1st Edition

Neutrices and External Numbers
A Flexible Number System

ISBN 9781498772679
Published June 24, 2019 by Chapman and Hall/CRC
360 Pages 4 B/W Illustrations

USD $180.00

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Book Description

Neutrices and External Numbers: A Flexible Number System introduces a new model of orders of magnitude and of error analysis, with particular emphasis on behaviour under algebraic operations. The model is formulated in terms of scalar neutrices and external numbers, in the form of an extension of the nonstandard set of real numbers. Many illustrative examples are given. The book starts with detailed presentation of the algebraic structure of external numbers, then deals with the generalized Dedekind completeness property, applications in analysis, domains of validity of approximations of solutions of differential equations, particularly singular perturbations. Finally, it describes the family of algebraic laws characterizing the practice of calculations with external numbers.


  • Presents scalar neutrices and external numbers, a mathematical model of order of magnitude within the real number system.
  • Outlines complete algebraic rules for the neutrices and external numbers
  • Conducts operational analysis of convergence and integration of functions known up to orders of magnitude
  • Formalises a calculus of error propagation, covariant with algebraic operations
  • Presents mathematical models of phenomena incorporating their necessary imprecisions, in particular related to the Sorites paradox


Table of Contents

1 Introduction to Elementary Nonstandard Analysis

2 Some models and calculations involving imprecisions.

3 Neutrices and external numbers

4 Advanced properties

5 Sequences. Convergence up to a neutrix

6 Functions of external numbers

7 Integration of functions of external numbers

8 Flexible systems of linear equations

9 Applications in asymptotics

10 Applications in other fields

11 External numbers as a complete arithmetical solid

A Background on Nonstandard Analysis

B Solutions to selected exercises

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Bruno Dinis is a postdoc at the Faculdade de Ciências, University of Lisbon, whose main area of interest is Mathematical Logic, Proof Theory, Nonstandard Analysis, and Philosophy of Mathematics.

Imme van den Berg is Associated Professor in Mathematics at the University of Évora, Portugal. His main area of interest lies in non-standard analysis. He is the author / co-author of 4 books and over 20 articles in the area of non-standard analysis.


The book Neutrices and External numbers: A Flexible Number System by Bruno Dinis and Imme van den Berg provides a remarkable presentation of the system of external numbers. It is a rigorous and convenient mathematical theory that provides a strong foundation for a complete doctrine of orders of magnitude — the art of neglecting—so present in the scientific approach. This subject is exposed in a pedagogical and comprehensive way by multiplying the points of view and the applications.

—Retired Professor Guy Wallet, La Rochelle University, France

[. . .] The latest addition to the outstanding CRC Press 'Monographs and Research Notes In Mathematics' series, Neutrices and External Numbers: A Flexible Number System by Bruno Dinis and Imme van den Berg (who is an Associated Professor in Mathematics at the University of Evora, Portugal and whose main area of interest lies in non-standard analysis) is enhanced for academia with an informative five page Foreword, as well as the inclusion of a twelve page Bibliography and a four page Index -- making is a significant and unreservedly recommended addition to professional, college, and university library Advanced Mathematics collections and supplemental curriculum lists. It should be noted for students, academia, and non-specialist readers with an interest in the subject that "Neutrices and External Numbers" is also available in a digital book format

Midwest Book Review