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