The Real and Complex Number System

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Contents 1 Introduction 1.1 Real Analysis 1.1.1 The Natural Numbers 1.1.2 The Integer Numbers 1.1.3 The Rational Numbers 1.2 is incomplete 1.3 Set Theory Conventions 2 Ordered Sets 3 Fields 4 The Real Field 5 The Extended Real Number System 6 The Complex Field 7 Euclidean Spaces 8 Dedekind Cuts

[edit] Introduction

[edit] Real Analysis

Real analysis is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.

The goal of this course is to prepare a student to acquire skills on the foundations of the basic theorems and results that shape the mechanisms of calculus and real analysis, and to progress towards a deeper understanding of mathematical ideas that will be a basis for further courses in Linear Algebra, Advanced Real Analysis, Complex Analysis, Functional Analysis, Partial Differential Equations, Lebesgue Theory, Calculus on Manifolds and the Integration of Differential Forms.

Our course begins with naive observations on previously known number systems:

[edit] The Natural Numbers

Natural Numbers arise as the simplest abstraction for the notion of quantity. They are usually conceived as "counting numbers" and for the purposes of this course (and unless otherwise stated), will begin with 1. If we try to abstract a little bit further, we can observe that as "counting numbers", they have the property of always having a succesor: that is, if you have a natural number, you can always find a new natural number, its succesor, such that no other natural number lies between that number and the former:

For example: If I give you the natural number 1729, the only possible successor would be 1730. If I say that 2008 is the succesor of a number, then the original number would be 2007.

The existence of a "succesor property", implies (as you should verify) that there is a least natural number: 1. Also, this property implies having a fundamental and simple algorithm for creating a new natural number from any given natural number.

What happens when we, somehow, "group" all possible natural numbers? We will not talk -for the moment- about "how many" natural numbers one can have (intuitively, you may say "¡infinite!", but we must be a little more precise and rigorous about the mathematical conception of "infinity"). But we can certainly assume it is possible to talk about the set that contains all of the natural numbers (since the succesor property, in a way, constructs all of them). We give that set a distinctive character: :

When talking about systems of numbers, it is helpful to write instead the set that contains them. We will continue developing more concepts of as we move forward, such as what it means for to be a partially ordered set, or a countable set; but we will not deal with their formal construction, which can be described in terms of Set Theory, and belongs to a different course.

[edit] The Integer Numbers

Integer Numbers can be thought of as an "extension" of the Natural Numbers, that includes all negative numbers plus the number zero. We call the set of all integer numbers . In the language of set theory, we can say that .

Many properties of the natural numbers are induced into the integers, but for the sake of brevity we will not develop further their formal construction, wich belongs to a more rigorous course on set theory and foundations of mathematics, and characterizes integers as equivalence classes of ordered pairs of natural numbers.

[edit] The Rational Numbers

Rational Numbers are also an "extension" of integers. They are formally constructed as equivalence classes of pairs of integers, and represent all the fractions that we know from basic algebra. The set of all rational numbers is called , and we will use them as pairs of integers m,n in the form m / n (as usual).

One would think that, adjusting the numerator and denominator of a rational number, any conceivable number could be expressed, from the smallest, to the largest (and indeed, ). But from the knowledge of solutions to algebraic equations, there are situations in which is not sufficient to express a valid solution to a given equation. Take the famous "pitagorean catastrophe", for instance:

Is there a rational number p such that p² = 2?

[edit] is incomplete

We are now going to tackle the question of the "completeness" of the rational numbers. Intuitively speaking, we want to find out if any number notion can be expressed using . Take a square, for instance. Isn´t it natural for us to wonder about the number that represents the "lenght" of the diagonal of this square?

Pithagoras' theorem tells us that this number is . But what the symbol tells us is "find a number x such that its square equals 2" In other words:

Thus, there is a strong reason for conjecturing a relationship between algebraic equations and the set of numbers we use to solve them (think about the complex numbers, for instance).

Let us show how no rational number can solve this equation (equivalent to saying: "let us show that the square root of two is not a rational number"): The proof is by contradiction. Let us first assume that is actually a rational number. Then, it can be written in the form fracmn, where both m and n are integers, and they are in their minimal proportion (i.e. already simplified). Thus, we can write:

m² = 2n²

Which shows that m² is an even number. That in turn implies that m² is divisible by 4 and that m is en even number

[edit] Set Theory Conventions

[edit] Ordered Sets

[edit] Fields

[edit] The Real Field

[edit] The Extended Real Number System

[edit] The Complex Field

[edit] Euclidean Spaces

[edit] Dedekind Cuts