Author	Ivanov, O. A. author
Title	Easy as ฯ? [electronic resource] : An Introduction to Higher Mathematics / by O. A. Ivanov
Imprint	New York, NY : Springer New York : Imprint: Springer, 1999
Connect to	http://dx.doi.org/10.1007/978-1-4612-0553-1
Descript	XVIII, 190 p. 1 illus. online resource

The present book is rare, even unique of its kind, at least among mathematics texts published in Russian. You have before you neither a textbook nor a monograph, although these selected chapters from elementary mathematics certainly constitute a fine educational tool. It is my opinion that this is more than just another book about mathematics and the art of teaching that subject. Without considering the actual topics treated (the author himself has described these in sufficient detail in of the book as a whole, the Introduction), I shall attempt to convey a general idea and describe the impressions it makes on the reader. Almost every chapter begins by considering well-known problems of elementary mathematics. Now, every worthwhile elementary problem has hidden behind its diverting formulation what might be called "higher mathematics," or, more simply, mathematics, and it is this that the author demonstrates to the reader in this book. It is thus to be expected that every chapter should contain subject matter that is far from elementary. The end result of reading the book is that the material treated has become for the reader "three-dimensional" as it were, as in a hologram, capable of being viewed from all sides

1 Induction -- 1.1 Principle or method? -- 1.2 The set of integers -- 1.3 Peanoโs axioms -- 1.4 Addition, order, and multiplication -- 1.5 The method of mathematical induction -- 2 Combinatorics -- 2.1 Elementary problems -- 2.2 Combinations and recurrence relations -- 2.3 Recurrence relations and power series -- 2.4 Generating functions -- 2.5 The numbers ?e, and n-factorial -- 3 Geometric Transformations -- 3.1 Translations, rotations, and other symmetries, in the context of problem-solving -- 3.2 Problems involving composition of transformations -- 3.3 The group of Euclidean motions of the plane -- 3.4 Ornaments -- 3.5 Mosaics and discrete groups of motions -- 4 Inequalities -- 4.1 The means of a pair of numbers -- 4.2 Cauchyโs inequality and the a.m.-g.m. inequality -- 4.3 Classical inequalities and geometry -- 4.4 Integral variants of the classical inequalities -- 4.5 Wirtingerโs inequality and the isoperimetric problem -- 5 Sets, Equations, and Polynomials -- 5.1 Figures and their equations -- 5.2 Pythagorean triples and Fermatโs last theorem -- 5.3 Numbers and polynomials -- 5.4 Symmetric polynomials -- 5.5 Discriminants and resultants -- 5.6 The method of elimination and Bรฉzoutโs theorem -- 5.7 The factor theorem and finite fields -- 6 Graphs -- 6.1 Graphical reformulations -- 6.2 Graphs and parity -- 6.3 Trees -- 6.4 Eulerโs formula and the Euler characteristic -- 6.5 The Jordan curve theorem -- 6.6 Pairings -- 6.7 Eulerian graphs and a little more -- 7 The Pigeonhole Principle -- 7.1 Pigeonholes and pigeons -- 7.2 Poincarรฉโs recurrence theorem -- 7.3 Liouvilleโs theorem -- 7.4 Minkowskiโs lemma -- 7.5 Sums of two squares -- 7.6 Sums of four squares. Eulerโs identity -- 8 The Quaternions -- 8.1 The skew-field of quaternions, and Eulerโs identity -- 8.2 Division algebras. Frobeniusโs theorem -- 8.3 Matrix algebras -- 8.4 Quaternions and rotations -- 9 The Derivative -- 9.1 Geometry and mechanics -- 9.2 Functional equations -- 9.3 The motion of a pointโparticle -- 9.4 On the number e -- 9.5 Contracting maps -- 9.6 Linearization -- 9.7 The Morse-Sard theorem -- 9.8 The law of conservation of energy -- 9.9 Small oscillations -- 10 The Foundations of Analysis -- 10.1 The rational and real number fields -- 10.2 Nonstandard number lines -- 10.3 โNonstandardโ statements and proofs -- 10.4 The reals numbers via Dedekind cuts -- 10.5 Construction of the reals via Cauchy sequences -- 10.6 Construction of a model of a nonstandard real line -- 10.7 Norms on the rationals -- References

1. Mathematics
2. Matrix theory
3. Algebra
4. Geometry
5. Number theory
6. Combinatorics
7. Mathematics
8. Combinatorics
9. Number Theory
10. Linear and Multilinear Algebras
11. Matrix Theory
12. Geometry