I now begin the escapade into different sorts of numbers. This may seem rather a strange adventure to anyone who hasn’t listened to too many professional mathematicians talks, or anyone who is too far removed from high school education. I would guess that most people who have read this have some particular picture of number in mind – likely one of the kinds of number I will talk about in this series. And if each person thinks themselves to have the correct notion of number – each and every person is wrong. For there is not one correct notion of number in modern mathematics. There are several – it all depends on context. This will be explained as the series goes on.
The Historical Starting Point: Natural Numbers
What are natural numbers? If you know what whole numbers are, you already have the right start. If you don’t remember, rest assured; these are the most intuitive numbers for humans, because they are the numbers of counting. One, two, three, and so on- these are the natural numbers. Simple enough, isn’t it? So one would think.
And yet, as we shall see as we continue onward into ever deeper realms of number, that the natural numbers only occasionally live up to their moniker of ‘natural’. Sometimes they really are the most sensible of numbers to think about, but sometimes they are not. Consider the following quotation…
“God made the integers; all else is the work of man.” – Leopold Kronecker
This is rather strange. Integers are not natural numbers! In fact, if you happen to already know this, the natural numbers are just about half of the integers. That may seem rather strange – and this strangeness will be addressed later.
The main point that given such a famous quote from such a famous mathematician, it is surprising that I don’t begin with the integers as my object of study. If Kronecker, who is by far my superior in ability, then why not adopt what appears to be his starting point? The reason for this is historical. For a great many people prior to Kronecker’s life, the “natural numbers” were far more ‘natural’ than the integers that Kronecker considers to be fundamental. It is a matter of mathematical history that it took quite a long time for the idea of “negative numbers” – which make up half of the integers, to show up on the scene as genuine numbers. Even zero, which is an integer, but not negative, took some time to show up on the mathematical scene.
Given that both zero and negative numbers took a long time to show up in mathematics, why include zero in the very first discussion of types of number? That is certainly a valid question. I imagine that a professional historian of mathematics would likely introduce zero separately from other numbers – because historically, this is highly significant. Modern readers would likely be highly surprised by how long it took humans to come up with the idea of zero as a number. And yet, as a mathematician myself, I think it is convenient to include zero in my first post about the nature of numbers, despite its historical oddities. There is, in the development of integers, three stages – the positive integers, zero, and then the negative integers. Due to the convenience of the modern perspective, it is easy to wrap zero into the modern perspective. It is, in my view, a remarkable accomplishment of the human intellect that zero need not be introduced as a topic in and of itself.
Anyways, on to a discussion of the so-called natural numbers.
The Positive Whole Numbers
The positive whole numbers are the basic numbers we use to count. These are 1, 2, 3, 4, and so on. The positive whole numbers are built upon the idea of “one after another” – there is the number 1, then the next number, then the next, and so on. As we shall see, this is actually not a concept that number systems always have. Even more special to the positive whole numbers is that there is a first positive whole number – the number 1. Fundamentally, the fact that the words first and next both make sense essentially define what the positive whole numbers are. The first means we have a starting point, the next means we have a list, one item after another, running off to infinity.
With the positive whole numbers, there are two basic operations we can perform. We can add two positive whole numbers together to obtain a new positive whole number, and we can multiply two together to get a new positive whole number. In some cases, you can also subtract and divide, but this doesn’t always work. The reason is that, sometimes, the subtraction of two positive whole numbers cannot be a positive whole number, and whole numbers divided by whole numbers might not be whole numbers. The subtraction and the division would be examples of this problem.
The solution to these problems will come in the future, where we will expand our definition of number in order to solve these complications. But for now, as long as we live in the world of positive whole numbers, subtraction and division should be viewed with some degree of suspicion.
The Number Zero
Along with the positive integers, we have the number zero. Zero is not positive, but neither is it what we will eventually call ‘negative’. Zero is, so to speak, the number that represents nothing at all.
Now, the number zero has two very special properties that deserve a pause. The first is that , that is, that , no matter what the number is. This is part of what defines the number zero. The second – which turns out to be precisely because of the first – is that no matter what is. Perhaps you’ve never had this identity explained before, but the logic behind it is actually pretty easy to lay out. We start by noticing that So, if we multiply both sides by , then . We can then use the distributive law to see that . Subtracting from each side, we conclude that .
We also know that division by zero is forbidden in all mathematics – and it is the fact that that explains why division by zero is not allowed. Since division doesn’t really fit into a discussion of whole numbers in a nice way, we will leave the problem of dividing by zero later on when we focus more on division. The curious reader, however, could quite possibly find the solution for themselves from the equation .
