TITLE: Blackboard bold, Bold, Fraktur, and Reserved Variable.
QUESTION [9 upvotes]: There seems to be an arbitrary choice of how one would want to represent the set of all real numbers. Most commonly, I've seen $\Bbb{R}$, followed by $\textbf{R}$, then by a reserved variable $R$, then, only once or twice, by $\mathfrak{R}$. I remember reading somewhere (regrettably, the knowledge of "where" has escaped me), from a supposed guy big in math, that the use of $\Bbb{R}$ is discouraged, and one should use $\textbf{R}$ instead.
What I would like to know, who is this guy, and what is the "proper" notation for the set of all real numbers? Is it just user preference, or is there some sort of unstated standard that mathematicians have for the set of all real numbers?

REPLY [3 votes]: Either $\mathbb R$ or $\textbf R$ is good. The choice depends on your answer to this question: do you like blackboard bold from your aesthetic perspective?
Don't use $R$ because it's commonly used to mean a particular ring at hand, e.g., if you're discussing primes in $\mathbb Z[\sqrt{15}]$ (or $\textbf Z[\sqrt{15}]$), then you say $R$ is that ring.
I wish someone had mentioned back in 2014 that $\mathfrak R$ looks a lot like $\Re$, if not exactly the same. The latter is a function that gives you the real part of a complex number, e.g., $\Re(2 + i) = 2$. Correspondingly $\Im(2 + i) = 1$ (or $i$?), so it's probably a good idea not to use either of those to mean other things.