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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?

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    $\begingroup$ As far as I know, it's just preference. Maybe you'll find comments about notation in this question, but I'm not sure. Check it, though $\endgroup$ – Ivo Terek Dec 17 '14 at 23:36
  • $\begingroup$ Probably because it's easier to type. **R** is easier than $\mathbb{R}$. $\endgroup$ – Billy Rubina Dec 17 '14 at 23:37
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    $\begingroup$ Somethimes $\mathbb{R}$ is referred to as "blackboard bolding", i.e., what happens when you try to write $\mathbf{R}$ with chalk. $\endgroup$ – Raclette Dec 17 '14 at 23:37
  • $\begingroup$ @Vÿska though it is the case that people use $\LaTeX{}$ for this, and the code would be \textbf{R} (and also $\Bbb{R}$ can be written as \Bbb{R}). $\endgroup$ – Conor O'Brien Dec 17 '14 at 23:38
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    $\begingroup$ It is user preference based on discipline and context. Notation nazis who are convinced there is a "right notation" exist but are usually politely ignored. Just do your best to be clear to your intended audience. $\endgroup$ – rschwieb Dec 18 '14 at 0:08
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As Robert Soupe says, either $\mathbf{R}$ or $\Bbb{R}$ is unlikely to confuse a reader without explanation.

Until the last decade or two of the 20th Century, $\mathbf{R}$ was common notation for the real numbers in print journals. As Raclette notes, $\Bbb{R}$ is called blackboard boldface because lecturers would put a double spine on the "R" to make it look boldface in chalk.

Once (La)TeX came into wide use among mathematical authors (by the mid-1990s), blackboard bold fonts were, predictably, created to duplicate the visual effect of blackboard writing. Young authors (and some older ones) seem to prefer blackboard bold to plain bold for the standard number systems (and/or they pick it up by osmosis on web sites such as this one), and the notation is now entrenched.

In terms of practicalities of LaTeX coding, the right answer is: Abstract your notation, including font choices, into the preamble, using semantic macros. (That is, make the code of your document body reflect mathematical meaning, not typographical appearance.)

For example, put commands such as \newcommand{\Number}[1]{\mathbf{#1}} \newcommand{\R}{\Number{R}} into the preamble of your LaTeX file, and use \R in the body. That way, you (or a journal editor) can change your notation to match house style in a trivial, predictable way.

You can even do a cheap version of this in your MSE posts by putting $\newcommand{\R}{\mathbf{R}}$ at the start of your post, and using \R in the body.

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    $\begingroup$ This is exactly what I teach in my LaTeX courses, with the only difference that I define \numberset instead of \Number! $\endgroup$ – egreg Dec 18 '14 at 14:04
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    $\begingroup$ On MSE, you need to be very careful in choosing the macro name. Macros on MSE are not localized within corresponding answer/question. If you redefine a macro on one answer, it can silently stop macro with same name working on question or other answers. You should make sure the macro name you choose will never be used by anyone on same page. $\endgroup$ – achille hui Dec 18 '14 at 14:21
  • $\begingroup$ @achillehui: Thank you for that information. Perhaps \Reals is a safer choice? I'm happy to edit my answer if it seems worthwhile to do so. $\endgroup$ – Andrew D. Hwang Dec 19 '14 at 13:50
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    $\begingroup$ @user86418 \Reals is definitely a much safer choice. I can't think of any sensible definition for it other than the real numbers $\mathbb{R}$. In any event, any long as the warning in my comment (or something equivalent) is around, your answer is perfectly fine. $\endgroup$ – achille hui Dec 19 '14 at 14:29
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The proper notation for the set of all real numbers is either $\mathbb{R}$ or $\textbf{R}$. It really comes down to your choice, and whichever you choose you can back with plenty of precedent. But most of the time people will understand what you mean without you having to explain it.

The problem with $R$ is that in the context of algebraic number theory it probably makes you think of an arbitrary ring or domain; in general people might be confused and think they skipped over your definition of a variable or a function.

If you're going to use $\mathfrak{R}$ to mean the reals you really need to say so at the beginning of the paper or book.

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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.

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    $\begingroup$ I've seen $\text{Re}(2+i)=2$ used before $\endgroup$ – Conor O'Brien Jul 16 '18 at 0:08

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