Elementary Mathematics from a Higher Standpoint: Volume I: Arithmetic, Algebra, Analysis

ISBN-13: 9783662494400 / Angielski / Miękka / 2016 / 312 str.

Felix Klein; Gert Schubring

Opis

This three-part treatment on mathematical knowledge needed by teachers was a pathbreaking book published by the German mathematician Felix Klein between 1924 and 1928 and it is still as relevant as it was 90 years ago. The books pulse with the excitement of the actual lectures they came from and combine geometric intuition with the precision of arithmetic formulas in an inspiring way.

The first volume begins with topics associated with arithmetic, including calculating with natural numbers, the first extension of the notion of numbers, special properties of integers, and complex numbers. Algebra-related subjects constitute the second part, which examines real equations with real unknowns as well as equations in the field of complex quantities. The final part explores elements of analysis, with discussions of logarithmic and exponential functions, geometric functions, and infinitesimal calculus."

Spis treści

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.

O autorze

Felix Klein (1849-1925) was a leading German mathematician whose research interests included group theory, complex analysis, and geometry. His work influenced many areas of mathematics and related subjects, ranging from mathematical physics to mathematical didactics.

To this day, Felix Klein is considered one of the most important mathematicians of the 19th century. 

Blurb

These three volumes constitute the first complete English translation of Felix Klein’s seminal series “Elementarmathematik vom höheren 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.

About the Author:
Felix Klein (1849-1925) was a leading German mathematician whose research interests included group theory, complex analysis, and geometry. His work influenced many areas of mathematics and related subjects, ranging from mathematical physics to mathematical didactics.

To this day, Felix Klein is considered one of the most important mathematicians of the 19th century.