4.1.1 Introduction

No doubt, most readers of this book think of real numbers as values on a number line, which has long been accepted by scientists and engineers as a model for measurements of length, mass, and time.

Although there is nothing wrong with this intuitive interpretation, it is the goal of this section to show how the real numbers can be logically created from more primitive number systems like the natural numbers, as well as introducing aspects of the real numbers decimal expansions, the least upper bound property, types of real numbers like rational, irrational, algebraic, and transcendental, and completeness properties.

Without going into the history of how numbers went from 1, 2, 3, … to the real numbers, there are two fundamental approaches to how to define the real numbers. First, we can state axioms that we believe characterize our interpretation of the real numbers. This is the here they are, the real numbers. This approach is called the synthetic approach, whereby axioms hopefully embody what we believe a “continuum” should be.

On the other hand, we can “construct” the real numbers, much like a carpenter builds a house. In this approach, we begin with what our distant ancestors gave us, the natural numbers 1, 2, 3, …. We then take these numbers and doing some “mathematical carpentry” replacing hammers and nails with a little set theory and predicate logic, to logically construct, step‐by‐step, the real numbers. This is the approach taken in this section, starting with the natural numbers we construct the integers, then the rational numbers, all the way to real numbers. The synthetic axiomatic approach of here they are will be left to the next section.

4.1.2 The Building of the Real Numbers

The construction of the real numbers begins with the simplest numbers, the natural numbers

then, by a series of steps we construct the integers

followed by the rational numbers

and finally, the real numbers ℝ.

4.1.3 Construction of the Integers: ℕ → ℤ

The way we construct the integers 0, ± 1, ± 2, … from the natural numbers 1, 2, 3, … is to define integers as pairs of nonnegative integers, where we interpret each pair (m, n) as the difference m − n. Thus, the pair (2, 5) of natural numbers is defined as −3, the pair (7, 1) as 6, and the pair (4, 4) as zero. If we then define addition, subtraction, and multiplication of these pairs (m, n) consistent with the arithmetic of the natural numbers, we have successfully defined the integers in terms of the natural numbers.

To carry out this program, we begin with the Cartesian product

of pairs of natural numbers, called grid points, which are illustrated in Figure 4.1.

We now define an equivalence relation “≡” between pairs of natural numbers by

This relation can easily be verified to be an equivalence relation and its verification is left to the reader. See Problem 1. We saw in Section 3.3 that equivalence relations on a given set partitions the set into disjoint equivalence classes. In this case, the equivalence classes consist of grid points on 45° lines as drawn in Figure 4.1. A few of these equivalence classes are listed in Table 4.1, designated by which shows a representative member of each equivalence class. Each one of these equivalence classes will define an integer. For example, the equivalence class defines zero, defines −1, defines +1.

Equivalence class	Integer definition
	−2
	−1
	0
	1
	2

We now define the integers ℤ as the collection of all these equivalences classes. In other words

which we write as

We can also define the nonnegative and negative integers as

We now must define addition, subtraction, and multiplication for the integers, consistent with the rules of the natural numbers. We define for p, q, r, s ∈ ℕ addition ⊕, subtraction ⊖, and multiplication ⊗ of integers (represented by pairs of natural numbers) by:

• Addition:
• Subtraction:
• Multiplication:

For example

• Addition:
• Subtraction:
• Multiplication:

When p > r and q > s, the pairs (p, r) and (q, s) correspond to p − q > 0 and r − s > 0, hence, the above definition of addition, subtraction, and multiplication of pairs of natural numbers reduces to identities of the natural numbers.

4.1.4 Construction of the Rationals: ℤ → ℚ

We now move up the number chain and construct the rational numbers ℚ from the integers ℤ using a strategy similar to what we did in constructing the integers as pairs of natural numbers.

To carry this out, define an equivalence relation on ℤ × (ℤ − {0}) by

where (m, n) and (m′, n′) are pairs of integers and we “interpret” them as m/n and m′/n′, respectively. This equivalence relation partitions the grid points in ℤ × (ℤ − {0}) into distinct equivalence classes as drawn in Figure 4.2. Observing the following equivalences

we see that the equivalence classes consist of the grid points on straight‐lines passing through the origin, where the equivalence class with representative (m, n) is assigned the rational number m/n. For example

A few equivalence classes with an arbitrary designated representative are shown in Table 4.2.

Equivalence class	Rational correspondence
	−1
	0

We now define the rational number p/q as

We now define the arithmetic operations on the newly formed rational numbers.

• Addition:
• Subtraction:
• Multiplication:

For example

• Addition:
• Subtraction:

4.1.5 How to Define Real Numbers

We now come to a fork in the road. There are several ways to define the real numbers and each has its merits and demerits. On the one hand, we could define real numbers in the way they were described to you in grade school, as decimal expansions. This is nice since all numbers, rational and irrational are more or less treated in the same way, and we normally perform arithmetic when numbers are written in this form, like 2.35 + 4.91. However, on the negative side, when real numbers are represented by expressions of the form

where the as and bs are decimal digits, it is the dot, dot, dot at the end of the expansion that causes problems theoretically. Another problem is that certain real numbers have more than one decimal expansion, like

Like we said, there is that nasty dot, dot, dot again.

Still another demerit for the decimal expansion interpretation of real numbers is that they do not relate visually to points on a continuous number line, which is how most people like to think of real numbers.

However, there are two more approaches for defining the real numbers, one due to Cantor and the other due to his close friend Richard Dedekind. Both of these approaches are based on defining the real numbers in terms of the rational numbers, much like we defined the rational numbers in terms of the integers. Cantor's approach defines real numbers limits of sequences of rational numbers, like in the definition of the irrational real number π being the limit of the sequence

Although this approach has intuitive appeal, it demands that the reader develop a background in sequences, convergence, null sequences, and other concepts from real analysis, not introduced in this book. Hence, we follow the second approach, that due to Richard Dedekind.

Although rational numbers have many desirable properties, they have from the perspective of calculus, the undesirable property that they have “gaps” like the gaps at and π. In fact, there are an infinite number of gaps, and as we have seen in Section 2.5, an uncountable number of gaps. The idea is to “fill in” these gaps, and arriving at a new number system called the real numbers.

4.1.6 How Dedekind Cuts Define the Real Numbers

Dedekind's 1872 idea appeals to our intuitive grasp of the rational numbers aligned on a line. His idea was to partition the rational numbers into two disjoint sets L  and  U satisfying the following two conditions.

The above definition of a Dedekind cut does not uniquely define a partition of the rational numbers, but three different partitions called Dedekind cuts. They are as follows.

Each Dedekind cut of Type 1 or Type 2 corresponds to a rational number, whereas a Dedekind cut of Type 3 correspond to a number that is not rational. These are numbers that fill all the gaps between rational numbers. These are called irrational numbers. This leads us to the Dedekind cut definition of the real numbers.

4.1.7 Arithmetic of the Real Numbers

Now that we have defined the real numbers, the next job is to Dedekind cuts to define addition, subtraction, multiplication, and division of the real numbers (as well as defining order relations like < and ≤) consistent with those of the rational numbers. Although we can define arithmetic operations of real numbers in this way, they are rather complicated, and we will not include them here. If we had to carry out even the easiest arithmetic, like adding using the Dedekind cut definition, it would a difficult task. As we all know, the way we do arithmetic involving irrational numbers, we replace them by a decimal digits, and find the approximate value . Of course, we cannot be 100% accurate like we are when we restrict arithmetic to the rational numbers, like

but that is the price of enlarging our number system to the real numbers.