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Before Eudoxus's theory of proportion there was a theory of irrationals based on continued fraction expansions, which Fowler calls anthyphairesis. Theaetetus is said to develop a classification of quadratic irrationals in this setting. Given positive $x$ and $y$, with $x/y$ being a square root of a rational number, $x+y$ is called a binomial, $x-y$ an apotome (if $x>y$), and $\sqrt{xy}$ a medial (if it is irrational). There are some interesting results described in book X of Euclid's Elements: the classes are mutually exclusive, for any binomial or apotome the split into $x$ and $y$ is unique, etc. Further classes are considered, apotome of a medial, binomial of a medial, and so on.

Theaetetus seems to deal with positive elements of $\mathbb{Q}(\sqrt{2},\sqrt{3},\dots)$, but I can not think of any algebraic or number theoretic classification scheme that would produce his classes. Are they related to something in modern algebra?

Conifold
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  • I'm making this a comment because of the "modern algebra" part, but they sure are related to the classification of quadrics (but I guess this classifies as "geometry" rather than algebra, although the frontier is dim between the two fields). – VicAche Nov 05 '14 at 20:51
  • @VicAche This is interesting, I thought quadrics were classified by types of quadratic forms rather than number theoretic properties. Could you elaborate. – Conifold Nov 07 '14 at 15:55
  • see "Quadric forms" entry in The Princeton Companion to Mathematics. (just search for it on google if you don't want to buy the book). I don't have enough time to sum it up here now. – VicAche Nov 08 '14 at 17:43
  • Our real numbers correspond to a ratios of quantities for the Greeks. So we should use y as the unit of length. So letting x/y=sqrt(r), then the binomial corresponds to 1+sqrt(r), the apotome to 1-sqrt(r) (with r<1), and the medial to the fourth root of r. I don't think there is much more to it than this. I can expand a little into an actual answer if this is what you had in mind. (Good illustration of the need for MathJax, though!) – Michael Weiss Nov 08 '14 at 22:50
  • @Michael Weiss I think sqrt(r)+/-sqrt(q) would also be binomial/apotome. A natural classification from modern point of view would be by minimal degree of a polynomial with rational coefficients that the number is a root of. But 1+sqrt(r) has degree 2 while sqrt(r)+sqrt(q) has degree 4, also x+/-y have the same degree, so it doesn't seem to be that. Agree on MathJax! – Conifold Nov 08 '14 at 23:43
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    I read the interesting paper of Fowler you refer to. He mentions several prominent mathematicians he consulted, and it looks like if he knew the answer to your question he would mention it. I recommend that you post your question on Math Overflow. – Alexandre Eremenko Nov 28 '14 at 13:40
  • I do not understand this question at all. There are no extant writings by Theaetetus, just a brief passage in one of Plato's work. We do not even know whether he is historical or just a literary fiction. Even less do we know whether he worked inside the "field" you mentioned - did Fowler really claim he did? And if he did, why are there no traces of his work in Euclid? –  May 07 '15 at 15:15
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    @Franz Lemmermeyer There is Plato's dialogue entirely devoted to Theaetetus and named after him, he is also mentioned by many other sources. So we know that he is as historical as Pericles. Euclid does not credit anyone in the Elements, not even himself, but books X and XII are ascribed to Theaetetus by others, just like book V is to Eudoxus. And I only wrote that he seems to deal with (what we now call) elements of a field, not that he knowingly "worked inside" a field. The details are described under the book X link. – Conifold May 07 '15 at 23:04
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    I just want to point out that the point of Fowler's book (The Mathematics of Plato's Academy) is explicitly to try to answer the question of the motivation of the classification and its possible significance. – Marius Kempe Apr 01 '19 at 09:14
  • Here are some modern commentaries on Euclid's book X: C. M. Taisbak, Coloured quadrangles: a guide to tenth book of Euclid's Elements, Museum Tusculanum Press, Copenhagen, 1962. D. H. Fowler, An invitation to read book X of Euclid's Elements, Hist. Math. 19 (1992) 233-364. W. R. Knorr, Euclid’s Tenth Book: An Analytic Survey, Historia Scientiarum 29 (1985), 17–35. – Alexandre Eremenko Oct 31 '18 at 01:51

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