NumbersA book about numbers sounds rather dull. This one is not. Instead it is a lively story about one thread of mathematics-the concept of "number" told by eight authors and organized into a historical narrative that leads the reader from ancient Egypt to the late twentieth century. It is a story that begins with some of the simplest ideas of mathematics and ends with some of the most complex. It is a story that mathematicians, both amateur and professional, ought to know. Why write about numbers? Mathematicians have always found it diffi cult to develop broad perspective about their subject. While we each view our specialty as having roots in the past, and sometimes having connec tions to other specialties in the present, we seldom see the panorama of mathematical development over thousands of years. Numbers attempts to give that broad perspective, from hieroglyphs to K-theory, from Dedekind cuts to nonstandard analysis. |
Contents
Natural Numbers Integers and Rational Numbers | 9 |
Real Numbers | 27 |
Complex Numbers | 55 |
The Fundamental Theorem of Algebra | 97 |
What is 𝛑? | 123 |
The pAdic Numbers | 155 |
Real Division Algebras | 179 |
Introduction | 181 |
CAYLEY Numbers or Alternative Division Algebras | 249 |
Composition Algebras HURWITZs TheoremVectorProduct Algebras | 265 |
Division Algebras and Topology | 281 |
Infinitesimals Games and Sets | 303 |
Nonstandard Analysis | 305 |
Numbers and Games | 329 |
Set Theory and Mathematics | 355 |
381 | |
Repertory Basic Concepts from the Theory of Algebras | 183 |
Hamiltons Quaternions | 189 |
The Isomorphism Theorems of FROBENIUS HOPF and GELFANDMAJOR | 221 |
387 | |
Portraits of Famous Mathematicians | |
Other editions - View all
Numbers Heinz-Dieter Ebbinghaus,Hans Hermes,Friedrich Hirzebruch,Max Koecher,Klaus Mainzer,Jürgen Neukirch,Alexander Prestel,Reinhold Remmert No preview available - 1996 |
Numbers Heinz-Dieter Ebbinghaus,Hans Hermes,Friedrich Hirzebruch,Max Koecher,Klaus Mainzer,Jürgen Neukirch,Alexander Prestel,Reinhold Remmert No preview available - 1996 |
Common terms and phrases
a₁ alternative algebras associative axiom BANACH algebra bilinear form CAUCHY CAYLEY Chapter coefficients cohomology commutative complex numbers composition algebra construction converges CONWAY games DEDEKIND Dedekind cut defined definition denoted differential division algebra divisors of zero epimorphism equation equivalent Euclidean EULER example exists field finite follows formula function fundamental sequence fundamental theorem GAUSS GELFAND-MAZUR theorem geometry homology homomorphism HOPF HOPF's hypothesis identity imaginary induction infinite integers introduced isomorphic Lemma linear mapping f Math mathematician mathematics matrices multiplication natural numbers nested intervals nth roots obtain octonions orthogonal p-adic numbers product rule proof prove quadratic algebra quaternions R-algebra rational numbers real numbers real polynomial relation representation residue class ring roots scalar product set theory statement subset theorem of algebra topological uniquely unit element vector bundles vector space