Number Systems

A Path into Rigorous Mathematics

Anthony Kay

• Publisher: CRC Press
• Year: 2021
• eBook ISBN: 9780429602245
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Number Systems: A Path into Rigorous Mathematics aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their proofs. The book continually seeks to build upon students' intuitive ideas of how numbers and arithmetic work, and to guide them towards the means to embed this natural understanding into a more structured framework of understanding.

The author's motivation for writing this book is that most previous texts, which have complete coverage of the subject, have not provided the level of explanation needed for first-year students. On the other hand, those that do give good explanations tend to focus broadly on Foundations or Analysis and provide incomplete coverage of Number Systems.

Features

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• Approachable for students who have not yet studied mathematics beyond school

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• Does not merely present definitions, theorems and proofs, but also motivates them in terms of intuitive knowledge and discusses methods of proof

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• Draws attention to connections with other areas of mathematics

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• Plenty of exercises for students, both straightforward problems and more in-depth investigations

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• Introduces many concepts that are required in more advanced topics in mathematics.

Chapter 1

Introduction: The Purpose of This Book

1.1 A Very Brief Historical Context

In the ancient world, mathematical ideas were developed for practical purposes: in relation to astronomical observing, which was vital for timekeeping (days, months, and years being defined by the positions of astronomical objects relative to observers on Earth); for accountancy, to keep records of possessions and for trading; and for surveying land. However abstraction, the consideration of mathematical concepts such as number, size, and shape independently of any application, also developed notably in ancient Greece. The idea of rigorous proof, that new concepts could be firmly and indisputably ascertained by a process of logical deduction based on previous knowledge, also appeared in ancient Greece. But the subsequent development of mathematics was not a smooth process of building theory rigorously on previously established knowledge; rather, concepts were often developed in a rather intuitive manner, and only much later proved rigorously. For example, the ideas of differential and integral calculus as originated by Isaac Newton and Gottfried von Leibniz in the late 17th century and further developed by Leonhard Euler and the Bernoullis in the 18th century were adequate for all applications needed at that time but involved poorly defined notions of infinitesimal (vanishingly small) changes in quantities. It was not until the 19th century that rigorous definitions of limit, continuity, convergence, and other notions that form the subject now known to mathematicians as Analysis were developed. All these notions are founded on the properties of numbers, but was there actually a rigorous definition of numbers? No; such definitions did not appear until the second half of the 19th century, in particular in the work of Richard Dedekind who asked as the title of one of his books, Was sind und was sollen die Zahlen? [What are numbers and what should they be?]. It is the objective of the present book to present a rigorous theory of numbers, not only for the intrinsic interest of the topic but also as a way into rigorous mathematics for students who have not previously encountered this approach.

1.2 The Axiomatic Method

We have already referred to the idea of new concepts being based on previous knowledge, but what is that previous knowledge based on? Clearly there needs to be a starting point. All mathematical reasoning must ultimately be founded on axioms. Axioms are statements which are defined to be true; some axioms are statements of what seems to be obvious, others are definitions of mathematical concepts or objects. In any case, they are the “rules of the game” which everyone accepts to be true. But we cannot set axioms arbitrarily; they need to give rise to “useful” mathematics, which relates to our intuitive concepts of number and all the other mathematical notions used in applications in natural sciences and many areas of human activity; and they need to be self-consistent, i.e. they must not allow contradictory deductions to be made.

Thus the axiomatic method which we adopt proceeds from the axioms by first proving a theorem based only on the axioms, then proving further theorems which may be based on previous theorems as well as on the axioms, building a structure of theoretical knowledge in which everything ultimately rests on the axioms. Some theorems will be described as lemmas, which are those that are not of much interest in themselves but are needed in order to prove later theorems, or as corollaries, which are those that follow almost immediately from the previous theorem.

All axioms, theorems, and any other statements in any mathematical argument are propositions. A proposition is a statement that can only be true or false, so excludes self-contradictory statements (for example, “This sentence is false”) and a huge variety of statements made in ordinary discourse. That is not to say that the truth or falsehood of a proposition must be known. Indeed there exist propositions in mathematics which are known to be undecidable: all that is known about such a proposition is that no process of logical deduction based on the accepted axioms is capable of proving either truth or falsehood. Below, we sometimes use a single letter P or Q to symbolise a general proposition; we also occasionally use the somewhat tautological phrase “P is true” for emphasis (it is tautological because writing, “Paris is in France” is sufficient to indicate the truth of the proposition that Paris is in France; we don't write, “ ‘Paris is in France’ is true.”). We shall also use the phrase, “P is false” for what in the standard notation of logic can be written $\neg P$ (“not P”). If a proposition concerns a class of objects, we may write $P(x)$ where x denotes a general object from the class; for example, if x denotes cities, $P(x)$ might be the proposition, “x is in France”.

