Arithmetic - Jsoftware

Arithmetic

Kenneth E. Iverson

Copyright ? 2002 Jsoftware Inc. All rights reserved.

Preface

Arithmetic is the basic topic of mathematics. According to the American Heritage Dictionary [1], it concerns "The mathematics of integers under addition, subtraction, multiplication, division, involution, and evolution."

The present text differs from other treatments of arithmetic in several respects:

The provision of simple but precise definitions of the counting numbers and other notions introduced.

The use of simple but precise notation that is executable on a computer, allowing experimentation and providing a simple and meaningful introduction to computer programming.

The introduction and significant use of fundamental mathematical notions (such as vectors, matrices, Heaviside operators, and duality) in simple contexts that make them easy to understand. This lays a firm foundation for a wealth of later use in mathematics.

Emphasis is placed on the use of guesses by speculation and criticism in the spirit of Lakatos, as discussed in the treatment of proofs in Chapter 5.

The thrust of the book might best be appreciated by comparing it with Felix Klein's Elementary Mathematics from an Advanced Standpoint [2]. However, I shun the corresponding title Arithmetic from an Advanced Standpoint because it would incorrectly suggest that the treatment is intended only for mature mathematicians; on the contrary, the use of simple, executable notation makes it accessible to any serious student possessing little more than a knowledge of the counting numbers.

Like Klein, I do not digress to discuss the importance of the topics treated, but leave that matter to the knowledge of the mature reader and to the faith of the neophyte.

Table of Contents

Introduction ..............................................................................1 A. Counting Numbers.......................................................................... 1 B. Integers ........................................................................................... 2 C. Inverses ........................................................................................... 2 D. Domains.......................................................................................... 3 E. Nouns and Verbs............................................................................. 3 F. Pronouns and Proverbs.................................................................... 3 G. Conjunctions................................................................................... 4 H. Addition And Subtraction............................................................... 5 I. Verb Tables ...................................................................................... 5 J. Relations .......................................................................................... 6 K. Lesser-Of and Greater-Of............................................................... 7 L. List And Table Formation............................................................... 7 M. Punctuation .................................................................................... 8 N. Insertion.......................................................................................... 9 O. Multiplication ................................................................................. 10 P. Power............................................................................................... 10 Q. Summary......................................................................................... 11 R. On Language................................................................................... 12

Properties of Verbs ..................................................................17 A. Valence, Ambivalence, And Bonds................................................ 17 B. Commutativity ................................................................................ 18 C. Associativity ................................................................................... 18 D. Distributivity................................................................................... 18 E. Symmetry ........................................................................................ 19 F. Display of Proverbs......................................................................... 20 G. Inverses........................................................................................... 20 H. Partitions......................................................................................... 20 I. Identity Elements and Infinity.......................................................... 21 J. Experimentation ............................................................................... 22 K. Summary of Notation ..................................................................... 22 L. On Language................................................................................... 22

Partitions and Selections.........................................................25 A. Partition Adverbs............................................................................ 25 B. Selection Verbs ............................................................................... 26

C. Grade and Sort ................................................................................ 28 D. Residue ........................................................................................... 28 E. Characters........................................................................................ 29 F. Box and Open.................................................................................. 30 G. Summary of Notation ..................................................................... 31 H. On Language .................................................................................. 31

Representation of Integers ......................................................33 A. Introduction .................................................................................... 33 B. Addition .......................................................................................... 34 C. Multiplication.................................................................................. 35 D. Normalization ................................................................................. 37 E. Mixed Bases.................................................................................... 39 F. Experimentation .............................................................................. 40 G. Summary of Notation ..................................................................... 41

Proofs ........................................................................................43 A. Introduction .................................................................................... 43 B. Formal and Informal Proofs............................................................ 47 C. Proofs and Refutations.................................................................... 48 D. Proofs.............................................................................................. 50

Logic ..........................................................................................57 A. Domain and Range ......................................................................... 57 B. Propositions .................................................................................... 58 C. Booleans ......................................................................................... 58 D. Primitives........................................................................................ 60 E. Boolean Dyads ................................................................................ 61 F. Boolean Monads.............................................................................. 62 G. Generators....................................................................................... 62 H. Boolean Primitives.......................................................................... 63 I. Summary of Notation ....................................................................... 63

Permutations ............................................................................65 A. Introduction .................................................................................... 65 B. Arrangements.................................................................................. 67 D. Products of Permutations................................................................ 69 E. Cycles.............................................................................................. 70 F. Reduced Representation .................................................................. 71 G. Summary of Notation ..................................................................... 72

Classification and Sets ............................................................75

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