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1860 Manual of Algebra describes a method which is now taught in upper secondary schools worldwide:

To rationalize the denominators of fractions which consist of binomial quadratic surds, use the following RULE: Multiply the numerator and denominator by a binomial surd, conjugate in form to that which appears in the denominator.

I wasn't able to find an attestation of the term conjugate in this context earlier (there are a lot of conjugate diameters noising the search results though, and I could miss something), but plenty of people seem to be using multiplication by conjugate ante litteram, cf. these 1813 and even 1702 examples (the notation in the latter one is different from the modern one, but it seems that the technique is the same).

Unfortunately prior to the turn of the 18th entury the math literature was written and published mainly in Latin, which I don't know. However, a single 1673 source on a related topic which doesn't appear to explicitly state this rule refers to 10th Book of Euclid's Elements, is the method actually as early as that? I tried to read Euclid as well as two modern retellings, but didn't find a formulation of this technique: the classic seems to describe related properties of irrational expressions rather than instructing how to solve problems here.

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  • $\begingroup$ I don't understand the 1860 text. I googled for the "multiplying by conjugates" and yes it is okay. I think it can be taught to people about 14-18 years, but I think in most countries of the world (incl. usa) it mostly does not happen. $\endgroup$ – peterh - Reinstate Monica Dec 30 '20 at 23:36
  • $\begingroup$ I also think that Euler (inventor of $e^{i\pi}=-1$) surely knew it. Maybe already Tartaglia did it. Knowing complexes it is trivial for a mathematician, imho. $\endgroup$ – peterh - Reinstate Monica Dec 30 '20 at 23:58
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    $\begingroup$ @peterh - Reinstate Monica: I think it can be taught to people about 14-18 years --- I don't understand what led to your comment, as rationalizing denominators is a very standard topic (although it's been slowly moving out of the curriculum in many countries for the past few decades with the introduction of calculators) and the underlying idea appears all over the place in mathematics (see here for several examples). $\endgroup$ – Dave L Renfro Dec 31 '20 at 11:43
  • $\begingroup$ @DaveLRenfro It requires also the knowledge of the complex numbers and how can we calculate with them. It is more useful if it is based on a real problem (like the 3rd grade equations). $\endgroup$ – peterh - Reinstate Monica Dec 31 '20 at 14:46
  • $\begingroup$ @peterh - Reinstate Monica: Are we talking about the same thing? Complex numbers are not required to either carry out or to conceptually understand rationalization based on difference of squares factorization. And I don't know what you mean by "3rd grade equations" ... cubic equations? Also, I'm U.S. based, and "3rd grade" as a school reference is at least 5 years before even linear equations typically appear in the curriculum, and in any event, I'm not sure what equations specifically pertain to in this discussion (nor do I know the meaning of "real problem" in this context). $\endgroup$ – Dave L Renfro Dec 31 '20 at 18:17
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As with many "who was first" questions there is no straightforward answer. This is what May described as priority chasing coming to grief in Historiographic vices:

"The hope of finding a 'first' comes to grief because of the historically dynamic character of ideas... If we describe a result with sufficient vagueness, there seems to be an endless sequence of those who had something within the vague specifications".

Book X of Elements is a geometric theory attributed to Theaetetus (c. 400 BC) of what we now express as quadratic irrationals. With some creativity one can read conjugates into it, specifically into X.112, but it requires a lot of modern rewriting as one can see by looking at the original. For the rewriting see e.g. History of Math by Merzbach and Boyer, p. 105 and Heath's commentary on X.112, p.246:

"X.112 is the equivalent of rationalising the denominators of the fractions $\frac{c^2}{\sqrt{A}+\sqrt{B}}$ and $\frac{c^2}{a+\sqrt{B}}$ by multiplying numerator and denominator by $\sqrt{A}-\sqrt{B}$ and $a-\sqrt{B}$ respectively. Euclid proves that $\frac{\sigma^2}{\rho+\sqrt{k}\rho}=\lambda\rho-\sqrt{k}\,\lambda\rho$ ($k < 1$), and his method enables us to see that $\lambda=\sigma^2/(\rho^2-k\rho^2)$. The proof is a remarkable instance of the dexterity of the Greeks in using geometry as the equivalent of our algebra. Like so many proofs in Archimedes and Apollonius, it leaves us completely in the dark as to how it was evolved. That the Greeks must have had some analytical method which suggested the steps of such proofs seems certain; but what it was must remain apparently an insoluble mystery."

Heath proceeds to rewrite the proof of X.112 in modern algebraic notation. What "seemed certain" to him is no longer believed by historians, but the use of conjugates did emerge in the algebraization of book X by Islamic commentators, e.g. Berggren credits Abu-Kamil (c. 900 AD) with it in Episodes in the Mathematics of Medieval Islam, p.133. But even his rationalizing is verbal, modern algebraic notation is a much later development.

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