Real Analysis/The real numbers

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 $ℝ$ . So, what is missing? Why do we need anything beyond the rationals?

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

$f:\mathbb{Q}\to \mathbb{Q};x\mapsto \{\begin{array}{cc}0& :x^2<2\\ 1& :x^2>2\end{array}$

Clearly this function has a dramatic jump in it around the rational $1.4$ , 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 $\mathbb{Q}$ , 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 $ℝ$ so that no matter how clever we try to be, if a function has a 'jump' in the way that $f$ does, then we will always be able to find a specific number at which it jumps.

The following sections describe the properties of $ℝ$ 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 $ℝ$, 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!

A constructive approach

With a constructive approach, we are not happy simply to assume exactly what we want, but rather we try to construct $ℝ$ 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 $\mathbb{Q}$ 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 $ℝ$ is a complete ordered field. This is in fact saying a great many things:

• That $ℝ$ is a totally ordered field.
• That $ℝ$ 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:

1. $ℝ$ is a field. For this, we require binary operations addition (denoted $+$) and multiplication (denoted $\times$) defined on $ℝ$ , and distinct elements $0$ and $1$ satisfying:
   1. $(ℝ,+,0)$ is a commutative group, meaning:
      1. $\mathrm{\forall }x,y,z\in ℝ:(x+y)+z=x+(y+z)$ (associativity)
      2. $\mathrm{\forall }x,y\in ℝ:x+y=y+x$ (commutativity)
      3. $\mathrm{\forall }x\in ℝ:x+0=x$ (identity)
      4. $\mathrm{\forall }x\in ℝ:\mathrm{\exists }y\in ℝ:x+y=0$ (inverse)
   2. $(ℝ\setminus \{0\},\times ,1)$ is a commutative group, meaning:
      1. $\mathrm{\forall }x,y,z\in ℝ\setminus \{0\}:(x\times y)\times z=x\times (y\times z)$ (associativity)
      2. $\mathrm{\forall }x,y\in ℝ\setminus \{0\}:x\times y=y\times x$ (commutativity)
      3. $\mathrm{\forall }x\in ℝ\setminus \{0\}:x\times 1=x$ (identity)
      4. $\mathrm{\forall }x\in ℝ\setminus \{0\}:\mathrm{\exists }y\in ℝ\setminus \{0\}:x\times y=1$ (inverse)
   3. $\mathrm{\forall }x,y,z\in ℝ:x\times (y+z)=(x\times y)+(x\times z)$ (distributivity)
2. $ℝ$ is a totally ordered set. For this we require a relation (denoted by $\le$) satisfying:
   1. $\mathrm{\forall }x\in ℝ:x\le x$ (reflexivity)
   2. $\mathrm{\forall }x,y,z\in ℝ:(x\le y\text{ and }y\le z)⟹x\le z$ (transitivity)
   3. $\mathrm{\forall }x,y\in ℝ:(x\le y\text{ and }y\le x)⟹x=y$ (anti-symmetry)
   4. $\mathrm{\forall }x,y\in ℝ:\text{either }x\le y\text{ or }y\le x$ (totality)
3. $ℝ$ is complete in this order (see below for details).
4. The field operations and order interact in the expected manner, meaning:
   1. $\mathrm{\forall }x,y,z\in ℝ:x\le y⟹(x+z)\le (y+z)$
   2. $\mathrm{\forall }x,y,z\in ℝ:(x\le y\text{ and }0\le z)⟹(x\times z)\le (y\times z)$

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 $ℝ$ is the only complete ordered field.

