The unique style of Qi Bo's content
-Wu Daren's Guide to Elementary Mathematics from a High Viewpoint.
In memory of Klein
-Qi introduced, forward-looking elementary mathematics.
Preface to the first edition
Preface to the third edition
English version order
order
The first part of arithmetic
Chapter 1 Operation of Natural Numbers
Introduction of the concept of school mileage in 1. 1
Basic law of 1.2 operation
The Logical Basis of 1.3 Integer Operation
The second chapter is the first expansion of the concept of number.
2. 1 negative number
2.2 score
2.3 irrational numbers
The third chapter is about the special properties of integers.
Chapter IV Complex Numbers
4. 1 common plural
4.2 Higher-order complex numbers, especially quaternions
4.3 Multiplication-Rotation and Extension of Quaternions
4.4 Complex teaching in middle schools
Attachment: On the Modern Development and General Structure of Mathematics
Part II Algebra
Chapter V Real Equations with Real Unknown Numbers
5. 1 single parameter equation
5.2 two-parameter equation
5.3 Equation with Three Parameters λ, μ and ν
Chapter VI Complex Field Equation
6. Basic Theorem of1Algebra
6.2 Complex parameter equation
The third part analyzes
Chapter VII Logarithmic Function and Exponential Function
7. Systematic discussion on1algebraic analysis
7.2 the historical development of theory
7.3 Middle School Logarithmic Theory
7.4 The viewpoint of functional theory
Chapter VIII Angle Function
8.1angle function theory
8.2 trigonometric function table
8.3 Application of Angle Function
Chapter 9 is about calculus itself
9. 1 General considerations in calculus
9.2 Taylor Theorem
9.3 Historical and teaching considerations
appendix
I. Transcendence of Numbers E and π
Ⅱ. Set theory
Table of Contents of Volume II
Preface to the first edition
Preface to the third edition
English version order
order
The fourth part is the simplest geometric manifold.
Chapter 10 Line segment, area and volume as relative quantities.
Chapter 11 grassmann determinant principle on the plane.
Chapter 12 grassmann's spatial principle
Chapter 13 Classification of Spatial Basic Graphics under Cartesian Coordinate Transformation
Chapter 14 Derived Manifolds
The fifth part geometric transformation
Chapter 15 Affine Transformation
Chapter 16 Projection transformation
Chapter 17 High-order Point Transformation
17. 1 inverse evolutionary change
17.2 Some more general mapping projections
17.3 the most general reversible single-valued continuous point transformation
Chapter 18 Transformation caused by the change of spatial elements
18. 1 double conversion
18.2 tangent transformation
18.3 some examples
Chapter 19 Imaginary number theory
The sixth part systematically discusses geometry and its foundation.
Chapter 20 Systematic discussion
20. 1 Overview of geometric structure
20.2 Invariant Theory of Linear Substitution
20.3 Application of Invariant Theory in Geometry
20.4 Kelly principle and systematization of affine geometry and metric geometry
Chapter 21 Geometric Basis
2 1. 1 Plane geometric system emphasizing motion
2 1.2 another development system of metric geometry-the function of parallel axioms
2 1.3 Euclidean geometric elements
Table of Contents of Volume III
Translator's words
Preface to the first edition
Preface to the third edition
order
The seventh part is the real variable function and its representation in rectangular coordinates.
Chapter 22 Explanation of Single Independent Variable X
22. 1 empirical precision and abstract precision, modern real number concept
22.2 Exact mathematics and approximate mathematics are also divided in pure geometry.
22.3 Intuition and thinking, from different aspects of geometry.
22.4 Use two theorems about point sets to illustrate.
Chapter 23 unary function x y=f(x)
23. The abstract determination and empirical determination of1function (function with concept)
23.2 On the guiding role of spatial intuition
23.3 Accuracy of natural laws (different views on material composition)
23.4 Properties of Empirical Curve: Connectivity, Direction and Curvature
23.5 How similar is the Cauchy definition of continuous function to the empirical curve?
23.6 Integrability of continuous functions
23.7 Existence Theorem of Maximum and Minimum Value
23.8 Four Generalized Derivatives
23.9 Weierstrass nondifferentiable function; An overview of its image
23. Non-differentiability of10 Wilstrass function
23. 1 1 "reasonable" function
Chapter 24 Approximate Representation of Functions
24. 1 Approximate the empirical curve with a reasonable function.
