Real Numbers - Revision history

Khanh: /* Why we need the real numbers */

2021-11-07T21:57:41Z

Why we need the real numbers

Khanh: /* The complete ordered field */

2021-11-07T21:56:03Z

The complete ordered field

Khanh: /* Resources */

2021-11-07T21:53:14Z

Resources

Lila: /* Completeness */

2021-10-20T00:48:46Z

Completeness

Lila at 00:47, 20 October 2021

2021-10-20T00:47:36Z

Lila: /* The axioms */

2021-10-12T14:54:05Z

The axioms

Lila: /* A constructive approach */

2021-10-12T14:51:03Z

A constructive approach

Lila: Created page with "==Why we need the real numbers== This is a good juncture to justify the subject of real analysis, which essentially reduces to justifying the necessity of studying $\R$
2021-10-12T14:44:28Z

Created page with "==Why we need the real numbers== This is a good juncture to justify the subject of real analysis, which essentially reduces to justifying the necessity of studying <math>\R</m..."

New page
==Why we need the real numbers==
This is a good juncture to justify the subject of real analysis, which essentially reduces to justifying the necessity of studying <math>\R</math> . So, what is missing? Why do we need anything beyond the rationals?

The first sign of trouble is square roots. Famously, <math>\sqrt2</math> is not rational – in other words, there is no rational number which squares to <math>2</math> (see the exercises). This fact has a curious consequence – consider the following function:

<math>f:\Q\to\Q\ ;\ x\mapsto\begin{cases}0&:x^2<2\\1&:x^2>2\\\end{cases}</math>

Clearly this function has a dramatic jump in it around the rational <math>1.4</math> , where it suddenly changes from being equal to zero and starts being equal to one. However, it's difficult (or even impossible) to pin down exactly where this jump happens. Any specific rational number is safely on one side or the other, and, indeed, in the standard [[Topology/Topological Spaces|Topology]] on <math>\Q</math> , this function is continuous (don't worry if that makes no sense to you).

It is this flaw which the real numbers are designed to repair. We will define the real numbers <math>\R</math> so that no matter how clever we try to be, if a function has a 'jump' in the way that <math>f</math> does, then we will always be able to find a specific number at which it jumps.

The following sections describe the properties of <math>\R</math> which make this possible.

==Different perspectives==
In order to prove anything about the real numbers, we need to know what their properties are. There are two different approaches to describing these properties – axiomatic and constructive.

===An axiomatic approach===
When we take an axiomatic approach, we simply make a series of assertions regarding <math>\mathbb R</math>, and assume that they hold.

The assertions that we make are called ''axioms'' – in a mathematical context this term means roughly 'basic assumption'.

The advantage of this approach is that it is then clear exactly what has been assumed, before proceeding to deduce results which rely only on those assumptions.

The disadvantage of this approach is that it might not be immediately clear that any object satisfying the properties we desire even exists!

===[[Real Analysis/Constructing the real numbers|A constructive approach]]===
With a constructive approach, we are not happy simply to assume exactly what we want, but rather we try to ''construct'' <math>\R</math> from something simpler, and then prove that it has the properties we want. In this way, what could have been axioms become theorems. There are several different ways to do this, starting from <math>\Q</math> and using some method to 'fill up the gaps between the rationals'.

All of these methods are fairly complex and will be put off until the next section.

==The axioms==
So, what are these axioms which we will need? The short version is to say that <math>\R</math> is a ''complete ordered field''. This is in fact saying a great many things:
That <math>\R</math> is a [[Real Analysis/Rational Numbers|totally ordered field]].
That <math>\R</math> is complete in this ordering (Note that the meaning of completeness here is not quite the same as the common meaning in the study of partially ordered sets).
That the algebraic operations (addition and multiplication) described by the field axioms interact with the ordering in the expected manner.