A Note About Zero
To a modern reader, it might seem like zero comes along pretty naturally with the other whole numbers. This is not so. The discovery of the concept of zero is, in reality, one of the major shifting points in the history of mathematics and took a long time. To demonstrate how difficult it was to conceptualize the number zero, I think it will suffice here that when someone finally did come up with the idea of zero, it wasn’t even treated as a number yet. Zero was at first mainly a positional idea – this means zero was acceptable to use to tell apart 13 from 103 and 1003, but “0” didn’t have any meaning yet in and of itself. This took quite a long time – and we should pause and appreciate the conceptual difficulty of actually treating “nothing” like a number.
The numbers 0, 1, 2, 3, … are the beginning of the journey through the many kinds of numbers that mathematicians think about. They have the advantage of being very simple to understand, but with that advantage comes the disadvantage that we can’t do as much with them – we can’t even really understand subtraction yet because subtractions like do not land in the world of natural numbers, and division ends up having even worse problems. In order to solve these problems, we will have to expand our concept of number. As we do so, the numbers will become gradually more complicated, but we will gain the advantage of being able to do more and more with them. This is the dynamic that always exists within number systems – the “big” systems will be hard to use but very versatile, and the small ones are easy to use but very limited in their use. As this series continues, take time to notice this pattern.
Appendix: Peano’s Axioms of Arithmetic
Between roughly the time periods 1850 and 1950, the study of mathematical logic or the foundations of mathematics was in vogue. The goal of this study was to make extremely clear what exactly we mean by foundational concepts like numbers and groupings of numbers. Since the counting numbers are among the most basic concepts in mathematics, this was a natural place to begin the attempt the process of axiomatic systematization.
But what exactly is this process? I find it helpful to imagine the process of teaching mathematics – either to a person or to a computer – as helpful. When you do this sort of teaching, there are two basic sorts of things you have to do – you have to teach someone what sorts of things you are working with, and you have to give them rules for how to work. For something like multiplication, you might be told that multiplication is like ‘repeated addition’, that you do multiplication with numbers, and then you are given some rules of how to multiply numbers together.
This is something like what axioms are. Axioms are meant to be a completely non-ambiguous set of rules for working within some framework. Because we want these rules to be non-ambiguous, many axioms (once understood) seem so obvious as to not warrant saying out loud. But this is exactly the point – we want to check our intuition by making everything as simple as possible and building up from the foundation.
Giuseppe Peano came up with the first well-known set of axioms that are designed to define the numbers 0, 1, 2, 3, 4, and so on – the natural numbers. Here are the axioms of so-called Peano arithmetic (as taken from Wikipedia, because I like how they are presented there):
- 0 is a natural number.
- For every natural number , .
- For all natural numbers and , if , then .
- For all natural numbers , and , if and , then .
- For all and , if is a natural number and , then is also a natural number.
- For every natural number , is a natural number.
- For all natural numbers and , if and only if .
- For every natural number , is false.
These axioms may look a bit confusing. I’ll give a quick explanation of what they actually mean.
(1) This identifies the “first number” – which is zero.
(2-4) These axioms clarify the meaning of the equals sign . (2) is called the identity property, (3) is called the reflexive property, and (4) is called the transitive property.
(5) This makes clear that things that are equal are either both natural numbers or both not natural numbers.
(6) The symbol is often called the “successor”. You should think of as . So, what (6) says is that if is a natural number, then so is .
(7) The equations and are essentially the same equation.
(8) The value is never zero to zero if is a natural number (since negatives are not natural numbers).
Why axiomatize something as simple as natural numbers? Doesn’t this make things too complicated?
Well, yes and no. Professional mathematics went through a very important phase in which they focused for over 100 years on setting up the “foundations” of mathematics. One reason this is so important is because it helps us really understand which ideas about whole numbers are basic and which are actually built up out of more basic pieces.
Here is an example: it is pretty intuitively clear that . However, notice that this is not actually stated anywhere in the 8 axioms. This is because you don’t actually have to – you can build up the proof of that equation from these 8 axioms along with a method of proof called induction (see my post here on that method). It is interesting that something as intuitively simple as is not actually something we have to assume to be true in advance – we can build that truth out of still more simple truths. It is one of the many goals of modern mathematics to more fully understand what is and is not one of the ‘simple truths’ and what happens when we limit ourselves to smaller (or expand to a larger) set of ‘simple truths’.