Further notation

Having defined these operations and relations on $ℝ$ , 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 $\times$ for multiplication, we may simply denote it by juxtaposition. In other words, we write $xy$ to denote $x\times y$.
• Since both multiplication and addition are associative, we omit unnecessary bracketing when several numbers are added or multiplied. In other words, rather than writing $(x+y)+z$ or $x+(y+z)$, which are equal, we simply write $x+y+z$ 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 $x+yz$ should be interpreted as $x+(yz)$, not as $(x+y)z$.
• The number $x+y$ is called the sum of $x$ and $y$.
• The number $xy$ is called the product of $x$ and $y$.
• The additive inverse of $x$ is written $-x$, and called the negative or negation of $x$. So, $x+(-x)=0$.
• The multiplicative inverse of $x$ is written $x^{-1}$, and called the reciprocal, or simply the inverse of $x$. So, $x(x^{-1})=1$.
• We define the binary operation of subtraction as follows: For $x,y\in ℝ$, we set $x-y=x+(-y)$. The number $x-y$ is called the difference of $x$ and $y$.
• 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, $a+b-c-d+e$ should be interpreted as $(((a+b)-c)-d)+e$.
• We define the binary operation of division as follows: For $x,y\in ℝ$, with $y=0$, we set $x/y=x(y^{-1})$. The number $x/y$ is called the quotient of $x$ and $y$, and is also denoted $\frac{x}{y}$.
• 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 $\frac{x}{y}$ notation, rather than the $x/y$ notation helps to avoid confusion.
• We define the binary operation of exponentation as follows: For $x\in ℝ$ and $n\in {\mathbb{N}}_0$ we define $x^n$ recursively by $x^0=1$ and $x^{n+1}=(x^n)x$. Then for $n\in \mathbb{Z}$, with $n<0$, we define $x^n=(x^{-1}{)}^{-n}$.
• Exponentation has a higher precedence than any of division, multiplication, addition and subtraction. For example, $a{b}^2+d^3$ should be interpreted as $(a(b^2))+(d^3)$.
• We write $x\ge y$ to mean $y\le x$.
• We write $x<y$ to mean $x\le y$ and $x=y$.
• We write $x>y$ to mean $y<x$.
• To abbreviate a collection of equalities or inequalities, they may be strung together. For example, the expression $a\le b=c=d<e$ should be interpreted as $a\le b$ and $b=c$ and $c=d$ and $d<e$.
• To say $x$ is positive means $x>0$.
• To say $x$ is negative means $x<0$.
• To say $x$ is non-positive means $x\le 0$.
• To say $x$ is non-negative means $x\ge 0$.
• We also introduce notation for several common varieties of subsets of $ℝ$. All of these subsets are called intervals:
  • $[a,b]=\{x\in ℝ:a\le x\le b\}$ (called the closed interval from $a$ to $b$)
  • $(a,b)=\{x\in ℝ:a<x<b\}$ (called the open interval from $a$ to $b$)
  • $[a,b)=\{x\in ℝ:a\le x<b\}$
  • $(a,b]=\{x\in ℝ:a<x\le b\}$
  • In all these cases, $a$ is called the lower limit of the interval, and $b$ is called the upper limit.
  • An excluded lower limit (as in the second and fourth cases) may be replaced by $-\mathrm{\infty }$ to indicate that there is no lower restriction. For example $(-\mathrm{\infty },b]=\{x\in ℝ:x\le b\}$.
  • Similarly, an excluded upper limit (as in the second and third cases) may be replaced by $\mathrm{\infty }$. For example, $(-\mathrm{\infty },\mathrm{\infty })=ℝ$.
  • Some specific intervals which appear frequently are the closed unit interval, or just unit interval, which is $[0,1]$, and ${ℝ}^{+}=(0,\mathrm{\infty })$, the positive real numbers.

Completeness

The rational numbers $\mathbb{Q}$ 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

Let $A\subseteq ℝ$. We say $b\in ℝ$ is an upper bound for $A$ if

$\mathrm{\forall }s\in A:s\le b$

For example, $3$ is an upper bound for $[0,1]$, as is $1$, but $\frac{1}{2}$ is not, because $1\in [0,1]$ and $1>\frac{1}{2}$. A set with an upper bound $b$ is said to be bounded above by $b$.

Least Upper Bound

We say $s$ is a least upper bound or supremum for $A$ if $s$ is an upper bound for $A$, and $b$ is any upper bound for $A$ then $s\le b$. More formally:

$(\mathrm{\forall }a\in A:a\le s)\text{ and }(\mathrm{\forall }b\in ℝ:((\mathrm{\forall }a\in A:a\le b)⟹(s\le b)))$

Similarly, we say $b\in ℝ$ is a lower bound for $A$ if

$\mathrm{\forall }a\in A:a\ge b$

and we say $i$ is a greatest lower bound or infimum for $A$ if:

$(\mathrm{\forall }a\in A:a\ge i)\text{ and }(\mathrm{\forall }b\in ℝ:((\mathrm{\forall }a\in A:a\ge b)⟹(i\ge b)))$

The supremum and infimum of a set $A$ are denoted $\sup A$ and $\inf A$ respectively.

The Least upper bound axiom

Now we are finally ready to state the last axiom:

• If $S\subseteq ℝ$ is non-empty and has an upper bound, then $S$ has a least upper bound in $ℝ$.

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 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.