24.2 Approximate expression of reasonable function with simple analytical formula
24.3 Lagrange interpolation formula
24.4 Taylor Theorem and Taylor Series
24.5 Approximate Expression of Integral and Derivative Functions by Lagrange Polynomials
24.6 About analytic function and its role in explaining nature
24.7 infinite trigonometric series interpolation method
Chapter 25 further expounds the trigonometric function representation of functions.
25. Error estimation in1empirical function representation
25.2 trigonometric series interpolation obtained by least square method
25.3 Harmonic Analyzer
25.4 examples of trigonometric series
25.5 Chebyshev's work on interpolation
Chapter 26 Binary Functions
26. 1 continuity
26.2 Examples of partial derivation of order reversal
26.3 Approximate representation of functions on a sphere by spherical function series.
26.4 Value Distribution of Spherical Function on Sphere
26.5 Error estimation of approximate representation of series with perfect spherical function
Part VIII Free Geometry of Plane Curves
Chapter 27 discusses plane geometry from the point of view of exact theory.
27. Several theorems about point sets in1
27.2 A point set generated by the inversion of two or more disjoint circles.
27.3 Properties of Limit Point Set
27.4 the concept of two-dimensional continuum and general curve
27.5 peano curve covering the whole square.
27.6 The narrow concept of curve: Jordan curve
27.7 A narrower concept of curve: regular curve.
27.8 Approximate representation of intuitive curve and conventional ideal curve
27.9 Perceptibility of Ideal Curve
27. 10 special ideal curve: analytic curve and algebraic curve, and grassmann geometry generation method of algebraic curve.
27. 1 1 expresses the experience graph with ideal graph; Pere's point of view
Chapter 28 continues to discuss plane geometry from the perspective of accurate theory.
28. 1 Continuous Inverse of Two Tangent Circles
28.2 Continuous inversion of tangent circles of three circles ("module diagram")
28.3 Standard Model of Tangent Circle of Four Circles
28.4 General paragraph of four-circle tangent circle
28.5 Characteristics of Non-analytic Curve Obtained
28.6 The premise of the whole discussion is Willoni's further idealization.
Chapter 29 Turning to Applied Geometry: A. Measurement
29. 1 Inaccuracy of all actual measurements, the practice of Sneijos project.
29.2 Accuracy is determined by redundant measurement, and the principle of least square method is expounded.
29.3 Legendre theorem about spherical triangle illustrates approximate calculation.
29.4 Importance of the shortest straight line on the earth's reference ellipsoid in measurement (about the hypothesis of differential equation theory)
29.5 About the leveling surface and its actual measurement
Chapter 30 continuation of applied geometry: B. descriptive geometry
30. 1 The hypothesis of an error theory in descriptive geometry is illustrated by drawing Pascal theorem.
30.2 Possibility of inferring ideal curve characteristics from empirical charts
30.3 the application of algebraic curve, algebraic knowledge to be used.
30.4 Propose the theorem to be proved: w ′+2t ″ = n (n-2).
30.5 Continuity Method Used in Proof
30.6 Conversion between Cn with and without two key points
30.7 Example of Flat Curve Conforming to Theorem
30.8 Example of odd curve
30.9 Give examples to illustrate the continuity method in the proof and the completion of the proof.
The ninth part represents ideal graphics as graphics and models.
1 The softening curve has no singularity, especially the shape of C3 (the projection of the curve and the plane part of its tangent plane).
Seven kinds of singularity of two torsion curves
3 general discussion on the shape of surface without singularity
4. Two key points about F3, especially its two-stage key point and single-stage key point.
Overview of 5f3 profile
Appeal: By observing nature, we should constantly revise traditional scientific conclusions.
Comparison of name translation
postscript