Real numbers – Serlo

Real numbers are the basics for this book, because we want to understand topics related to these numbers like real sequences or real functions. But before we begin with studying the concepts of analysis, we have to ask: "What are real numbers?"

This isn't an easy question. So let's start by looking at different ideas and concepts of how to define these real numbers.

Ways to describe real numbers

We all already have an intuitive idea of what real numbers are, even though it is often hard to put it into words. For example one often imagines the real numbers as points on the number line:

The real numbers visualized by points on the real line

Our next task now is to translate this intuitive idea of real numbers into mathematical expressions. There are two approaches how to do this: either by using axioms or by constructing the real numbers.

The axiomatic characterisation

In the axiomatic description of mathematics, we lay the foundation with the axioms on which we built the theorems step by step.

Instead of accurately specifying what the real numbers are, the axiomatic characterisation instead describes the properties of real numbers. We want to do this, by making this kind of a statement: "The real numbers are the set of objects, which have the following properties:<list of the properties of real numbers>". These basic properties we will define through axioms (missing). Short recap: An axiom is a statement (missing), which we accept to be true, without proof. Every mathematical model that fulfils these properties or axioms can be used as a model for real numbers. Statements which are proven with the help of axioms are called theorems.

When formulating these axioms we must make sure that they don't contradict each other. In other words we need to make sure that there exist at least one model, which fulfils all axioms. A bad example for a model would be, that $1<0$ and $1\geq 0$ cannot be true at the same time. Making sure to not get too abstract is also good. We want the axioms to represent our intuitive idea of real numbers.

We also need to be aware that we need to have enough axioms to characterise the real numbers. If we have too few, there could be other structures which fulfil all requirements as a model, but wouldn't fit our intuitive idea of real numbers. for example it isn't enough for there just to be the possibility to add $+$ two real numbers $x$ and $y$ resulting in $x+y=y+x$. With only that the whole numbers would also be a model for the real numbers, which doesn't fit with our idea of real numbers. So just having addition isn't enough.

The final thing to pay attention to, is to avoid redundancy. So if a statement can be proven through other axioms, it shouldn't be an axiom. So for example: if the statement E can be proven by only using the axioms A1 and A2, then E doesn't need to be defined as an axiom.

The construction-based approach

If we want to construct the real numbers, we have to start with the rational numbers as building blocks. Then by using a specific method/process we create new objects, based on the rational numbers, which then we define as the real numbers.

In contrast to the axiomatic characterisation, which talks about the properties of real numbers, the construction-based approach has the advantage, that we can precisely determine what the real numbers are. They are exactly the objects which we created through our construction process. We especially don't have to define the properties with axioms, because the properties can be derived from the properties of the constructed objects.

There are different methods to construct the real numbers. The resulting structures are none the less equivalent: they have the same properties(these structures are called "isomorph").

The relationship between the approaches

Both approaches lead to the same result. If one can prove all the properties defined by the axioms for the constructed objects, then all other properties derived from the axioms are also true for the constructed objects. Vice versa all theorems proved only based on the axioms in the model, can be recreated in the axiomatic description.

So it is unimportant which approach we choose. We will start with the axiomatic approach, since it is easier to understand.

Summary: The axiomatic approach to characterising the real numbers

To determine the axioms of the real numbers, we can do the following:

1. Develop an intuitive idea: We first need an intuitive idea of the real numbers. We can for example go back to the idea, of real numbers being points on the number line.
2. Define the axioms: Now all axioms need to be defined. When doing that we need to pay attention to the following:
   • Comprehensible: The axioms should make sense, meaning every axiom should be intuitive.
   • Free from contradiction:The axioms need to be free from contradiction. This can be implicitly shown, by it being possible to construct an actual model of the real numbers, because this is only possible when the axioms don't contradict themselves.
   • Sufficiency: Once all axioms are fulfilled for a model, then that model should correspond to our intuitive idea of real numbers. We should be able to derive all theorems for real numbers from the axioms.
   • To avoid redundancy: No axiom should be derive from the other axioms.
3. To define terms and prove theorems:

Building upon these axioms, we will introduce the core ideas of analysis as well as the theorems of analysis.

In analysis all theorems are proven by using the axioms of the real numbers. Here we will have to introduce concepts and theorems which you already know from school mathematics - like the equation $1\cdot 0=0$. During these proofs it is really important, that we only use properties of real numbers, which we either defined in the axioms or have already proven. Things that you know from school are off limits, until you have proven them based on the axioms!

The axioms of real numbers can be divided in three categories: The "field axioms",the "ordering axioms" and the "Archimedean property".

The field axioms

The field axioms define the addition and multiplication, which is the arithmetic structure of the real numbers. The real numbers are objects, that can be added and multiplied by using the properties of addition and multiplication which you are used to. The essential properties of these operations are the field axioms. The division and subtraction are derived from multiplication and addition. So in total this group of axioms describe how you can calculate with real numbers.