1.3 The Place of Number Systems within Mathematics

The Natural Numbers which we use to count discrete objects are the first mathematical concept that we encounter as children, so we naturally regard numbers as the most fundamental aspect of mathematics. However, the ancient Greeks regarded geometrical concepts (lines and shapes, with their lengths and angles) as fundamental, whereas modern mathematicians tend to regard the concept of sets as underlying the whole of mathematics. In particular, our axioms for number systems will be presented in the language of Set Theory. So, before starting on the axiomatic theory of number systems, we present an informal review of the required background in Set Theory in Chapter 2; a student who already has some knowledge of sets, relations and functions and their notation could probably go straight into Chapter 3 on Natural Numbers, and only refer back to Chapter 2 if and when they encounter an unfamiliar concept from Set Theory.

A single number system is not sufficient for all purposes. Natural Numbers are fine for counting objects, but not for accountancy where one may be in credit or debit. The Integers, which suffice for the latter purpose, cannot deal with dividing objects into equal parts. So we need fractions (Rational Numbers), but these are not adequate for measuring quantities that vary continuously (length, weight, etc.). The Real Numbers which suffice for such measurements are then found not to be capable of providing solutions to perfectly well-formed equations, so we then define the Complex Numbers.

Leopold Kronecker thought that “Die ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk” [God made the whole numbers, all else is the work of humansⁱ]. However, the approach in this book follows the philosophy of Dedekind, that “Die Zahlen sind freie Schöpfungen des menschlichen Geistes” [Numbers are free creations of the human mind], since for each of the five number systems mentioned above we shall write down axioms which were originally devised by human thinkers. No kind of number is supposed to simply exist without the need for such axioms to be laid down, although all number systems must demonstrate utility in describing the natural world and/or facilitating human activity.

Having defined our number systems, what do we do with them? We have mentioned some practical applications of numbers above, but what is the next step in the process of building a body of theory by logical deduction? There are various directions that one can take. Most obvious is Number Theory, which investigates the properties of the Natural Numbers and the Integers in depth. Secondly, there is Analysis, which deals with functions of Real Numbers and of Complex Numbers, laying the rigorous foundations for differential and integral calculus and then extending to consider multi-dimensional spaces. Importantly, Analysis replaces vague concepts of “infinitely large” or “infinitesimally small” with rigorous definitions based on the properties of Real Numbers; infinity is not a number, and statements like “$1÷0=\infty$” have no place in mathematics! Thirdly, the study of arithmetic operations in number systems yields some of the basic concepts of Abstract Algebra, in which one investigates the structures arising from operations on sets of objects without reference to the kind of object in the set. Some of the elementary concepts in all these areas of mathematics will be discussed in this book where they arise naturally, but readers wishing to investigate any of these topics in greater depth should find a textbook devoted to the topic.

1.4 Mathematical Writing, Notation, and Terminology

Any mathematical argument, even if written mostly or entirely in symbols, should be capable of being read as grammatically correct English. Every symbol has a meaning which can be expressed as a word or words. For example, = means “equals”; and “$A=B$” can be read as a simple sentence, “A equals B”. Note here that A and B are names given to some mathematical objects, so do not need to be expressed in words. We have here assumed an intuitive notion of equality: two mathematical expressions are equal if and only if they are representations of the same object. Equality can be defined more formally as an equivalence relation (see ...

Table of contents

• Cover
• Half Title
• Title
• Copyright
• Contents
• Preface and Acknowledgments
• 1 Introduction: The Purpose of This Book
• 2 Sets and Relations
• 3 Natural Numbers, ℕ
• 4 Integers, ℤ
• 5 Foundations of Number Theory
• 6 Rational Numbers, ℚ
• 7 Real Numbers, ℝ
• 8 Quadratic Extensions I: General Concepts and Extensions of ℤ and ℚ
• 9 Quadratic Extensions II: Complex Numbers, ℂ
• 10 Yet More Number Systems
• 11 Where Do We Go from Here?
• A How to Read Proofs: The “Self-Explanation” Strategy
• Bibliography
• Index