Timeline for Are the rationals on a unit circle dense?
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18 events
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| Jan 16, 2017 at 16:24 | comment | added | lulu | @WillOrrick Thanks for this. I'm intrigued, and will go read the old texts. Personally, I find it very difficult to orient myself in the classical way of thinking. But of course that's my weakness, not theirs. | |
| Jan 16, 2017 at 14:19 | comment | added | Will Orrick | Diophantus, writing, it is believed, in about 250 CE, was explicitly concerned with rational solutions of equations, and he employed an algebraic notation, apparently of his own invention. The relevant calculation is Problem 8 in Book II of Arithmetica. Diophantus illustrates the method by dividing $4^2$ into two squares. He notes that $16-x^2$ is square and writes it as $(mx-4)^2$. Taking $m=2$ and equating the two gives $x=16/5$ and hence $4^2=(16/5)^2+(12/5)^2$. I am not sure whether Diophantus looked at rational points on ellipses. | |
| Jan 16, 2017 at 13:57 | comment | added | Will Orrick | ... makes it unlikely that they would have formulated the question as we do today. | |
| Jan 16, 2017 at 13:55 | comment | added | Will Orrick | I am not aware of any evidence that Euclid looked at rational points on ellipses, but you certainly shouldn't take that as definitive. There was work, both prior to Euclid, and at the same time or slightly after, by Archimedes, relating to certain Pell equations, which implies knowledge of integer points on hyperbolas, but I have not looked at this, so I can't say what techniques were used or whether the Greeks viewed it in those terms. At any rate, in 300 BCE, the Greeks did not regard what we call rational numbers as numbers, which... | |
| Jan 16, 2017 at 13:40 | comment | added | Will Orrick | ...might let $x=p(rp)$ and $y=q(rq)$, where $p$, $rp$, $q$, and $rq$ are positive integers. Then you get the triple $$(rpq, \frac{r}{2}(p^2-q^2),\frac{r}{2}(p^2+q^2)).$$ This method is given in a Lemma that is stated in between Propositions 28 and 29 in Book X. Book X contains 115 propositions in total, and its main concern is the classification and relationships of various kinds of irrational numbers. It is believed that much of this material goes back to Theaetetus, who appears in one of Plato's dialogs, but little is known about the origins and motivations for its development. | |
| Jan 16, 2017 at 13:16 | comment | added | Will Orrick | The method Euclid gives in Book X of Elements starts with two lines (lengths), $x>y$. Proposition 6 in Book II relates two squares to a rectangle, and in modern, algebraic terms would be $$xy+\left[\frac{1}{2}(x-y)\right]^2=\left[\frac{1}{2}(x+y)\right]^2.$$ If $x$ and $y$ are positive integers of the same parity, then two terms in this expression are perfect squares. Euclid imposes the condition that the remaining term, $xy$, be a perfect square by requiring that $x$ and $y$ be similar plane numbers, that is, areas of similar rectangles with positive integer side lengths. So you... | |
| Jan 15, 2017 at 12:05 | comment | added | lulu | @WillOrrick Oh, you are likely right. My reasoning was based on Euclid's generating mechanism for Pythagorean triples...where did that come from if not from this? But, in truth, I don't know the history. Is there evidence that Euclid also handled rational points on ellipses? | |
| Jan 15, 2017 at 8:37 | comment | added | Will Orrick | I'm coming late to this, and the comments you are responding to are gone, but I find it hard to see this device in Euclid's derivation. It is clearly there in Diophantus's derivation, however. | |
| Jan 14, 2017 at 22:05 | comment | added | lulu | What are you asking? Euclid certainly used this device, phrased differently, to classify all the Pythagorean triples. | |
| Jan 14, 2017 at 21:52 | comment | added | lulu | Oh, for the most part he did. But don't forget: density wouldn't have occurred to him as a problem....density in what? There was an understanding that things like $\sqrt 2$ weren't rational, though they were constructible, and a suspicion that things like $\pi$ or $\sqrt[3] 2$ might be even worse. But there wasn't a very fleshed out Geometric notion behind such ideas. | |
| Jan 14, 2017 at 21:39 | comment | added | lulu | @j4nbur53 Well, you don't need much topology here. And since you want a statement about density you'll need some. All I really need is continuity....nothing too powerful. Alternatively, you can just use the fact that, in my construction, $\tan \frac {\theta}2$ is the $y$-intercept of the line I draw, so all I really need is the density of the rationals on the interval $[0,1]$. | |
| Jan 14, 2017 at 21:36 | comment | added | lulu | @NoahSchweber Thanks! Reciprocated. | |
| Jan 14, 2017 at 21:32 | comment | added | Noah Schweber | Indeed (and incidentally I upvoted this answer and your comment). | |
| Jan 14, 2017 at 21:23 | comment | added | lulu | May be worth pointing out that, of course, my construction is exactly the same as that of @NoahSchweber . He projects from $(1,0)$ while I project from $(-1,0)$ but any rational point on the circle would do the job. I just added enough details to show that, in fact, every rational point on the circle arises this way. | |
| Jan 14, 2017 at 21:15 | comment | added | Noah Schweber | @j4nbur53 The map $f:x\mapsto \tan{x\over 2}$ is continuous and surjective onto $\mathbb{R}$, so the $f$-preimage of any dense subset of $\mathbb{R}$ (e.g. $\mathbb{Q}$) is dense in the domain. | |
| Jan 14, 2017 at 21:12 | comment | added | user4414 | The intermediate claim is interesting, but the last "it is clear" part might need some explanation. | |
| Jan 14, 2017 at 21:05 | history | edited | lulu | CC BY-SA 3.0 |
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| Jan 14, 2017 at 20:49 | history | answered | lulu | CC BY-SA 3.0 |