Arithmetic, Algebra, Analysis
著者
書誌事項
Arithmetic, Algebra, Analysis
（Elementary mathematics from a higher standpoint, v. 1）
Springer, c2016
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注記
"Translation of the 4th German editon"Elementarmathematik vom höheren Standpunkte aus", vol.1 by Felix Klein, Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band 14, Verlag von Julius Springer, Berlin 1933. A previous English language edition, Felix Klein"Elementary Mathematics from an Advanced Standpoint- Arithmetic, Algebra, Analysis", translated by E.R. Hedrick and C.A. Noble, New York 1931, was based on the 3rd German edition and published by Dover Publications"--T.p. verso
Includes bibliographical references (p. xii) and indexes
内容説明・目次
内容説明
These three volumes constitute the first complete English translation of Felix Klein's seminal series "Elementarmathematik vom hoeheren Standpunkte aus". "Complete" has a twofold meaning here: First, there now exists a translation of volume III into English, while until today the only translation had been into Chinese. Second, the English versions of volume I and II had omitted several, even extended parts of the original, while we now present a complete revised translation into modern English.
The volumes, first published between 1902 and 1908, are lecture notes of courses that Klein offered to future mathematics teachers, realizing a new form of teacher training that remained valid and effective until today: Klein leads the students to gain a more comprehensive and methodological point of view on school mathematics. The volumes enable us to understand Klein's far-reaching conception of elementarisation, of the "elementary from a higher standpoint", in its implementation for school mathematics.This volume I is devoted to what Klein calls the three big "A's": arithmetic, algebra and analysis. They are presented and discussed always together with a dimension of geometric interpretation and visualisation - given his epistemological viewpoint of mathematics being based in space intuition. A particularly revealing example for elementarisation is his chapter on the transcendence of e and p, where he succeeds in giving concise yet well accessible proofs for the transcendence of these two numbers. It is in this volume that Klein makes his famous statement about the double discontinuity between mathematics teaching at schools and at universities - it was his major aim to overcome this discontinuity.
目次
First Part: Arithmetic. I. Calculating with Natural Numbers.- II. The First Extensions of the Notion of a Number.- III. Concerning Special Properties of Integers.- IV. Complex Numbers.- V. Concerning the Modern Development and the General Structure of Mathematics.- Second Part: Algebra. I. Real Equations with Real Unknowns.- II. Equations in the Field of Complex Quantities.- Third Part: I. Logarithmis and Exponential Functions.- The Goniometric Functions.- III. Concerning Infitesimal Calculus Proper.- Supplement: I. Transcendence of the Numbers e and pi.- II. Set Theory.- Appendix.- Index of Names.- Index of Contents.
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