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Essays on the Theory of Numbersby Richard Dedekind
Synopses & ReviewsPublisher Comments:This volume contains the two most important essays on the logical foundations of the number system by the famous German mathematician J. W. R. Dedekind. The first presents Dedekind's theory of the irrational numberthe Dedekind cut ideaperhaps the most famous of several such theories created in the 19th century to give a precise meaning to irrational numbers, which had been used on an intuitive basis since Greek times. This paper provided a purely arithmetic and perfectly rigorous foundation for the irrational numbers and thereby a rigorous meaning of continuity in analysis. The second essay is an attempt to give a logical basis for transfinite numbers and properties of the natural numbers. It examines the notion of natural numbers, the distinction between finite and transfinite (infinite) whole numbers, and the logical validity of the type of proof called mathematical or complete induction. The contents of these essays belong to the foundations of mathematics and will be welcomed by those who are prepared to look into the somewhat subtle meanings of the elements of our number system. As a major work of an important mathematician, the book deserves a place in the personal library of every practicing mathematician and every teacher and historian of mathematics. Authorized translations by "Vooster " V. Beman. Synopsis:Two classic essays by great German mathematician: one provides an arithmetic, rigorous foundation for the irrational numbers, the other is an attempt to give the logical basis for transfinite numbers and properties of the natural numbers. Synopsis:Two classic essays by great German mathematician: one provides an arithmetic, rigorous foundation for the irrational numbers, the other is an attempt to give the logical basis for transfinite numbers and properties of the natural numbers.
Table of ContentsI. Continuity and Irrational Numbers
Preface 1. Properties of Rational Numbers 2. Comparison of the Rational Numbers with the Points of a Straight Line 3. Continuity of the Straight Line 4. Creation of Irrational Numbers 5. Continuity of the Domain of Real Numbers 6. Operations with Real Numbers 7. Infinitesimal Analysis II. The Nature and Meaning of Numbers Prefaces 1. Systems of Elements 2. Transformation of a System 3. Similarity of a Transformation. Similar Systems 4. Transformation of a System in Itself 5. The Finite and Infinite 6. Simply Infinite Systems. Series of Natural Numbers 7. Greater and Less Numbers 8. Finite and Infinite Parts of the NumberSeries 9. Definition of a Transformation of the NumberSeries by Induction 10. The Class of Simply Infinite Systems 11. Addition of Numbers 12. Multiplication of Numbers 13. Involution of Numbers 14. Number of the Elements of a Finite System What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
Other books you might likeRelated SubjectsMetaphysics » General Science and Mathematics » Mathematics » Advanced Science and Mathematics » Mathematics » Algebra » General Science and Mathematics » Mathematics » Foundations and Logic Science and Mathematics » Mathematics » General Science and Mathematics » Mathematics » Number Theory 

