Real numbers – "Math for Non-Geeks"

verweis=Math for Non-Geeks

Real numbers are the basis for this book, because we want to understand topics related to these numbers like real sequences or real functions. But before we begin 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 to doing this: either by using axioms or by constructing the real numbers.

The axiomatic characterisation

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 a statement of the form: "The real numbers are a set of objects with the following properties." These basic properties we will define through axioms (missing). Short recap: An axiom is a statement (missing) that we accept as 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 exists at least one model that fulfils all axioms. For example, the requirements $1 < 0$ and $1 \geq 0$ would contradict each other. 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, requiring only a commutative addition operation, $x + y = y + x$ , is not enough. The whole numbers also satisfy that requirement but do not fit our idea of the 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^[1]. The resulting structures are nonetheless equivalent: they are isomorphic as ordered fields.

The relationship between the approaches

Both approaches lead to equivalent structures if the constructed objects satisfy all the axioms. Every property derived from those axioms then also holds for the constructed real numbers.

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:

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.
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 shown by constructing a model of the real numbers, which is possible only if the axioms are consistent.
- 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 derived from the other axioms.
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 real 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 properties used here comprise the field axioms, the ordering axioms, the completeness axiom, and the Archimedean property.

The field axioms

The field axioms define addition and multiplication, which form 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 $ℝ$ we can define the operations $+ : ℝ \times ℝ \to ℝ$ and $\cdot : ℝ \times ℝ \to ℝ$ . They satisfy the following properties:

Properties of the addition:
- Associative law of addition: For all real numbers $x, y, z$ we have $x + (y + z) = (x + y) + z$ .
- Commutative law of addition: For all real numbers $x$ and $y$ we have $x + y = y + x$ .
- Existence of a zero: There is at least one real number $0 \in ℝ$ , 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 $x, y, z$ we have $x \cdot (y \cdot z) = (x \cdot y) \cdot z$ .
- Commutative law of multiplication: For all real numbers $x$ and $y$ we have $x \cdot y = y \cdot x$ .
- Existence of a one: There is at least one real number $1 \in ℝ$ 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 $x, y, z$ we have $x \cdot (y + z) = x \cdot y + x \cdot z$ .

The ordering axioms

The ordering axioms describe a linear order of the real numbers. The real numbers are therefore objects that can be compared with each other. Of two different real numbers, one is larger than the other. 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 defines 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
$\forall x \in ℝ : 0 < x \dot{\lor} 0 = x \dot{\lor} 0 < - x$
Closure of positive numbers under addition: For all real numbers $a$ and $b$ , if $0 < a$ and $0 < b$ , then $0 < a + b$ is also true. In characters: $\forall a, b \in ℝ : 0 < a \land 0 < b ⟹ 0 < a + b$
Closure of positive numbers under multiplication: For all real numbers $a$ and $b$ , if $0 < a$ and $0 < b$ , then $0 < a b$ is also true. In characters: $\forall a, b \in ℝ : 0 < a \land 0 < b ⟹ 0 < a b$

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:

$x < y : ⟺ 0 < y - x$

So $x$ is smaller than $y$ exactly when 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:

$x \leq y : ⟺ x < y \lor x = y$
$x \geq y : ⟺ y < x \lor x = y$
$x > y : ⟺ y < x$

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 is that among 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 anymore. These gaps do not exist in the real numbers, because the completeness axiom denies the existence of such gaps. A nested sequence of nonempty closed bounded real intervals has at least one real number common to every interval.

A real number can be approximated by a nested sequence of closed bounded intervals, each contained in the previous one, whose lengths converge to 0:

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 nested sequence of nonempty closed bounded intervals $I_{1}$ , $I_{2}$ , $I_{3}$ , ... there exists a real number that lies in all the intervals. If their lengths converge to 0, that number is unique.

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 $x, 2 x, 3 x, 4 x, \dots$ ever grows beyond $y$ . So for all natural numbers $n$ there would be $n x < y$ .

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

Definition (The Archimedean axiom)

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

$\forall x > 0, y > 0 \exists n \in ℕ : n x \geq y$

Using the completeness axiom and the Archimedean property, 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, finite floating-point numbers used in computer calculations are rational^[2]. Such calculations usually feature errors, but their errors can usually be made sufficiently small.

Einzelnachweise

↑ For example, there exists the method of the Cauchy sequences and the method of the Dedekind cuts.
↑ See the Wikipedia article on floating-point numbers.

Complex numbers →

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[1] For example, there exists the method of the Cauchy sequences and the method of the Dedekind cuts.

[2] See the Wikipedia article on floating-point numbers.

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