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Elementary Mathematics From an Advanced Standpoint: Arithmetic, Algebra, Analysis: Felix Klein

This book, a translation of the first of Klein's three volumes entitled Elementarmathematik vom höheren Standpunkte aus, is a series of lectures that Klein gave for teachers of mathematics in secondary schools. The material is presented under the headings of arithmetic, algebra, analysis, and a supplement. The section on arithmetic treats the extensions of the number system and the laws of operation, beginning with integers and ending with complex numbers and quaternions. The treatment seeks to explain the how and why of the subject. As an example, we note the discussion of the little understood rule of signs: "minus times minus gives plus." The section on algebra is devoted to the solution of equations. First, some geometric methods are explained for investigating the real roots of rational integral equations containing parameters. Then complex roots are considered, especially of those equations whose solutions lead to a consideration of the groups of motions connected with the regular bodies. Free use is made of Riemann surfaces and other parts of the theory of functions of a complex variable. The section on analysis is devoted to the logarithmic, exponential, and trigonometric functions, and a discussion of the infinitesimal calculus proper. A wide variety of subjects is treated, however, in connection with these general topics: the construction of the early logarithmic and trigonometric tables, expansions in Fourier series, Taylor's Theorem, and Newton's and Lagrange's interpolation formulas will serve as samples. Finally, the supplement contains proofs of the transcendence of e and ir, and a discussion of assemblages.

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