Understand Mathematics, Understand Computing: Discrete Mathematics That All Computing Students Should Know

by Arnold L. Rosenberg, Denis Trystram

Overview

In this book the authors aim to endow the reader with an operational, conceptual, and methodological understanding of the discrete mathematics that can be used to study, understand, and perform computing. They want the reader to understand the elements of computing, rather than just know them. The basic topics are presented in a way that encourages readers to develop their personal way of thinking about mathematics. Many topics are developed at several levels, in a single voice, with sample applications from within the world of computing. Extensive historical and cultural asides emphasize the human side of mathematics and mathematicians.

By means of lessons and exercises on “doing” mathematics, the book prepares interested readers to develop new concepts and invent new techniques and technologies that will enhance all aspects of computing. The book will be of value to students, scientists, and engineers engaged in the design and use of computing systems, and to scholars and practitioners beyond these technical fields who want to learn and apply novel computational ideas.

Product Details

ISBN-13:	9783030583750
Publisher:	Springer International Publishing
Publication date:	12/05/2020
Edition description:	2020
Pages:	550
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Prof. Arnold Rosenberg is a distinguished university professor emeritus at the University of Massachusetts, Amherst. He also held research positions at Northeastern University and Colorado State University, a professorship at Duke University, and a staff research position at IBM Watson Research Center. He was elected a fellow of the ACM in 1996 for his work on graph-theoretic models of compuation, emphasizing theoretical studies of parallel algorithms and architectures, VLSI design and layout, and data structures. In 1997, he was elected as a fellow of the IEEE for fundamental contributions to theoretical aspects of computer science and engineering.

Prof. Denis Trystram is a distinguished professor at the Grenoble Institute of Engineering, an honorary member of the Institut Universitaire de France (IUF), and he works at the Laboratoire d'Informatique de Grenoble (LIG) in the team-project DataMove-INRIA. His research interestst include the design and analysis of efficient algorithms for optimizing resource use in parallel and distributed systems, approximation algorithms for scheduling and packing problems, and algorithms for data analytics. Both authors have considerable teaching and practical experience in the application of discrete mathematics approaches to computing tasks.

Table of Contents

Introduction.- “Doing” Mathematics: A Toolkit for Mathematical Reasoning.- Sets and Their Algebras: The Stem Cells of Mathematics.- Numbers I: The Basics of Our Number System.- Arithmetic: Putting Numbers to Work.- Summations: Complex Operations from Simple Components.- The Vertigo of Infinity: Handling the Very Large and the Infinite.- Numbers II: Building the Integers and Building with the Integers.- Recurrences: Rendering Complex Structure Manageable.- Numbers III: Operational Representations and Their Consequences.- The Art of Counting: Combinatorics, Probability, and Statistics.- Graphs I: Representing Relationships Mathematically.- Graphs II: Graphs Within Computation and Communication.- Solutions to Exercises.- App. A, Pairing Functions.- App. B, A Deeper Look at the Fibonacci Numbers.- App. C, Two Recurrence-Defined Number Families.- App. D, Signed-Digit Numerals: Carry-Free Addition.- App. E, The Diverse Delights of de Bruijn Networks.- List of Symbols.- References.- Index.

What People are Saying About This

From the Publisher

“The breadth of today's technology is so wide and the variety of programming languages so large, that we can easily feel far removed from the foundations that support the modern technical edifice that is computing. For example, when we write code we create and invoke methods. Increasingly, we build solutions by combining and calling cloud microservices. We query databases. We search and filter information using online services. In utilizing different technologies these regular tasks appear to be quite dissimilar, but in fact they are fundamentally alike. They are all instances of mathematical functions, that is: mappings between sets. It turns out that when we have a mathematical grounding in the concept of sets we are empowered to discover unifying abstractions and powerful simplifications in our solutions. [This] is a wonderful guide to the mathematical connections that underpin computing and it shows you where to look.” (Peter Rodgers, Founder, 1060 Research)

“This book is a must-read for anyone who wishes to understand the mathematical foundations of the modern computing enterprise. The book is exceptionally accessible to a diverse audience of students, practitioners, scientists, hardware designers, and software professionals. It uses the most powerful techniques for teaching by approaching each topic in multiple different ways and connecting abstract math with concrete applications. The authors accomplish the rare feat of complementing mathematical rigor with intuitive explanations and visual examples. The result is a true joy to read with its conversational prose and interesting historical asides that bring the topics to life.” (Ramesh K. Sitaraman, UMass Amherst)

Editorial Reviews

There are a lot of things I like about this book … . The basic concept of the book, the overall structure of the book, and the material presented are all excellent: it deserves a better implementation.” (David J. Littleboy, SIGACT News, Vol. 54 (3), 2023)

“The text is written in an easy to read format which generously incorporates narratives from the history of mathematics as well as rigorous proofs of the concepts presented. The appendices and references to other texts provide the reader with numerous sources of supplementary information for those wishing to delve into a subject at a deeper level … . chapters are organized and clearly labeled to express which sections are appropriate for a beginning learner, an intermediate learner, or the specialist.” (Tom French, MAA Reviews, October 3, 2021)

“Each chapter comes with several exercises from easy to difficult, the latter with complete solutions in the appendix. To accommodate the book to readers with different backgrounds and goals, the authors provide a guide which gives paths through the book for several courses. The exposition is always clear and motivating, no prerequisites are presumed, all terms and concepts are defined precisely, and there are many look-and-see proofs.” (Dieter Riebesehl, zbMATH 1465.68004, 2021)

From the Publisher