Which R to use to indicate the real numbers?

Question

In general, when people are talking about set of real numbers, they usually use one of the following notations:

$$\tag{1} x \in R.$$ $$\tag{2} x \in \textbf{R}.$$ $$\tag{3} x \in \mathbb{R}.$$ $$\tag{4} x \in \Re.$$

I understand that it might be convient to use one notation over the other (e.g. when the letter $R$ assigned a different meaning already, like a region of integration).

However I am curious whether there is an actual distinction between the notation, like a historical reason, or is it just preference which notation to use?

The last one seems to be used quite often as wel..

In Operations Research (4) is used a lot to specify the real numbers. — , Jan 07 '17 at 10:39
@Glasses, one would more conventionally use \Re instead of \mathfrak R tho. — J. M. ain't a mathematician, Jan 07 '17 at 10:40
@J.M.isn'tamathematician thanks didn't know that shortcut :) ! — , Jan 07 '17 at 10:42
Note that the third of these is by far the easiest to draw by hand (e.g. on a blackboard or paper), if one only double-bars the leftmost vertical line. — Patrick Stevens, Jan 07 '17 at 10:50
There is no right choice. It's about context, and as long as the reader is aware of the meaning of the symbol it's fine. In set theory you can find $\Bbb R$ used for denoting a forcing poset quite often. — Asaf Karagila, Jan 07 '17 at 11:01
Several related questions have been posted on this site: Blackboard bold, Bold, Fraktur, and Reserved Variable, Why do mathematicians use this symbol $\mathbb R$ to represent the real numbers? Who invented or used very first the double lined symbols $\mathbb{R},\mathbb{Q},\mathbb{N}$ etc and even on meta Blackboard bold. — Martin Sleziak, Jan 07 '17 at 12:04

Nosrati · Accepted Answer · 2017-01-07T11:09:48.777

Nr. (1) usually used for a region in analysis, and is a ring in algebra also,

Nr. (2) not conventional in new mathematical books for reals. Some people use Bold characters for vectors, like A, B, ...

Nr. (3) is the best character, preferred by many mathematicians. This character specific number sets like $\mathbb{R}$ for real numbers, $\mathbb{C}$ for complex numbers, $\mathbb{Q}$ for rationals, $\mathbb{Z}$ for integers, $\mathbb{N}$ for natural numbers, and so on. It is the best character for this purpose.

Nr. (4) designed for real part of a complex number in computer editors, but mathematicians don't used it vastly. They used bold $Re$ (Re) instead of it.

score 7 · Answer 2 · answered Jan 07 '17 at 10:37

7

it is $$\mathbb{R}$$ the right sign

answered Jan 07 '17 at 10:37

Dr. Sonnhard Graubner

97,058

goodlike answer (+1)! – tired Jan 07 '17 at 10:50

score 3 · Answer 3 · answered Jan 07 '17 at 10:42

3

I prefer $\mathbf{R}$ in print, leaving $\mathbb{R}$ for writing on paper or blackboard.

It's an aesthetic choice, also made by Serre and Knuth for instance:

https://en.wikipedia.org/wiki/Blackboard_bold

See a discussion at

https://groups.google.com/forum/#!msg/sage-devel/aM3wW0AwMoY/KwNSMcPxeLcJ

answered Jan 07 '17 at 10:42

lhf

221,500

3

In StackExchange one uses $\Bbb R$, as Dr. Sonnhard Graubner indicates (a competitive sick has put him a downvote). – Ataulfo Jan 07 '17 at 10:49
I use $\Bbb R$ here too, to avoid noise. – lhf Jan 07 '17 at 10:50

Which R to use to indicate the real numbers?

3 Answers3