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I am really curious about and struggling with finding when the word "Real number" began to be used as an official terminology to refer to both rational and irrational numbers.

In Wiki, it says the adjective "Real" was introduced by Descartes, and Leo Corry says, after Descartes's introduction

"terms like "imaginary" and "real" were sweepingly adopted by mathematicians of the following generations"

However, I cannot find any pieces of literature where "Real number" refers to both rational and irrational numbers before Dedekind's time. I think they just used "numbers", not "real numbers." So, is Dedekind the one who officially used that term when he gave a rigorous definition of the real number system?

I am also aware that even before Dedekind gave the definition, many mathematicians, such as Newton and Euler, didn't differentiate irrationals from rationals when they were using them. I am really curious when this specific term "Real numbers" started to be used as an official title to refer to rationals and irrationals before Dedekind.

I'd really appreciate it if you'd share your knowledge and information!

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  • $\begingroup$ youtube.com/watch?v=VUdFdlQNfpg $\endgroup$ Jul 5 at 13:48
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    $\begingroup$ As you can see from the complete answer below, the real issue is not real vs imaginary, but the fact that still at Cauchy time "numbers" are mainly integers vs magnitudes (as per Euclid). $\endgroup$ Jul 6 at 9:25
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It is hard to say what "official" means exactly, it is not like there was a bureau of terminological standards. But "real numbers", "real values" and "real quantities" were certainly widely used long before Dedekind, and the erasure of the distinction between rationals and irrationals where it is not relevant is even older, it goes back to Stevin (1585), see How were irrational numbers accepted by mathematicians? It was supported by Stevin's normalization of arbitrary decimals and the idea of generation of "quantities" by continuous motion, upon which Newton, Euler and Cauchy founded calculus. Both served as an early substitute for the arithmetical constructions of real numbers a la Bolzano, Meray, Dedekind, Cantor, etc.

Descartes did not employ "real numbers" as a term, but he did start the tradition of inserting the adjective to distinguish them from imaginary numbers. Here is the relevant quote from La Geometrie (1637):

"For the rest, neither the false nor the true roots are always real, sometimes they are only imaginary, that is to say one may imagine as many as I said in each equation, but sometimes there exists no quantity corresponding to those one imagines."

Since by "roots" he means intersections of arbitrary polynomial curves with the axis he clearly did not distinguish rational from irrational. Newton, Leibniz, Euler and others continued in the same vein, with "real" becoming a term. Leibniz's unpublished 1675 paper (see A Contribution of Leibniz to the History of Complex Numbers by McClennon) discusses the cubic as follows:

"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have three real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible"; "How can it be that a real quantity, a root of the proposed equation, is expressed by the intervention of the imaginary?"

Euler's Introductio in analysin infinitorum (1748) uses "real numbers", "real values" and so on routinely. In chapter I (Bruce's translation) we read, for example:

"Thus, even if this function $\sqrt{(9-zz)}$ at no time is able to receive real numbers substituted in place of $z$ greater than the number three, yet imaginary values of $z$ may be attributed"; "The three values of $Z$ for that one corresponding value of $z$ either are all real or one will be real, while the two remaining are imaginary."

Considering the general influence of Euler's textbooks on the subsequent notation and terminology his use can be said to make it "official". By the way, when Euler uses "numbers" without qualification he means integers, not real numbers, see e.g. chapter XVI.

Cauchy's Cours d'Analyse (1821), another influential textbook, adds "real" in the same context, but Cauchy typically talks about (variable) "quantities" or "functions" and their values, not "numbers". In section 7.1 (Bradley-Sandifer's translation) he writes:

"In general, we call an imaginary expression any symbolic expression of the form $\alpha+\beta\sqrt{-1}$, where $\alpha$ and $\beta$ denote real quantities"; "Given this, any imaginary equation is just the symbolic representation of two equations involving real quantities".

In Note III he even uses "real" without direct relation to the imaginary, in the intermediate value theorem:

"If the two quantities $f(x_0)$ and $f(X)$ have opposite signs we can satisfy the equation $f(x)=0$ with one or several real values of $x$ contained between $x_0$ and $X$".

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  • $\begingroup$ Super appreciate your quotes and the references you introduced! Can you also let me know any of Newton's remarks where he said "real" number/quantity/value? I'd really appreciate it! $\endgroup$ Jul 6 at 22:56
  • $\begingroup$ Also, I am wondering if Steven presented irrational numbers in the form of infinite decimal expressions. I am aware that he treated rational numbers and irrationals equally, but as far as I know, what he did was to express rational numbers as decimal fractions. I am not sure if I am correct. $\endgroup$ Jul 6 at 22:59
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    $\begingroup$ @withgrace1040 On Stevin see Katz and Katz, Stevin numbers and reality. Newton did not have much use for the adjective because he did not deal with complex numbers at any length. Even in The method of fluxions, where he mentions "complex quantities" in passing, there are no "real quantities". Of the method of first and last ratios of quantities in Principia does not have them either. $\endgroup$
    – Conifold
    Jul 7 at 23:18

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