In more detail, we assert the following:
#<math>\R</math> is a field. For this, we require binary operations ''addition'' (denoted <math>+</math>) and ''multiplication'' (denoted <math>\times</math>) defined on <math>\R</math> , and distinct elements <math>0</math> and <math>1</math> satisfying:
##<math>(\R,+,0)</math> is a commutative group, meaning:
###<math>\forall x,y,z\in\R:(x+y)+z=x+(y+z)</math> (associativity)
###<math>\forall x,y\in\R: x+y=y+x</math> (commutativity)
###<math>\forall x\in\R: x+0=x</math> (identity)
###<math>\forall x\in\R:\exists y\in\R:x+y=0</math> (inverse)
##<math>(\R\setminus\{0\},\times,1)</math> is a commutative group, meaning:
###<math>\forall x,y,z\in\R\setminus\{0\}:(x\times y)\times z=x\times (y\times z)</math> (associativity)
###<math>\forall x,y\in\R\setminus\{0\}:x\times y=y\times x</math> (commutativity)
###<math>\forall x\in\R\setminus\{0\}:x\times1=x</math> (identity)
###<math>\forall x\in\R\setminus\{0\}:\exists y\in\R\setminus\{0\}:x\times y=1</math> (inverse)
##<math>\forall x,y,z\in\R:x\times(y+z)=(x\times y)+(x\times z)</math> (distributivity)
#<math>\R</math> is a totally ordered set. For this we require a relation (denoted by <math>\le</math>) satisfying:
##<math>\forall x\in\R: x\le x</math> (reflexivity)
##<math>\forall x,y,z\in\R:(x\le y\text{ and }y\le z)\implies x\le z</math> (transitivity)
##<math>\forall x,y\in\R:(x\le y\text{ and }y\le x)\implies x=y</math> (anti-symmetry)
##<math>\forall x,y\in\R:\text{either }x\le y\text{ or }y\le x</math> (totality)
#<math>\R</math> is complete in this order (see below for details).
#The field operations and order interact in the expected manner, meaning:
##<math>\forall x,y,z\in\R:x\le y\implies(x+z)\le(y+z)</math>
##<math>\forall x,y,z\in\R:(x\le y\text{ and }0\le z)\implies(x\times z)\le(y\times z)</math>

This is a substantial list, and if you are not used to axiomatic mathematics (or even if you are!) it may seem somewhat daunting, especially since we have yet to give details of what completeness means. This is amongst the longest list of axioms in any region of mathematics, but if you examine each in turn, you will find that they all state things which you have probably taken for granted as 'the way numbers behave' without a second thought.

These axioms are so exacting that there is a sense in which they specify the real numbers precisely. In other words <math>\R</math> is the ''only'' complete ordered field.

===Further notation===
Having defined these operations and relations on <math>\R</math> , we need to introduce more notation to aid in talking about them. Hopefully all these conventions should be familiar to you, but it is important to formally present them all to avoid confusion following from misunderstanding of notation:

Rather than writing <math>\times</math> for multiplication, we may simply denote it by juxtaposition. In other words, we write <math>xy</math> to denote <math>x\times y</math>.
* Since both multiplication and addition are associative, we omit unnecessary bracketing when several numbers are added or multiplied. In other words, rather than writing <math>(x+y)+z</math> or <math>x+(y+z)</math>, which are equal, we simply write <math>x+y+z</math> to denote their common value.
* To further save writing of brackets, by convention, multiplication has a higher precedence than addition. So, for example, the expression <math>x+yz</math> should be interpreted as <math>x+(yz)</math>, not as <math>(x+y)z</math>.
* The number <math>x+y</math> is called the ''sum'' of <math>x</math> and <math>y</math>.
* The number <math>xy</math> is called the ''product'' of <math>x</math> and <math>y</math>.
* The additive inverse of <math>x</math> is written <math>-x</math>, and called the ''negative'' or ''negation'' of <math>x</math>. So, <math>x+(-x)=0</math>.
* The multiplicative inverse of <math>x</math> is written <math>x^{-1}</math>, and called the ''reciprocal'', or simply the ''inverse'' of <math>x</math>. So, <math>x(x^{-1})=1</math>.
* We define the binary operation of ''subtraction'' as follows: For <math>x,y\in\mathbb R</math>, we set <math>x-y=x+(-y)</math>. The number <math>x-y</math> is called the ''difference'' of <math>x</math> and <math>y</math>.
* Subtraction has the same precedence as addition (less than that of multiplication), and when the two operations are mixed without bracketing, left-associativity is implied. For example, <math>a+b-c-d+e</math> should be interpreted as <math>(((a+b)-c)-d)+e</math>.
* We define the binary operation of ''division'' as follows: For <math>x,y\in\mathbb R</math>, with <math>y\not=0</math>, we set <math>x/y=x(y^{-1})</math>. The number <math>x/y</math> is called the ''quotient'' of <math>x</math> and <math>y</math>, and is also denoted <math>\frac{x}{y}</math>.
* Division has a higher precedence than that of addition or subtraction, but there is no simple convention as to how to handle mixed multiplication and division. Using the <math>\frac{x}{y}</math> notation, rather than the <math>x/y</math> notation helps to avoid confusion.
* We define the binary operation of ''exponentation'' as follows: For <math>x\in\mathbb R</math> and <math>n\in\mathbb N_0</math> we define <math>x^n</math> recursively by <math>x^0=1</math> and <math>x^{n+1}=(x^n)x</math>. Then for <math>n\in\mathbb Z</math>, with <math>n<0</math>, we define <math>x^n=(x^{-1})^{-n}</math>.
* Exponentation has a higher precedence than any of division, multiplication, addition and subtraction. For example, <math>ab^2+d^3</math> should be interpreted as <math>(a(b^2))+(d^3)</math>.
* We write <math>x\geq y</math> to mean <math>y\leq x</math>.
* We write <math>x<y</math> to mean <math>x\leq y</math> and <math>x\not=y</math>.
* We write <math>x>y</math> to mean <math>y<x</math>.
* To abbreviate a collection of equalities or inequalities, they may be strung together. For example, the expression <math>a\leq b=c=d<e</math> should be interpreted as <math>a\leq b</math> and <math>b=c</math> and <math>c=d</math> and <math>d<e</math>.
* To say <math>x</math> is ''positive'' means <math>x>0</math>.
* To say <math>x</math> is ''negative'' means <math>x<0</math>.
* To say <math>x</math> is ''non-positive'' means <math>x\leq 0</math>.
* To say <math>x</math> is ''non-negative'' means <math>x\geq 0</math>.
* We also introduce notation for several common varieties of subsets of <math>\mathbb R</math>. All of these subsets are called ''intervals'':
<math>[a,b]=\{x\in\mathbb R:a\leq x\leq b\}</math> (called the ''closed interval'' from <math>a</math> to <math>b</math>)
<math>(a,b)=\{x\in\mathbb R:a<x<b\}</math> (called the ''open interval'' from <math>a</math> to <math>b</math>)
<math>[a,b)=\{x\in\mathbb R:a\leq x<b\}</math>
<math>(a,b]=\{x\in\mathbb R:a<x\leq b\}</math>
In all these cases, <math>a</math> is called the ''lower limit'' of the interval, and <math>b</math> is called the ''upper limit''.
An excluded lower limit (as in the second and fourth cases) may be replaced by <math>-\infty</math> to indicate that there is no lower restriction. For example <math>(-\infty,b]=\{x\in\mathbb R:x\leq b\}</math>.
Similarly, an excluded upper limit (as in the second and third cases) may be replaced by <math>\infty</math>. For example, <math>(-\infty,\infty)=\mathbb R</math>.
Some specific intervals which appear frequently are the ''closed unit interval'', or just ''unit interval'', which is <math>[0,1]</math>, and <math>{\mathbb R}^+=(0,\infty)</math>, the positive real numbers.

