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I would like to know why the real numbers are called “the real numbers.” I would also like to know the meaning of “real” in the phrase “real number.”

Further questions and clarifications:

  • I’d like to know both why the phrase is used now and why it was initially introduced (should these two differ), though I’m most interested in the latter question.
  • When did the phrase “real numbers” come to denote the reals? In what language did this terminology first appear? Who first used it? Did it quickly see widespread acceptance, or was there pushback?
  • The most likely explanation, to my ears, is that “real” is meant to mark a contrast with “imaginary”: real numbers can be given physical interpretations (by representing physical magnitudes such as charge, force, etc.; functions on reals could represent the dynamics of physical systems) whereas imaginary numbers cannot, hence the terminology. But this explanation implies that the phrase “real numbers” only came to denote real numbers after the introduction of imaginary numbers. Perhaps the historical evidence suggests otherwise? This explanation also suggests that there are no physical interpretations of imaginary numbers—this sounds doubtful to me, though perhaps it would have seemed obvious back when the terms were introduced.
  • Note that many other languages use a synonym of “real” to denote the real numbers (e.g. French: “nombre réel," German: “reelle Zahl," Chinese: “实数”).

The rationale for “rational number” is pretty clear (though I don’t actually know the history). “Natural number” is less transparent. Any historical/etymological insights on that one welcome too.

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    $\begingroup$ Possibly better suited for hsm.stackexchange.com. Also this website: jeff560.tripod.com/r.html (Earliest Known Uses of Some of the Words of Mathematics) claims that "real number" was introduced by Descartes, in French of course. An the entry under "imaginary" (jeff560.tripod.com/i.html) has much more info. $\endgroup$
    – Sam Hopkins
    Commented Apr 9 at 19:45
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    $\begingroup$ Meta: I don't understand the downvote(s) on this question. The origin of the term “real number” is a perfectly good and sensible question on the history of mathematics / mathematical terminology, and while it may indeed be more appropriate for HSM StackExchange (which could be a reason to close it here), downvoting should be used in lieu of voting to close. $\endgroup$
    – Gro-Tsen
    Commented Apr 9 at 20:30
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    $\begingroup$ Almost certainly, "real" numbers were named after the word "imaginary" was used to describe the square roots of negative real numbers, in order to distinguish reals from imaginaries. $\endgroup$
    – Daniel Asimov
    Commented Apr 9 at 20:33
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    $\begingroup$ Before the distinction imaginary/real numbers arose, real numbers were referred to by Stevin as "continuous" numbers, in distinction with rationals (source) $\endgroup$
    – Carlo Beenakker
    Commented Apr 9 at 21:02
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    $\begingroup$ What you call "synonym" would rather be "literal translation". Also note that a priori the word was not translated from English, but to English. $\endgroup$
    – YCor
    Commented Apr 10 at 7:25

2 Answers 2

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I believe the distinction between real and imaginary number was introduced first by Descartes; e.g.,

Au reste, tant les vraies racines que les fausses ne sont pas toujours réelles, mais quelquefois seulement imaginaires

(see here, page 47).

The idea of arbitrary number (rational or otherwise) represented by an unending decimal was popularized even earlier by Stevin but I am not aware of him referring to such numbers as "real". Some details on Stevin and real numbers can be found in

Katz, K.; Katz, M. "Stevin numbers and reality." Foundations of Science 17 (2012), no. 2, 109-123. https://doi.org/10.1007/s10699-011-9228-9 and https://arxiv.org/abs/1107.3688 and https://www.ams.org/mathscinet-getitem?mr=2935194

Cauchy was, of course, aware of this approach to the real numbers. Therefore in my view the widespread idea that his proofs of such results as the intermediate value theorem are somehow incomplete or lacking, is exaggerated.

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    $\begingroup$ Only the first sentence of this answer has anything to do with the question asked (and this fact was also mentioned in the first comment, with references). $\endgroup$
    – Sam Hopkins
    Commented Apr 10 at 12:12
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    $\begingroup$ The fact that real numbers were known before Descartes (though not under that name) is significant. $\endgroup$
    – Mikhail Katz
    Commented Apr 10 at 12:32
  • $\begingroup$ For anyone else on here as ignorant of French as I am, here is the translation that google gave me: "Moreover, both the true roots and the false ones are not always real, but sometimes only imaginary." $\endgroup$
    – Will Brian
    Commented Apr 10 at 12:57
  • $\begingroup$ @WillBrian, as explained there on page 41, "false" means "negative". $\endgroup$
    – Mikhail Katz
    Commented Apr 10 at 13:02
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    $\begingroup$ @WillBrian, the negatives were not considered "genuine numbers" until quite late; I believe in Cauchy one still finds discussions of signs that make it appear that the negatives are "false" somehow :-) $\endgroup$
    – Mikhail Katz
    Commented Apr 10 at 13:19
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An excellent explanation is given in the short paper of

It was published in the Math Intelligencer, which is available by subscription only, but here is a youtube movie where the author just reads this paper:

https://www.youtube.com/watch?v=VUdFdlQNfpg

[[Added by David Roberts: This article comes with a disclaimer

The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them

]]

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    $\begingroup$ Any chance you could summarise the main points of the explanation in your Answer? $\endgroup$
    – Paul Taylor
    Commented Apr 10 at 12:57
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    $\begingroup$ Wildberger attributes complex numbers to "The Europeans" without any dates. This is not a very informative way of writing history. $\endgroup$
    – Mikhail Katz
    Commented Apr 10 at 13:07
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    $\begingroup$ The issue here is not complex numbers but real numbers, and he doesn't tell us who invented those. Your post does not answer the question. $\endgroup$
    – Mikhail Katz
    Commented Apr 10 at 13:11
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    $\begingroup$ The summary seems to be "satire" and I've heard better. I gave up after 10 minutes out of 21. I'm glad I didn't pay the subscription. Hardly an Answer to a reasonable Question. $\endgroup$
    – Paul Taylor
    Commented Apr 10 at 13:18
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    $\begingroup$ @MikhailKatz: no thanks. your Answer is much more to the point. $\endgroup$
    – Paul Taylor
    Commented Apr 10 at 13:23

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