Elementary Mathematics from an Advanced Standpoint Arithmetic, Algebra, Analysis

by Klein, Felix

• ISBN13: 9780486434803
• ISBN10: 048643480X
• eBook ISBN(s): 9780486165967
• Format: Paperback
• Copyright: 2004-07-26
• Publisher: Dover Publications
• More Book Details

Summary

"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.

Table of Contents

Introduction	1	(5)
First Part: Arithmetic
I. Calculating with Natural Numbers	6	(16)
1. Introduction of Numbers in the Schools	6	(2)
2. The Fundamental Laws of Reckoning	8	(2)
3. The Logical Foundations of Operations with Integers	10	(7)
4. Practice in Calculating with Integers	17	(5)
II. The First Extension of the Notion of Number	22	(15)
1. Negative Numbers	23	(5)
2. Fractions	28	(3)
3. Irrational Numbers	31	(6)
III. Concerning Special Properties of Integers	37	(18)
IV. Complex Numbers	55	(32)
1. Ordinary Complex Numbers	55	(3)
2. Higher Complex Numbers, especially Quaternions	58	(7)
3. Quaternion Multiplication-Rotation and Expansion	65	(10)
4. Complex Numbers in School Instruction	75	(2)
Concerning the Modern Development and the General Structure of Mathematics	77	(10)
Second Part: Algebra
I. Real Equations with Real Unknowns	87	(14)
1. Equations with one parameter	87	(1)
2. Equations with two parameters	88	(6)
3. Equations with three parameters λ μ ν	94	(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