===Completeness===
The rational numbers <math>\mathbb Q</math> satisfy all of the axioms above which have been explained in detail, and so if we are to escape the problem which we described [[Real analysis/The real numbers#Why we need the real numbers|above]] then we clearly need something more. This 'something more' is ''completeness''. There are several equivalent ways of describing completeness, but most of them require us to know about [[Real analysis#Sequences|Sequences]], which we do not introduce until the next chapter, so for the moment we can only give one definition.

====Upper bounds====

Let <math>A\subseteq\mathbb R</math>. We say <math>b\in\mathbb R</math> is an ''upper bound'' for <math>A</math> if
:<math>\forall s\in A:s\leq b</math>
For example, <math>3</math> is an upper bound for <math>[0,1]</math>, as is <math>1</math>, but <math>\frac{1}{2}</math> is not, because <math>1\in[0,1]</math> and <math>1>\frac{1}{2}</math>. A set with an upper bound <math>b</math> is said to be ''bounded above by <math>b</math>''.
=====Least Upper Bound=====
We say <math>s</math> is a ''least upper bound'' or ''supremum'' for <math>A</math> if <math>s</math> is an upper bound for <math>A</math>, and <math>b</math> is any upper bound for <math>A</math> then <math>s\leq b</math>. More formally:
:<math>(\forall a\in A:a\leq s)\mbox{ and }(\forall b\in\mathbb R:((\forall a\in A:a\leq b)\implies(s\leq b)))</math>
Similarly, we say <math>b\in\mathbb R</math> is a ''lower bound'' for <math>A</math> if
:<math>\forall a\in A:a\geq b</math>
and we say <math>i</math> is a ''greatest lower bound'' or ''infimum'' for <math>A</math> if:
:<math>(\forall a\in A:a\geq i)\mbox{ and }(\forall b\in\mathbb R:((\forall a\in A:a\geq b)\implies(i\geq b)))</math>
The supremum and infimum of a set <math>A</math> are denoted <math>\sup A</math> and <math>\inf A</math> respectively.

====The Least upper bound axiom====

Now we are finally ready to state the last axiom:
* If <math>S\subseteq\mathbb R</math> is non-empty and has an upper bound, then <math>S\ </math> has a least upper bound in <math>\mathbb R</math>.
This is the axiom of the real numbers that finally satisfies what was lacking in the rationals: completeness. It is worth noting at this point, to avoid possible confusion, that in the study of general partially ordered sets, the definition of completeness is that ''every'' subset has a least upper bound, and there is no condition that they be non-empty or bounded above.
Nevertheless, we really do wish to impose these two conditions in this case.

====Other completeness axioms====
There are other equivalent ways to state the completeness axiom, but they involve [[Real analysis/Sequences|sequences]], so we shall delay them until after the discussion of that topic. Because of the existence of these other forms, this axiom is sometimes called the ''least upper bound axiom''.

==Resources==
* [https://en.wikibooks.org/wiki/Real_Analysis/The_real_numbers The Real Numbers], Wikibooks: Real Analysis

@@ Line 66: / Line 66: @@
 The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.
-First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element ''z'', {{nowrap|''z'' + 1}} is larger).
+First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element ''z'', ''z'' + 1 is larger).
 Additionally, an order can be Dedekind-complete. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.
@@ Line 73: / Line 73: @@
 But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the ''largest'' Archimedean field in the sense that every other Archimedean field is a subfield of <math>\mathbb{R}</math>. Thus <math>\mathbb{R}</math> is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.
 ===Further notation===

@@ Line 148: / Line 148: @@
 But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the ''largest'' Archimedean field in the sense that every other Archimedean field is a subfield of <math>\mathbb{R}</math>. Thus <math>\mathbb{R}</math> is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.
-==Resources==
+== Licensing ==
-* [https://en.wikibooks.org/wiki/Real_Analysis/The_real_numbers The Real Numbers], Wikibooks: Real Analysis
+Content obtained and/or adapted from:
-* [https://en.wikipedia.org/wiki/Real_number Real number], Wikipedia
+* [https://en.wikibooks.org/wiki/Real_Analysis/The_real_numbers The Real Numbers, Wikibooks: Real Analysis] under a CC BY-SA license

