# Who invented multiplying by the conjugate to rationalize denominators and when?

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.

• 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. Dec 30 '20 at 23:36
• 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. Dec 30 '20 at 23:58
• @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). Dec 31 '20 at 11:43
• @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). Dec 31 '20 at 14:46
• @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). Dec 31 '20 at 18:17

"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."