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Looking at the Catalogue of Elementary Mathematics from a High Angle
The first volume catalogue

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