Elementary Mathematics from an Advanced Standpoint; Arithmetic, Algebra, Analysis: Felix Klein: 9780486434803: Book: Technology: eCampus.com

Makes the reader feel the inspiration that comes from listening to a great mathematician.�Bulletin, American Mathematical Society
A distinguished mathematician and educator enlivens abstract discussions of arithmetic, algebra, and analysis by means of graphical and geometrically perceptive methods. His three-part treatment begins with topics associated with arithmetic, including calculating with natural numbers, the first extension of the notion of number, special properties of integers, and complex numbers. Algebra-related subjects constitute the second part, which examines real equations with real unknowns and equations in the field of complex quantities. The final part explores elements of analysis, with discussions of logarithmic and exponential functions, the goniometric functions, and infinitesimal calculus. 1932 ed. 125 figures.

Discusses�calculating with natural numbers, the first extension of the notion of number, special properties of integers, and complex numbers; algebra-related subjects such as real equations with real unknowns and equations in the field of complex quantities. Also explores elements of analysis, with discussions of logarithmic and exponential functions, the goniometric functions, and infinitesimal calculus. 1932 edition. 125 figures.

Introduction

(5)

First Part: Arithmetic

I. Calculating with Natural Numbers

(16)

1. Introduction of Numbers in the Schools

(2)

2. The Fundamental Laws of Reckoning

(2)

3. The Logical Foundations of Operations with Integers

(7)

4. Practice in Calculating with Integers

(5)

II. The First Extension of the Notion of Number

(15)

1. Negative Numbers

(5)

2. Fractions

(3)

3. Irrational Numbers

(6)

III. Concerning Special Properties of Integers

(18)

IV. Complex Numbers

(32)

1. Ordinary Complex Numbers

(3)

2. Higher Complex Numbers, especially Quaternions

(7)

3. Quaternion Multiplication-Rotation and Expansion

(10)

4. Complex Numbers in School Instruction

(2)

Concerning the Modern Development and the General Structure of Mathematics

(10)

Second Part: Algebra

I. Real Equations with Real Unknowns

(14)

1. Equations with one parameter

(1)

2. Equations with two parameters

(6)

3. Equations with three parameters λ μ ν

(7)

II. Equations in the field of complex quantities

101

(43)

A. The fundamental theorem of algebra

101

(3)

B. Equations with a complex parameter

104

(40)

1. The "pure" equation

110

(5)

2. The dihedral equation

115

(5)

3. The tetrahedral, the octahedral, and the icosahedral equations

120

(4)

4. Continuation : Setting up the Normal Equation

124

(6)

5. Concerning the Solution of the Normal Equations

130

(3)

6. Uniformization of the Normal Irrationalities by Means of Transcendental Functions

133

(5)

7. Solution in Terms of Radicals

138

(3)

8. Reduction of Genral Equations to Normal Equations

141

(3)

Third Part: Analysis

I. Logarithmic and Exponential Functions

144

(18)

1. Systematic Account of Algebraic Analysis

144

(2)

2. The Historical Development of the Theory

146

(9)

3. The Theory of Logarithms in the Schools

155

(1)

4. The Standpoint of Function Theory

156

(6)

II. The Goniometric Functions

162

(45)

1. Theory of the Goniometric Functions

162

(7)

2. Trigonometric Tables

169

(6)

A. Purely Trigonometric Tables

170

(2)

B. Logarithmic-Trigonometric Tables

172

(3)

3. Applications of Goniometric Functions

175

(32)

A. Trigonometry, in particular, spherical trigonometry

175

(11)

B. Theory of small oscillations, especially those of the pendulum

186

(4)

C. Representation of periodic functions by means of series of goniometric functions (trigonometric series)

190

(17)

III. Concerning Infinitesimal Calculus Proper

207

(30)

1. General Considerations in Infinitesimal Calculus

207

(16)

2. TAYLORS Theorem

223

(11)

3. Historical and Pedagogical Considerations

234

(3)

Supplement

I. Transcendence of the Numbers e and π

237

(13)

II. The Theory of Assemblages

250

(19)

1. The Power of an Assemblage

251

(11)

2. Arrangement of the Elements of an Assemblage

262

(7)

Index of Names

269

(2)

Index of Contents

271


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