This is a first post in a series in which I’d like to discuss various different types of numbers that mathematicians study. This may sound strange – and that is fair. After all, aren’t all numbers just… numbers? Why would some numbers be so different from other numbers? What sense does that make? I’m honestly not entirely sure whether people already are aware of the kind of answer a mathematician would give to this question. Perhaps everything I say here will be common knowledge. But even if this is so, it is worth saying. Speaking a bit loosely, just because you know something doesn’t mean you know it. This is a lesson I’ve learned in my pursuit of my PhD again and again. I try to avoid ever thinking I really understand anything beyond absolute basics in mathematics any longer, because there is such incredible depth to every single area of mathematical study that, quite frankly, only years of constant work in the topics can lead to mastery. Before I begin discussing all the important differences between different types of numbers, I’d like to briefly lay out a conceptual framework for why we’d even ask questions like this as mathematicians.
In thinking about this topic, what immediately came to mind is an analogy with science that ought to be helpful. Naively, you might ask why there would have to be different areas of science devoted to differently sized objects? Why should size matter so much? If you pretend for a moment that you know nothing about modern science – you have no idea about atoms or galaxies or anything like that – why would you think there would be massive differences between the physics of the small and the physics of the large? It seems like there wouldn’t be much of a reason to think so. Yet, in modern physics, it doesn’t take you very long to realize how big a difference there is. In fact, the most developed theory of the large-scale universe, called general relativity, is in fundamental conflict with the most developed theory of the very small – called quantum mechanics. And this conflict isn’t just that they describe very different things – it is that they rely on some contradictory assumptions about the nature of reality. Clearly, then, things are in great conflict.
There is something like this in mathematics – although not contradictory, you get extremely different pictures when you look at different number systems. That is why mathematicians care so much about number systems – the picture always changes when you change number system.
There are perhaps three types of numbers that are useful for briefly illustrating this idea (each of which I will discuss in greater detail in the future). These would be whole numbers, fractional numbers, and real numbers. Whole numbers are numbers like -7, 3, and 0. Fractional numbers include the whole numbers, but include new numbers like and . The real numbers include fractions, but add onto that even more numbers like or . (By the way, if you are wondering why I say these are not fractions, that is the right question to ask. This is not obvious and will be discussed eventually in this series.)
Let me ask one question to begin with – in each of these three systems, are there “gaps” between them? The answer for the whole numbers is yes – there is no whole number between 2 and 3. For the other two number systems, the answer is no. You can, for example, take the average of two fractions and the new number will be a different fraction. In fact, you can continue this process of finding new fractions over and over again. And yet, as shall be shown in later posts, not all numbers been 2 and 3, say, are fractions. Some of them (in fact, most of them!) are not fractions. Somehow, the entire collection of real numbers is much more smoothed out than the collection of all the fractions.
When you work out the mathematical consequences of these differences, it really shows. The properties of these new sets of numbers are completely different. As you enter into larger and larger realms of numbers, not only do the properties of the numbers shift, but you don’t even get to ask the same questions any more, because the changes when you move from one number system to the next an be so drastic that some of the old questions in smaller systems of numbers no longer make a lot of sense. For instance, with whole numbers questions about factorizing whole numbers can be very difficult, but that same question has no significant meaning for fractions or real numbers. If you want to talk about smoothness, then the real numbers is the place to be, the set of fractions is a nice middle ground, and the whole numbers know nothing at all about smoothness.
So, in case there is any wonder, this is why we talk about different number systems. And as we learn to ask different types of questions about numbers, we can come up with new types of numbers – the complex numbers (which are an extended version of the real numbers) will be a key example of this idea. There is a very important property of numbers called algebraic closure that comes up when you study polynomial equations that is not satisfied by the real numbers or smaller sets of numbers, but is true for these complex numbers. I anticipate also going through additional systems of numbers like algebraic numbers, transcendental numbers, and even the mysterious quaternions. As we grow in understanding of both the similarities and differences between different number systems, we can grow in understanding of mathematics itself. This is, then, enough of a reason to ask about number systems.
I will begin with natural numbers, then whole numbers, then fractions and real numbers, finally to complex numbers, and then beyond. This is the most natural progression. And I hope it is a fruitful one.