tag:blogger.com,1999:blog-11295132.post6369504533228653968..comments2014-10-20T08:43:52.658-07:00Comments on A Neighborhood of Infinity: On representing some real numbers exactlyDan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger16125tag:blogger.com,1999:blog-11295132.post-2694197481134730502012-01-26T06:28:36.100-08:002012-01-26T06:28:36.100-08:00Here I have represented the set of All Real Number...Here I have represented the set of All Real Numbers Exactly. Comments most Welcome. <br /><br />http://auminfinitecosmoses.com/categories/view/34<br /><br />http://auminfinitecosmoses.com/Narayanan Raghunathanhttp://auminfinitecosmoses.com/noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-69064466405495604462011-10-27T11:50:29.808-07:002011-10-27T11:50:29.808-07:00Here's another link to PERIODS by Kontsevich a...Here's another link to <a href="http://www.ihes.fr/~maxim/TEXTS/Periods.pdf" rel="nofollow">PERIODS by Kontsevich and Zagier (PDF)</a>, the one you list is now dead.Craighttp://www.blogger.com/profile/12170572784481628483noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-28056001931538047472010-04-30T15:19:13.425-07:002010-04-30T15:19:13.425-07:00@sigfpe
I should have guessed you knew it from th...@sigfpe<br /><br />I should have guessed you knew it from the title of the blog, but oh well.<br /><br />@leithaus<br /><br />One thing that convinced me that the distinction between algebraic and transcendental numbers isn't as arbitrary as it looks is Liouville's theorem on diophantine approximation and the Thue-Siegel-Roth theorem. Basically, it's easier to approximate transcendentals by rationals than it is to approximate algebraic numbers by rationals. <br /><br />@Twan van Laarhoven:<br /><br />It seems the numbers in R\D are necessary for measure theory, somehow. It seems you can't get a good theory of probability otherwise.Roberthttp://www.blogger.com/profile/10746976399050925428noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-87988215612417568772010-04-30T10:12:52.784-07:002010-04-30T10:12:52.784-07:00Dear Dan, This brings us back to Conway Games -- w...Dear Dan, This brings us back to Conway Games -- which significantly extended our access to the definable numbers. More interestingly, the current ontology of numbers is almost entirely an accident of history. We tend to think of these dividing lines, whole, rational, algebraic, constructible, real, complex as if they were given from above. There are many other ways to organize our conception of quantity. Here's an alternative proposal (among an infinitude of them). It turns out that every Conway Game has a full and faithful representation as a π-calculus process. One of the more interesting aspects of concurrency is the emerging type theory of behavior. Types give us a different way to classify programs. It would be quite interesting to investigate the organization of quantity in terms of the inhabitants of various types in a behavioral type system for the π-calculus. <br /><br />More generally, Conway Games themselves have a presentation as a monad, call it G. Given a monad for a collection, say Set, we can construct a logic for classifying sets of numbers via a distributive law from d: Set G -> G Set. That is, we get formulae (= types) that denote sets of numbers in terms of the operations on Conway Games. Best wishes, --gregleithaushttp://www.blogger.com/profile/01069099703796397027noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-7183724417855717602010-04-28T22:15:04.443-07:002010-04-28T22:15:04.443-07:00@Robert,
I did my PhD on (theta functions on) Rie...@Robert,<br /><br />I did my PhD on (theta functions on) Riemann surfaces so I guessed the connection with periods of Riemann surfaces. But to make the article accessible I traced back to the etymology further back to something a wider audience would understand.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-26163109949340449392010-04-28T22:08:00.750-07:002010-04-28T22:08:00.750-07:00I had the same idea as Yoshinaga after I read the ...I had the same idea as Yoshinaga after I read the Kontsevitch and Zagier paper, and I was about to embark on it (actually I intended to use some of Bishop's constructivist stuff to do it, as well as the Hironaka theorem). Good thing I happened to come here and didn't duplicate work.<br /><br />The name "period" comes from algebraic geometry, specifically, the theory of Abelian integrals. (See Griffiths & Harris, Principles of Algebraic Geometry, as the wikipedia article is not that good). However, I think the use of the term period in that theory does come from elliptic functions, so you are right if we take the transitive closure. <br /><br />They're a sort of cohomological thingy (the value of an integral along a path on a (complex) algebraic curve depends on the path if the genus > 0).Roberthttp://www.blogger.com/profile/10746976399050925428noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-58016968064493024012010-04-26T11:03:29.308-07:002010-04-26T11:03:29.308-07:00@tracking-is-for-the-weak Interval arithmetic can ...