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I was looking at Wikipedia's wonderful table of the history of certain mathematical symbols, and there was a certain glaring omission: the use of ℝ to apply to the set of all real numbers. They have references for ℕ, ℤ, ℚ, and fascinatingly ℂ makes an appearance even though ℝ does not. Is there anyone with a source on who was the first to use it and when? I think it would be great to update Wiki with this crucial information (but also post it here).

Rivers McForge
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    Related: Origin / first use of $\mathbb{Z}$ (blackboard bold Z)? AND Who designed the mathematical blackboard bold letters of AMS, and when? AND History of blackboard bold? (sci.math thread 5-8 October 2003). – Dave L Renfro Aug 20 '23 at 19:58
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    Please document your preliminary research. Where have you looked, what did you find? So far I have not been able to find use of $\mathbb{R}$ prior to 1970. Rudin used $R$, Dieudonne used $\mathbf{R}$. – njuffa Aug 20 '23 at 21:03
  • I have done zero research on my own, other than to notice this mysterious Wikipedia gap. I certainly could research it myself. However, if someone else already knows the answer, I see no point in reinventing the wheel; and additionally, I imagine having the answer recorded publicly on this site would be of broader value than satiating my own private curiosity. – Rivers McForge Aug 21 '23 at 02:39
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    @RiversMcForge Quite a few questions on this stack require extensive searching to produce a good answer. Therefore it is considered good form to document one's preliminary research in the question to avoid duplication of effort. For what it is worth, "This question shows research effort" is one of the canonical reasons for questions to receive upvotes. – njuffa Aug 21 '23 at 04:03
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    This may strike some users as odd, but I post questions not for the purpose of farming upvotes, but because I actually would like to know the answer. My question explains all of the research effort that I did (such as it was) in the body; it is clear, and arguably useful to the purpose of this site; and it is not directly duplicative of any of the questions that I could find in my HSMStack search. The sci.math thread @DaveLRenfro linked suggests that Gunning can be credited with the first textbook usage, and I would be happy to accept that as an answer/let him farm me for upvotes :) – Rivers McForge Aug 23 '23 at 01:15
  • @RiversMcForge On the Stack Exchange sites it is perfectly acceptable to supply an answer to one's own question; one can credit commenters with pointing to relevant information. – njuffa Aug 24 '23 at 18:09

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This addresses a narrow form of the question: when did the "blackboard bold" version of letter forms of $\mathbb R$ first appear in printed works?

Before the AMS gave us the mathbb fonts in the 1980s, Springer Verlag was using its own version, for instance in the journal Mathematische Annalen. A spot check in old issues showed its use in the paper by Gramsch, B. Funktionalkalkül mehrerer Veränderlichen in lokalbeschränkten Algebren appearing in Mathematische Annalen, in 1967, vol. 174, pp.311–344. On p. 323 of that paper we see the equivalent of $\mathbb{C}$ and on p. 331 we see the equivalent of $\mathbb R$. (Thanks to user njuffa for supplying the location of a non-paywall library version of this paper.) The design of the letter-form looks, I think, as if it were inspired by the typewriter overstruck I R that one sees in photo-offset lecture notes of the era, such as on p.23 of the 1967 Dynamical Theories of Brownian Motion by Edward Nelson, in the "Mathematical Notes" series of Princeton University Press.

The adoption of this practice must be the result of a matching of tastes of both the mathematical authors and the production staffs of the publishers. Both Springer and Princeton University Press took pride in their typography, and hear-say makes it clear that the mathematical typists at PUP did so to an exceptional degree.

The wider sense of the question, when did distinctive typographic forms (bold face, sans-serif bold, etc) for the symbols for the integers, the real numbers, etc, first appear in print, is addressed in the comments to the OP, and an approximately correct answer is: with Bourbaki, decades earlier.

kimchi lover
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  • OK, it's volume 174. Bernhard Gramsch, "Funktionalkalkül mehrerer Veränderlichen in lokalbeschränkten Algebren." Mathematische Annalen, Vol. 174, No. 4, Dec. 1967, pp. 311-344. On p. 323 there is $\mathbb{C}$ and on p. 331 there is $\mathbb{R}$. – njuffa Aug 27 '23 at 00:33
  • @njuffa You are right, and I'm sorry to have inconvenienced you with the volume mistake. – kimchi lover Aug 27 '23 at 01:17