@@ Line 6: / Line 6: @@
 <math>f:\Q\to\Q\ ;\ x\mapsto\begin{cases}0&:x^2<2\\1&:x^2>2\\\end{cases}</math>
-Clearly this function has a dramatic jump in it around the rational <math>1.4</math> , where it suddenly changes from being equal to zero and starts being equal to one. However, it's difficult (or even impossible) to pin down exactly where this jump happens. Any specific rational number is safely on one side or the other, and, indeed, in the standard [[Topology/Topological Spaces|Topology]] on <math>\Q</math> , this function is continuous (don't worry if that makes no sense to you).
+Clearly this function has a dramatic jump in it around the rational <math>1.4</math> , where it suddenly changes from being equal to zero and starts being equal to one. However, it's difficult (or even impossible) to pin down exactly where this jump happens. Any specific rational number is safely on one side or the other, and, indeed, in the standard Topology on <math>\Q</math> , this function is continuous (don't worry if that makes no sense to you).
 It is this flaw which the real numbers are designed to repair. We will define the real numbers <math>\R</math> so that no matter how clever we try to be, if a function has a 'jump' in the way that <math>f</math> does, then we will always be able to find a specific number at which it jumps.

@@ Line 31: / Line 31: @@
 ==The axioms==
 So, what are these axioms which we will need? The short version is to say that <math>\R</math> is a ''complete ordered field''. This is in fact saying a great many things:
-*That <math>\R</math> is a [[Real Analysis/Rational Numbers|totally ordered field]].
+*That <math>\R</math> is a totally ordered field.
 *That <math>\R</math> is complete in this ordering (Note that the meaning of completeness here is not quite the same as the common meaning in the study of partially ordered sets).
 *That the algebraic operations (addition and multiplication) described by the field axioms interact with the ordering in the expected manner.
@@ Line 97: / Line 97: @@
 ===Completeness===
-The rational numbers <math>\mathbb Q</math> satisfy all of the axioms above which have been explained in detail, and so if we are to escape the problem which we described [[Real analysis/The real numbers#Why we need the real numbers|above]] then we clearly need something more.  This 'something more' is ''completeness''.  There are several equivalent ways of describing completeness, but most of them require us to know about [[Real analysis#Sequences|Sequences]], which we do not introduce until the next chapter, so for the moment we can only give one definition.
+The rational numbers <math>\mathbb Q</math> satisfy all of the axioms above which have been explained in detail, and so if we are to escape the problem which we described above then we clearly need something more.  This 'something more' is ''completeness''.  There are several equivalent ways of describing completeness, but most of them require us to know about sequences, which we do not introduce until the next chapter, so for the moment we can only give one definition.
 ====Upper bounds====
@@ Line 121: / Line 121: @@
 ====Other completeness axioms====
-There are other equivalent ways to state the completeness axiom, but they involve [[Real analysis/Sequences|sequences]], so we shall delay them until after the discussion of that topic.  Because of the existence of these other forms, this axiom is sometimes called the ''least upper bound axiom''.
+There are other equivalent ways to state the completeness axiom, but they involve sequences, so we shall delay them until after the discussion of that topic.  Because of the existence of these other forms, this axiom is sometimes called the ''least upper bound axiom''.
 ==Resources==
 * [https://en.wikibooks.org/wiki/Real_Analysis/The_real_numbers The Real Numbers], Wikibooks: Real Analysis

@@ Line 24: / Line 24: @@
 The disadvantage of this approach is that it might not be immediately clear that any object satisfying the properties we desire even exists!
-===[[Real Analysis/Constructing the real numbers|A constructive approach]]===
+===A constructive approach===
 With a constructive approach, we are not happy simply to assume exactly what we want, but rather we try to ''construct'' <math>\R</math> from something simpler, and then prove that it has the properties we want. In this way, what could have been axioms become theorems. There are several different ways to do this, starting from <math>\Q</math> and using some method to 'fill up the gaps between the rationals'.