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Elementary Mathematics from an Advanced Standpoint Arithmetic, Algebra, Analysis

ISBN-10: 048643480X
ISBN-13: 9780486434803
Edition: 2004
Authors: Felix Klein
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Description: “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  More...

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Book details

List price: $18.95
Copyright year: 2004
Publisher: Dover Publications, Incorporated
Publication date: 7/26/2004
Binding: Paperback
Pages: 304
Size: 5.25" wide x 8.50" long x 0.75" tall
Weight: 1.804
Language: English

“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.

Introduction
Arithmetic
Calculating with Natural Numbers
Introduction of Numbers in the Schools
The Fundamental Laws of Reckoning
The Logical Foundations of Operations with Integers
Practice in Calculating with Integers
The First Extension of the Notion of Number
Negative Numbers
Fractions
Irrational Numbers
Concerning Special Properties of Integers
Complex Numbers
Ordinary Complex Numbers
Higher Complex Numbers, especially Quaternions
Quaternion Multiplication--Rotation and Expansion
Complex Numbers in School Instruction
Concerning the Modern Development and the General Structure of Mathematics
Algebra
Real Equations with Real Unknowns
Equations with one parameter
Equations with two parameters
Equations with three parameters [lambda], [mu], [nu]
Equations in the field of complex quantities
The fundamental theorem of algebra
Equations with a complex parameter
The "pure" equation
The dihedral equation
The tetrahedral, the octahedral, and the icosahedral equations
Continuation: Setting up the Normal Equation
Concerning the Solution of the Normal Equations
Uniformization of the Normal Irrationalities by Means of Transcendental Functions
Solution in Terms of Radicals
Reduction of Genral Equations to Normal Equations
Analysis
Logarithmic and Exponential Functions
Systematic Account of Algebraic Analysis
The Historical Development of the Theory
The Theory of Logarithms in the Schools
The Standpoint of Function Theory
The Goniometric Functions
Theory of the Goniometric Functions
Trigonometric Tables
Purely Trigonometric Tables
Logarithmic--Trigonometric Tables
Applications of Goniometric Functions
Trigonometry, in particular, spherical trigonometry
Theory of small oscillations, especially those of the pendulum
Representation of periodic functions by means of series of goniometric functions (trigonometric series)
Concerning Infinitesimal Calculus Proper
General Considerations in Infinitesimal Calculus
Taylors Theorem
Historical and Pedagogical Considerations
Supplement
Transcendence of the Numbers e and [pi]
The Theory of Assemblages
The Power of an Assemblage
Arrangement of the Elements of an Assemblage
Index of Names
Index of Contents

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