This time I'm just going to complain about the term, "the real numbers". There are two problems with this term. First, the real numbers aren't "real" in any meaningful sense. Second, the fact that some numbers are called "real" implies that all other numbers are somehow "unreal", when they are not any more unreal than the real numbers themselves.
The real numbers aren't real. We have lots of experience with the real world. That's where we live, after all. We spend hours there every day. We use numbers all the time, whether we're spending money, or measuring something, or checking the time, or whatever else we might be doing. The numbers we use are never real numbers.
We always round off numbers. Even if we're being precise about the time, we only give it to the nearest second, not the nearest hundredth. Even careful physics experiments which measure the speed of light or the mass of the Higgs boson or whatever have to always state the uncertainty in the results. The real numbers imply that numbers are never rounded off, and beyond the fact that we don't do that, we can't do that.
You could make the claim that the natural numbers exist in some way in the real world. After all, we count things all the time, and the natural numbers are the counting numbers. Even then, there are limits. We may care about an exact count for small numbers but once the numbers get big enough, we round them off. A chemist can tell you that there are 6.02×1023 carbon atoms in 12 g of graphite, but probably can't tell you all of the next 20 digits of that number, and even if they could, they wouldn't bother most of the time.
And that's before we get into the fact that there are more natural numbers than there are things in the universe to count. I'd like to post sometime about really big numbers, and the fact that it's easy to come up with numbers that are so big that they are both incomprehensible and useless. It's a fun topic, but in the real world we don't talk about incomprehensibly big numbers on that scale.
The point, simply, is that thinking of even the natural numbers as real world numbers is misleading, and our experience with real world numbers can lead us to sloppy or incorrect thinking about numbers in a purely mathematical context.
At the same time, the fact that some numbers are "real" implies that other numbers are "unreal". Since my next post will discuss numbers which are not real numbers, I feel a need to preemptively stick up for them. Not real should not imply not useful. Although I won't get into the applications of complex numbers, the applications do exist. Don't think that because the complex numbers aren't real numbers, you can just ignore them.
In short, the real numbers is a stupid term for stupid people and we should change it. So there.