatial bodies 20
Conversation. Integral values 21
Chapter 11. ELEMENTS OF THE THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS 21
Lesson 1. Probability and its properties
Lesson 2. Retests
Lesson 3. Random value
Conversation. Origin of probability theory
Chapter 12. EQUATIONS AND INEQUALITIES 22
Lesson 1. Equivalence of the equation
Lesson 2. Basic techniques for solving equations 23
Lesson 3. Systems of Equations 23
Lesson 4. Solving inequalities
Conversation. Resolvability algebraic equations 24
Answers 24.
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3rd ed. - M.: 2017.- 256 p. M.: 2014.- 256 p.
The textbook was written in accordance with the program for studying mathematics in institutions of NPO and SPO and covers all the main topics: number theory, roots, powers, logarithms, lines and planes, spatial bodies, as well as the basics of trigonometry, analysis, combinatorics and probability theory. For students in institutions of primary and secondary vocational education.
Format: pdf(2017, 256s.)
Size: 8.6 MB
Watch, download:drive.google
Format: pdf(2014, 256s.)
Size: 52.6 MB
Watch, download:drive.google
Table of contents
Basic notation 3
Preface 4
Chapter 1. DEVELOPMENT OF THE CONCEPT OF NUMBER 7
Lesson 1. Integers and rational numbers 7
Lesson 2. Real numbers 11
Lesson 3. Approximate calculations 15
Lesson 4. Complex numbers 18
Conversation. Numbers and roots of equations 22
Chapter 2
Lesson 1 Review 26
Lesson 2. Root nth degree 29
Lesson 3. Degrees 33
Lesson 4. Logarithms 37
Lesson 5. Exponential and logarithmic functions 40
Lesson 6. Exponential and logarithmic equations and inequalities 46
Conversation. Calculating powers and logarithms 49
Chapter 3. LINES AND PLANES IN SPACE 52
Lesson 1. Mutual arrangement of lines and planes 52
Lesson 2. Parallelism of lines and planes 56
Lesson 3. Angles between lines and planes 58
Conversation. Geometry of Euclid 61
Chapter 4. COMBINATORICS 66
Lesson 1. Combinatorial constructions 66
Lesson 2. Rules of combinatorics 69
Lesson 3. Number of orbits 72
Conversation. From the history of combinatorics 77
Chapter 5. COORDINATES AND VECTORS 79
Lesson 1 Review 79
Lesson 2. Coordinates and vectors in space 83
Lesson 3. Dot product 85
Lesson 4. Perpendicularity of lines and planes 88
Conversation. Vector space 90
Chapter 6. BASICS OF TRIGONOMETRY 93
Lesson 1: Angles and rotation 93
Lesson 2, Trigonometric operations 98
Lesson 3. Transforming trigonometric expressions 103
Lesson 4. Trigonometric functions 109
Lesson 5. Trigonometric Equations 114
Conversation. From the history of trigonometry 120
Chapter 7. FUNCTIONS AND GRAPHICS 122
Lesson 1: Overview of General Concepts 122
Lesson 2. The scheme of the study of function 127
Lesson 3. Function transformations and actions on them 131
Lesson 4. Symmetry of functions and transformation of their graphs 136
Lesson 5: Function Continuity 139
Conversation. Development of the concept of function 141
Chapter 8, Polyhedra and Round Bodies 143
Lesson 1. Dictionary of geometry 143
Lesson 2. Parallelepipeds and prisms 145
Lesson 3. Pyramids 148
Lesson 4. Round bodies 151
Lesson 5. Regular polyhedra 154
Conversation. Platonic Solids 157
Chapter 9. BEGINNINGS OF MATHEMATICAL ANALYSIS 159
Lesson 1. Process and its modeling 159
Lesson 2 Sequences 165
Lesson 3. The concept of a derivative 171
Lesson 4. Differentiation formulas 176
Lesson 5. Derivatives of elementary functions 180
Lesson 6. Applying the derivative to the study of functions 183
Lesson 7. Applications 187
Lesson 8. Antiderivative 193
Conversation. Taylor Formula 195
Chapter 10. THE INTEGRAL AND ITS APPLICATIONS 198
Lesson 1. Areas of plane figures 198
Lesson 2. Newton-Leibniz theorem 201
Lesson 3. Spatial bodies 207
Conversation. Integral quantities 213
Chapter 11. ELEMENTS OF THE THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS 219
Lesson 1. Probability and its properties 219
Lesson 2: Retesting 222
Lesson 3. Random variable 225
Conversation. The origin of the theory of probability 228
Chapter 12. EQUATIONS AND INEQUALITIES 230
Lesson 1 Equivalence of Equations 230
Lesson 2. Basic techniques for solving equations 233
Lesson 3 Systems of Equations 238
Lesson 4. Solving inequalities 242
Conversation, Solvability of algebraic equations 247
Answers 249
Foreword
Mathematics for 2500 years of its existence has accumulated the richest tool for the study of the world around us. However, as Academician A.N. Krylov, an outstanding Russian mathematician and shipbuilder, noted, a person turns to mathematics "not to admire innumerable treasures." First of all, he needs to get acquainted with "centuries of proven tools and learn how to use them correctly and skillfully."
This book will teach you how to use mathematical tools such as functions and their graphs, geometric figures, vectors and coordinates, derivative and integral. Although you may have had your first exposure to some of these concepts earlier, the book presents x anew. This is convenient for those who have forgotten the previously studied material a little, and is useful for everyone, since even familiar things will reveal new aspects and connections.
To facilitate the work with the textbook, the most important provisions and formulations are highlighted. Illustrations play an important role, therefore it is necessary to carefully consider the drawing related to the text for a better understanding of the text (even in ancient times they used this method of studying mathematics - they drew a drawing and said: “Look!”).
In addition to the undoubted practical value of the mathematical knowledge obtained, the study of mathematics leaves an indelible mark on the soul of every person. With mathematics, many associate objectivity and honesty, the desire for truth and the triumph of reason. Many people have a lifetime of self-confidence, which arose when overcoming the undoubted difficulties that they encountered in the study of mathematics. Finally, most of you are open to the perception of the harmony and beauty of the world that mathematics has absorbed, so you should not approach each page of the textbook, each task with an assessment of whether it will be used in the new life that awaits you after graduation .
The topics to which the textbook is devoted are number theory, spatial bodies, fundamentals mathematical analysis, the beginnings of the theory of probability - have not only applied value. They contain rich ideas, familiarization with which is necessary for every person.
It is hoped that the study of mathematics, which / textbook should help, will allow you to make sure high level of their abilities, will strengthen the desire to continue their education and bring many joyful minutes of communication with the "unshakable laws that mark the entire order of the universe."
30.12.2016 22:52
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