Describing real numbers in terms of natural numbers

Introduction

One can describe real numbers using Euclidian geometry as follows. Take a line. Take a point on the line, that point, the origin, corresponds to \(0\). Take another point, that point corresponds to \(1\) and defines a unit of measurement and direction. Then \(\dfrac{1}{2}\) corresponds to a point halfway between \(0\) and \(1\), and \(\sqrt{2}\) corresponds to a point at about \(1.4142...\).

While the geometric approach works for practical applications, it raises a chicken and egg question. What comes first? Numbers or geometry?

Dedekind, in the 19th century, came with an ingenious device to think of the real numbers in terms of the natural numbers, which contributed to the arithmetization of analysis.

Things and numbers

Numbers are a creation of the human mind, based on the ability to identify separate things, give them labels, recognise when we face the same thing again or a different thing, and relate things to other things.

Concretely, when faced with three sheep and three trees, we can recognise each sheep and tree as a distinct thing, we can tether each sheep to a tree forming pairs and conclude that each sheep and tree belongs to exactly one such pairs. Should one sheep free itself we can still conclude that it’s the same sheep, and it’s not one of the others still tethered to the trees. We can come with the word “three” as a label for situations when we can do this very specific pairing between distinct things.

Natural numbers

We start with zero and a relation of successor (or next, or follows). For example 1 is the successor of 0, 2 is the successor of 1 (also 2 is the successor of the successor of 0) and so on. This requires from the human mind the idea of going on forever: the idea of infinity.

A acute observer would question: how do we know that it does not loop at some point? This can be addressed with rules such:

zero is not the successor of another natural number.
if the successor of two numbers are the same, then those numbers are the same.

The first rule ensures that the loop does not happen at zero. It also guards against the situation that we’ll see with integers where negative numbers precede zero. The second rule ensures that there is no loop later by ensuring that (other than zero) a natural number is the successor of just another natural number.

Then the acute observer can say: hey, but in addition to the chain of numbers generated from zero, there could be other chains of loops. This can be addressed by saying that for the natural numbers they all have to belong to the chain of generated by the successor from zero, there are no other natural numbers.

And with this our detailed definition of the natural numbers is complete. We can then define:

addition as repeated application of the successor
multiplication as the repeated application of the addition
exponentiation as the repeated application of the multiplication

And these operations have the nice property that applying them to natural numbers results in a natural number.

However the counterpart operations (subtraction, division, root) are more complicated.

Integers

Take subtraction for example. When faced with \(3 - 5\) we could say it’s impossible, because if there are three people in a bus, you can’t have five get off the bus at the next stop. But what if we wanted to be able to do such operations to handle the money of an account getting into debit. Then we need negative numbers.

Formally we can introduce a new system of numbers that consists of a pair of sign and a natural number. For the sign we need thee things/values: positive, negative and none. Only zero can and must have no sign (“none” value for sign).

To the natural number 3, we can put in correspondence the integer 3, but they are distinct entities, the integers are pairs of a sign and a natural number, in this particular case for the integer 3 we talk of a pair where the sign is positive and the natural number in the pair is 3.

Rational numbers

This leads to division. When faced with trying to divide 1 into 2 parts we could either say it’s impossible, as is the case of the child of divorcing parents, but we might want to be able to do so when sharing a cake.

Formally we can introduce a new system of numbers that consist of triplets of a sign and two natural numbers (the numerator and denominator). We still have the restriction that the denominator cannot be zero, but other than that now also divisions are possible between arbitrary numbers.

When we write a fraction like \(\dfrac{1}{2}\), we think of a triplet where the sign is positive, the first natural number of the triplet is 1, and the second natural number of the triplet is 2.

Real numbers

This leads to the counterpart of exponentiation. Take square root for example.

We know since the ancient Greeks that any rational numbers have a common measure. For example for \(\dfrac{1}{2}\) and \(\dfrac{2}{3}\) the common measure is \(\dfrac{1}{6}\). But also from the ancient Greeks we know that the side of the square and it’s diagonal do not have a common measure. If the side is of length \(1\), the diagonal is \(\sqrt{2}\) which is not a rational number, it’s irrational.

To avoid falling back to geometrical descriptions, Dedekind’s idea is to define the real numbers (rational and irrational numbers) as follows.

A real number is a pair of infinite sets of rational numbers such that all the rational numbers from the first set are smaller than the rational numbers from the second set and together the two sets comprise all the rational numbers. Thus the set of the rational numbers is cut into two to define a real number: the Dedekind cut.

Concretely for \(\dfrac{1}{2}\) the first set contains all the rationals smaller than \(\dfrac{1}{2}\), the second set contains all the rationals greater than \(\dfrac{1}{2}\). As for \(\dfrac{1}{2}\) we can choose to put it in either set (where it acts as either the greatest or the smallest number in the set).

For \(\sqrt{2}\) the first set contains all the negative rationals and all positive rationals of which square is smaller than \(2\), while the second set contains all the positive rationals of which square is greater than \(2\). But neither of the two sets has a greatest or smallest number in the set.

While in the case of \(\dfrac{1}{2}\) we put a rational in correspondence with a real number, the rational is a triplet of sign and two natural numbers, while the real number is pair of infinite sets of rational numbers.

Complex numbers

This leads to the question of taking square root of negative numbers. Again we can take the view that it’s not possible or we define new kinds of numbers where the operation is possible.

Enter the complex numbers that are defined as pairs of real numbers. The first number is called the real part, the second number is the imaginary part. Both components of a complex number are pairs of infinite sets of rational numbers, which themselves are triplets of sign and two natural numbers.

Conclusion

This approach, particularly the ingenuous definition of real numbers by Dedekind cuts achieve definitions that do not rely on geometry.

But while we started on a solid footing observing the ability of the human mind to identify different things, it then introduced as a base for numbers ideas of sets and in particular treating the infinite as an actual thing as is the case for the definition of a real number as a pair of infinite sets.

Treating the infinite as an actual (e.g. each real number involves a set of infinite rationals) as opposed to a potential (e.g. you can always have another natural number) is a distinction and controversy that goes at least as far back as Aristotle.

References

R. Dedekind: Continuity and Irrational Numbers (1872)