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|title=Basic Mathematics
|title=Basic Mathematics
|image=[[File:Lang Basic Mathematics Cover.jpg]]
|image=[[File:Lang Basic Mathematics Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/Serge_Lang Serge Lang]
|author=[https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov]
|language=English
|language=English
|series=
|series=
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|isbn13=978-0387967875
|isbn13=978-0387967875
}}
}}
The textbook '''''Basic Mathematics''''' by [https://en.wikipedia.org/wiki/Serge_Lang Serge Lang] provides an overview of mathematical topics usually encountered through the end of high school/secondary school, specifically arithmetic, algebra, trigonometry, logic, and geometry. It serves as a solid review no matter how far along one may be in their studies, be it just beginning or returning to strengthen one's foundations.
The textbook '''''Linear Algebra''''' by [https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov] provides a thorough introduction to linear algebra.
 
Reading the Foreword and the Interlude is recommended for those unfamiliar with reading math texts.


== Table of Contents ==
== Table of Contents ==
Line 23: Line 21:
! Chapter/Section # !! Title !! Page #
! Chapter/Section # !! Title !! Page #
|-  
|-  
! colspan="3" | PART I: ALGEBRA
! colspan="3" | Chapter 1: Determinants
|-
|-
! colspan="3" | Chapter 1: Numbers
| 1.1 || Number Fields || 1
|-
|-
| 1 || The integers || 5
| 1.2 || Problems of the Theory of Systems of Linear Equations || 3
|-
|-
| 2 || Rules for addition || 8
| 1.3 || Determinants of Order \(n\) || 5
|-
|-
| 3 || Rules for multiplication || 14
| 1.4 || Properties of Determinants || 8
|-
|-
| 4 || Even and odd integers; divisibility || 22
| 1.5 || Cofactors and Minors || 12
|-
|-
| 5 || Rational numbers || 26
| 1.6 || Practical Evaluation of Determinants || 16
|-
|-
| 6 || Multiplicative inverses || 42
| 1.7 || Cramer's Rule || 18
|-
| 1.8 || Minors of Arbitrary Order. Laplace's Theorem || 20
|-  
| 1.9 || Multiplicative inverses || 23
|-
| colspan="2" | Problems || 28
|-  
|-  
! colspan="3" | Chapter 2: Linear Equations
! colspan="3" | Chapter 2: Linear Equations

Revision as of 16:12, 21 September 2021

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Basic Mathematics
Lang Basic Mathematics Cover.jpg
Information
Author Georgi Shilov
Language English
Publisher Springer
Publication Date 1 July 1988
Pages 496
ISBN-10 0387967877
ISBN-13 978-0387967875

The textbook Linear Algebra by Georgi Shilov provides a thorough introduction to linear algebra.

Table of Contents

Chapter/Section # Title Page #
Chapter 1: Determinants
1.1 Number Fields 1
1.2 Problems of the Theory of Systems of Linear Equations 3
1.3 Determinants of Order \(n\) 5
1.4 Properties of Determinants 8
1.5 Cofactors and Minors 12
1.6 Practical Evaluation of Determinants 16
1.7 Cramer's Rule 18
1.8 Minors of Arbitrary Order. Laplace's Theorem 20
1.9 Multiplicative inverses 23
Problems 28
Chapter 2: Linear Equations
1 Equations in two unknowns 53
2 Equations in three unknowns 57
Chapter 3: Real Numbers
1 Addition and multiplication 61
2 Real numbers: positivity 64
3 Powers and roots 70
4 Inequalities 75
Chapter 4: Quadratic Equations
Interlude: On Logic and Mathematical Expressions
1 On reading books 93
2 Logic 94
3 Sets and elements 99
4 Notation 100
PART II: INTUITIVE GEOMETRY
Chapter 5: Distance and Angles
1 Distance 107
2 Angles 110
3 The Pythagoras theorem 120
Chapter 6: Isometries
1 Some standard mappings of the plane 133
2 Isometries 143
3 Composition of isometries 150
4 Inverse of isometries 155
5 Characterization of isometries 163
6 Congruences 166
Chapter 7: Area and Applications
1 Area of a disc of radius r 173
2 Circumference of a circle of radius r 180
PART III: COORDINATE GEOMETRY
Chapter 8: Coordinates and Geometry
1 Coordinate systems 191
2 Distance between points 197
3 Equation of a circle 203
4 Rational points on a circle 206
Chapter 9: Operations on Points
1 Dilations and reflections 213
2 Addition, subtraction, and the parallelogram law 218
Chapter 10: Segments, Rays, and Lines
1 Segments 229
2 Rays 231
3 Lines 236
4 Ordinary equation for a line 246
Chapter 11: Trigonometry
1 Radian measure 249
2 Sine and cosine 252
3 The graphs 264
4 The tangent 266
5 Addition formulas 272
6 Rotations 277
Chapter 12: Some Analytic Geometry
1 The straight line again 281
2 The parabola 291
3 The ellipse 297
4 The hyperbola 300
5 Rotation of hyperbolas 305
PART IV: MISCELLANEOUS
Chapter 13: Functions
1 Definition of a function 313
2 Polynomial functions 318
3 Graphs of functions 330
4 Exponential function 333
5 Logarithms 338
Chapter 14: Mappings
1 Definition 345
2 Formalism of mappings 351
3 Permutations 359
Chapter 15: Complex Numbers
1 The complex plane 375
2 Polar form 380
Chapter 16: Induction and Summations
1 Induction 383
2 Summations 388
3 Geometric series 396
Chapter 17: Determinants
1 Matrices 401
2 Determinants of order 2 406
3 Properties of 2 x 2 determinants 409
4 Determinants of order 3 414
5 Properties of 3 x 3 determinants 418
6 Cramer's Rule 424
Index 429
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