vol 4: Glossary
Natural Numbers
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Natural numbers
[Hazewinkel, Natural number] [Borowski, Natural number]
The natural numbers are the counting numbers represented by the symbols 0, 1, 2, 3, ... . The natural numbers are defined formally by Peano's axioms. Since each new natural number if formed by adding one to its predecessor there no last natural number, and so the natural numbers are said to be infinite (latin for endless). Nevertheless, we can imagine a container or set containing all the natural numbers. This infinite set is called N.
N is the foundation for the transfinite numbers constructed by Georg Cantor.
One may also construct the natural numbers set theoretically. begin with the empty set whose cardinal is 0. We then construct a set whose only member is the empty set, cardinal 1. Then a set, cardinal 2, containing the empty set and the set whose cardinal is one. And so on.
The natural numbers have a natural order, the order of their creation from one another. The earlier members of this order are said to be smaller than later members. Peano's axioms treat the numbers as though the larger follows the smaller. The set theoretical construction sees the larger numbers containing the smaller.
Books
| Borowski, Ephraim J, & Johnathan M Borwein, Collins Dictionary of Mathematics, Harper Collins 1089 'It is the immodest hope of the authors that this dictionary will not only prove valuable as a reference book for students of mathematics at all levels from secondary schools to a master's degree, but also offer much to interest a more general readership. Amazon back |
| Burton, David M, Elementary Number Theory, Allyn and Bacon 1976 Preface: '"Plato said God is a geometer. Jacobi changed this to God is an arithmetician. Then came Kronecker and fashioned the memorable expression, God created the natural numbers, and all the rest is the work of man. Felix Klein." The purpose of the present volume is to give a simple account of classical number theory, as well as to impart some of the historical background in which the subject evolved.' Amazon back |
| Chaitin, Gregory J, Algorithmic Information Theory, Cambridge UP 1987 Foreword: 'The crucial fact here is that there exist symbolic objects (i.e., texts) which are "algorithmically inexplicable", i.e., cannot be specified by any text shorter than themselves. Since texts of this sort have the properties associated with random sequences of classical probability theory, the theory of describability developed ... in the present work yields a very interesting new view of the notion of randomness.' J T Schwartz Amazon back |
| Davenport, H, The Higher Arithmetic, Hutchinson University Library Introduction: 'A peculiarity of the higher arithmetic [theory of numbers] is the great difficulty which has often been experienced in proving simple general theorems which have been suggested quite naturally by numerical evidence. "It is just this," said Gauss "which gives the higher arithmetic that magical charm which has made it the favourite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics."' p 7. Amazon back |
| Deavours, Cipher, Cryptology: Machines, History and Methods, Artech House 1989 Preface: 'The articles in this volume are divided into five sections. Cryptologic Personalities, History, Analysis, Cryptographic Machines and Ciphers - Historical and Challenging. The Editors offer this second collection of Cryptographia's greatest hits to the reader in the hope that it will both amuse and instruct.' [Cryptologia Department of Mathematical Sciences, United States Military Academy, West Point NY 10996-9902, USA.Cryptologia] Amazon back |
| Frege, Gottlob, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Nature of Number, Northwestern UP 1980 Jacket: 'The book represents the first philosophically sound discussion of the concept of number in Western civilisation. It influenced profoundly developments in the philosophy of mathematics, general ontology and mathematics.' Amazon back |
| Hazewinkel, Michiel, Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.' Amazon back |
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