Why do mathematicians use this symbol $\mathbb R$ to represent the real numbers? So, I'm wondering why mathematicians use the symbols like $\mathbb R$, $\mathbb Z$, etc... to represent the real and integers number for instance. I thought that's because these sets are a kind of special ones. The problem is I've already seen letters like $\mathbb K$ to represent a field in some books just to say an example. So, someone knows why we use these kind of symbols?
Thanks
 A: They used to be written with bold capital letters. If they did not invent it, the members of the Bourbaki group certainly popularized that convention.
Since drawing bold letters is rather hard with chalk or with a pen, the so called blackboard-bold variants which you mention are a natural replacement. Similarly, when you have a typewritter, double striking does not do much to get a bold-like letter, but you can overprint two with a slight space. In fact, this was done in TeX before the blackboard bold fonts became normally available (and is still done by some in fact!)
A: Things that are the most frequently used have their own special symbols.
If you just write plain $R$ nobody knows if you're talking about a general ring or the real numbers. When you have $\mathbb{R}$ singled out for the real numbers, then there is no confusion. That's all.
$\mathbb{Z}$ is less mysterious when someone tells you that the German word for "number" begins with a Z.
Using $\mathbb{K}$ or $\mathbb{F}$ for fields also happens from time to time, but this does not really fit in with the pattern of naming the "main number sets" with mathbb script.
A: Mathematical tradition, we can say that.
But! In mathematics you can denote anything by any symbol, supposed it is correctly introduced.
My analysis teacher also liked to use $\Bbb K$, because he only considered the cases $\Bbb K=\Bbb R$ and $\Bbb K=\Bbb C$.
A: See the section "Letters for the sets of rational and real numbers" at http://jeff560.tripod.com/nth.html
A: $\mathbb {R,Z}$ etc. are imitating the way we write bold R, Z on a blackboard (hence the name, blackboard bold). It can be argued that when TeXing (not actually writing on a blackboard), you should write $\mathbf {R,Z}$ instead (since that's what $\mathbb {R,Z}$ are meant to represent on a blackboard, in the first place!), and I, for one, do just that most of the time.
They are written in bold to make the name distinct, because $R,Z$ may be used to represent other, more locally defined objects, while bold letters are rarely used as local variables. As to why are the particular letters are used, the $\bf R$ is probably self-explanatory, while $\bf Z$ originates from German (Zahlen).
$\bf K$ as a dummy field name also comes from German (Körper), and in this case bold is likely used to imitate $\bf R,C$ and to indicate that it is "the" background field when it is fixed in the context, so it is, at least locally, as fundamental as $\bf R,C$ are (e.g. in linear algebra and algebraic geometry). It is less often used in that way when we consider many distinct fields ard rings, like in abstract algebra (where letters starting with $K$, and continuing with $L$, and sometimes $M,N$, are still often used to denote fields, but are rarely bolded).
