Understanding Decimal Addition

In this chapter the authors use decimal addition to introduce important ideas relating to natural numbers, Peano’s axioms and computer arithmetic.

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System Architecture An Ordinary Engineering Discipline

System Architecture

Wolfgang J. Paul Christoph Baumann Petro Lutsyk Sabine Schmaltz •



System Architecture An Ordinary Engineering Discipline

123

Wolfgang J. Paul FR 6.1 Informatik Universität des Saarlandes Saarbrücken, Saarland Germany

Petro Lutsyk FR 6.1 Informatik Universität des Saarlandes Saarbrücken, Saarland Germany

Christoph Baumann School of Computer Science and Communication KTH Royal Institute of Technology Stockholm Sweden

Sabine Schmaltz FR 6.1 Informatik Universität des Saarlandes Saarbrücken, Saarland Germany

ISBN 978-3-319-43064-5 DOI 10.1007/978-3-319-43065-2

ISBN 978-3-319-43065-2

(eBook)

Library of Congress Control Number: 2016952820 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents 1

Introduction

2

Understanding Decimal Addition 2.1 Experience Versus Understanding . . . . 2.2 The Natural Numbers . . . . . . . . . . . 2.2.1 2 + 1 = 3 is a Definition . . . . . 2.2.2 1 + 2 = 3 is a Theorem . . . . . 2.2.3 9 + 1 = 10 is a Brilliant Theorem 2.3 Final Remarks . . . . . . . . . . . . . . . 2.4 Exercises . . . . . . . . . . . . . . . . .

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7 7 8 9 9 11 13 14

Basic Mathematical Concepts 3.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Implicit Quantification . . . . . . . . . . . . . . . 3.1.2 Numbers and Sets . . . . . . . . . . . . . . . . . 3.1.3 Sequences, Their Indexing and Overloading . . . 3.1.4 Bytes, Logical Connectives, and Vector Notation . 3.2 Modulo Computation . . . . . . . . . . . . . . . . . . . . 3.3 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Geometric Sum