Definition (Field axioms)

On the real numbers $\mathbb {R}$ we can define the operations $+:\mathbb {R} \times \mathbb {R} \to \mathbb {R}$ and $\cdot :\mathbb {R} \times \mathbb {R} \to \mathbb {R}$ . They satisfy the following properties:

• Properties of the addition:
  • Associative law of addition: For all real numbers $x,y,z$ there is $x+(y+z)=(x+y)+z$.
  • Commutative law of addition: For all real numbers $x$ and $y$ there is $x+y=y+x$.
  • Existence of a zero: There is at least one real number $0\in \mathbb {R}$, such that $0+x=x$ for all real numbers $x$.
  • Existence of a negative: For any real number $x$ there is at least one real number $-x$ with $x+(-x)=0$.
• Properties of the multiplication:
  • Associative law of multiplication: For all real numbers $x,y,z$ there is $x\cdot (y\cdot z)=(x\cdot y)\cdot z$.
  • Commutative law of multiplication: For all real numbers $x$ and $y$ there is $x\cdot y=y\cdot x$.
  • Existence of a one: There is at least one real number $1\in \mathbb {R}$ with $1\neq 0$, such that $1\cdot x=x$ for all real numbers $x$.
  • Existence of an inverse: For any real number $x\neq 0$ there is at least one real number $x^{-1}$ with $x\cdot x^{-1}=1$.
• Distributive law: For all real numbers $x,y,z$ gilt $x\cdot (y+z)=x\cdot y+x\cdot z$.

The ordering axioms

The ordering axioms describe a linear order of the real numbers. The real numbers are therefore objects, which one can be compared to each other. The comparison of two different real numbers results either in one of the numbers being larger in comparison or the other way around. This shows a clear connection between the structures of the real numbers and a line since all points on a line are ordered in a natural fashion. This group of axioms is imperative for the construction of a mental picture of the real numbers as points on the number line. This group of axioms also define how the operations of addition and multiplication are connected to the ordered structure.

The ordering of the real numbers can be described by knowing what all positive numbers are. If $x$ is a positive number, we write $0<x$. The positivity of the real numbers is thereby defined via the order axioms:

Definition (Ordering axioms)

The ordering axioms are

• Trichotomy of positivity: For all real numbers $x$ either $0<x$ or $x=0$ or $0<-x$ holds. Using the abbreviations "$\forall$" meaning "for all" and "${\dot {\lor }}$" meaning "either or" we can write this as
  $\forall x\in \mathbb {R} :0<x\ {\dot {\lor }}\ 0=x\ {\dot {\lor }}\ 0<-x$
• Completeness with respect to addition: For all real numbers $a$ and $b$, if $0<a$ and $0<b$, then $0<a+b$ is also true. In characters:
  $\forall a,b\in \mathbb {R} :0<a\land 0<b\implies 0<a+b$
• Completeness with respect to multiplication: For all real numbers $a$ and $b$, if $0<a$ and $0<b$, then $0<ab$ is also true. In characters:
  $\forall a,b\in \mathbb {R} :0<a\land 0<b\implies 0<ab$

With the help of the positivity property $0<x$ we can define the less-than relation:

Definition (less-than relation)

The less-than relation $x<y$ is defined by the following equivalence:

$x<y:\iff 0<y-x$

So there is exactly $x$ smaller than $y$ if the difference $y-x$ is positive. We can define all other order relations via the less-than relation:

Definition (Further order relations based on the less-than relation.)

For the real numbers, the relations $\geq$, $\leq$ and $>$ are also defined via the following equivalences:

• $x\leq y:\iff x<y\lor x=y$
• $x\geq y:\iff y<x\lor x=y$
• $x>y:\iff y<x$

The completeness axiom

The completeness axiom describes the transition or more precisely the definitive difference between the rational and real numbers. While both the above-mentioned groups of axioms are still fulfilled by the rational numbers, this isn't true any more for the completeness axiom. The reason for this is, that in the number range of the rational numbers, there exist "gaps" like ${\sqrt {2}}$. These gaps can be approximated by rational numbers to any degree, but the gaps aren't rational numbers any more. These gaps do not exist in the real numbers, because the completeness axiom denies the existence of such gaps. If it is possible to approximate something to any degree through real numbers, then this "something" exists and is a real number.

The approximation of a real number can be realized with nested intervals. These are a series of intervals, which are subsets of each other and their length converging towards 0:

Nested intervals

An interval nesting is used to approximate real numbers. Each interval restricts the range in which the number to be approximated lies and in the course of the interval nesting this range becomes smaller and smaller. The interval nesting principle guarantees that at least one number is approximated by each interval nesting:

Definition (General interval nesting principle)

For every general interval nesting $I_{1}$, $I_{2}$, $I_{3}$... there exists a real number that lies in all intervals and is thus approximated by all intervals.

This completeness axiom says that the set of real numbers has no "gaps". In order to describe that the real numbers are the smallest possible extension to fill the gaps of the rational numbers, we have to add another axiom to the interval nesting principle. For this we must exclude infinitely small and infinitely large numbers. A positive number $y$ would be infinitely large compared to a positive number $x$ if $y$ was larger than all multiples of $x$, that is, if none of the multiples $x,2x,3x,4x,\ldots$ ever grows beyond $y$. So for all natural numbers $n$ there would be $nx<y$.

Let us now rule this out. For every two positive numbers $x$ and $y$ there should be at least one natural number $n$ with $nx\geq y$. This is precisely the property described by Archimedes' axiom:

Definition (The Archimedian axiom)

For all real numbers $x,y>0$ there is a natural number $n$ such that $nx\geq y$. With the abbreviations "$\forall$" meaning "for all" and "$\exists$" meaning "there is", this reads as

$\forall x>0,y>0\,\exists n\in \mathbb {N} :nx\geq y$

Using the completeness axiom, one can show that real numbers can be arbitrarily approximated by rational numbers. This property is essential because it makes it possible to calculate with rational numbers instead of real numbers. For example, computer calculations are usually performed only with rational numbers. Such calculations usually feature errors, but their errors can usually be made sufficiently small.