Volume IV , issue 2 ( 2001 ) | back |

Selected chapters from algebra, VI. Infinite sets | 71$-$104 |

**Abstract**

This paper is the sixth part of the publication ``Selected chapters from algebra'', the first five having been published in previous issues of The Teaching of Mathematics, Vol\. I (1998), 1--22, Vol\. II, 1 (1999), 1--30, Vol\. II, 2 (1999), 65--80, Vol\. III, 1 (2000), 15--40, Vol\. III, 2 (2000), 63--82 and Vol\. IV, 1 (2001), 1--34. It is an English translation of: I. R. Shafarevich, {ıt Izbranye glavy algebry. Glava VI. Beskonechnye mnozhestva}, Matematicheskoe obrazovanie, N~3--4(6--7), 1998, Moskva, str\. 2--36.

**Keywords:** Finite and infinite sets, countable sets,
continuum, algebraic and transcendental numbers, thin sets, normal numbers,
strengthened inequality of Chebyshev.

**MSC Subject Classification:** 00A35

Solving equations, an elegant legacy | 105$-$144 |

**Abstract**

Solving equations. The problems, techniques, and viewpoints are our legacy. One theme throughout this lecture is that classical and modern mathematics are tightly intertwined, that contemporary mathematics contributes real insight and techniques to understand traditional problems. This is a considerably expanded version of a lecture---intended for undergraduates---that I gave at both the University of Montreal and the University of Pennsylvania. I thank R.~Horn and H.~Wilf for valuable suggestions. A shorter version appeared in the {ıt American Math.~Monthly}, {\bf 105}, Jan.~1998, pp.~1--21.}

**Keywords:** Equation, solution, duality, symmetry,
existence, iteration, variational methods, fixed point

**MSC Subject Classification:** 00A35