@tracking-is-for-the-weak Interval arithmetic can be an ingredient that goes into an implementation of the computable reals. For example, any computable real can be represented exactly as a computable stream of nested intervals (with some provisos).sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-55348688306127820302010-04-26T10:54:37.641-07:002010-04-26T10:54:37.641-07:00You write "given any algebraic number there i...You write "given any algebraic number there is always an infinite number of algebraic equations that it satisfies". This is true, but there is a unique best one: there is only one irreducible equation it satisfies. This is algorithmic, although I suppose it might sometimes be best not to do the actual factoring necessary.Dylan Thurstonhttp://www.blogger.com/profile/10602306570319761901noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-67938201039124022792010-04-26T06:40:15.818-07:002010-04-26T06:40:15.818-07:00I really prefer intervals. Not as accurate and to ...I really prefer intervals. Not as accurate and to ensure accuracy you need to do interval constraints and solving, but interval arithmetic allows you represent where your real number actually is.<br /><br />http://webhome.cs.uvic.ca/~vanemden/research/intConstrInd.htmltracking-is-for-the-weakhttp://openid.aliz.es/tracking-is-for-the-weaknoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-12427275591145973372010-04-26T03:55:10.021-07:002010-04-26T03:55:10.021-07:00If you really want to approximate the square root ...If you really want to approximate the square root of 2 within 1/1000, I suggest the fraction 41/29. ;-)<br /><br />Great post, by the way.Leon Smithhttp://www.blogger.com/profile/06462854866941248768noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-53498202249104761692010-04-26T01:12:45.348-07:002010-04-26T01:12:45.348-07:00I love learning about numbers and this is an excel...I love learning about numbers and this is an excellent article, thanks! I hadn't heard of periods or definables before but they ring bells. <br /><br />I'm wondering whether the periods arise from what Wildberger refers to as quadrance within what he defines as the rational numbers of his rational trigonometry (which seem to include real algebraic numbers). He doesn't solve them, instead they are but algebraic area representations - unless I've misunderstood.Craighttp://www.blogger.com/profile/12170572784481628483noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-75035297823384830942010-04-25T22:18:34.936-07:002010-04-25T22:18:34.936-07:00shreevatsa,
It's true to say that for *any* r...shreevatsa,<br /><br />It's true to say that for *any* representation of real numbers as finite sequences of symbols (such that different reals don't get the same representation) we always leave out uncountably many reals. I think this says something important about the meaning of "there exists" in mathematics. Classical mathematicians are happy to say something exists, even when it can't be constructed. But <a href="http://en.wikipedia.org/wiki/Constructivism_(mathematics)" rel="nofollow">constructive mathematicians</a> only like to say something exists if they can actually construct it.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-88318159543841957912010-04-25T22:05:38.854-07:002010-04-25T22:05:38.854-07:00Great post, as always!
[BTW, your link on Chaitin...Great post, as always!<br /><br />[BTW, your link on Chaitin's constant goes to Tarski's undefinability theorem instead.]<br /><br />A possibly dumb question: I'd like to say that there are only countably many definable numbers (since each is represented by a finite string), but can we make such a statement, since the "set" D is not definable? [And if it's true, we're left with the surprising fact that although "most" real numbers are undefinable (since only countably many are definable), we cannot give a single example of an undefinable number!]shreevatsahttp://shreevatsa.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-29557840995624271952010-04-25T21:12:23.238-07:002010-04-25T21:12:23.238-07:00Thanks for asking Jared. In the text I did define ...Thanks for asking Jared. In the text I did define real algebraic numbers but my notation was non-standard. See the update I just added.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-89881272146651330302010-04-25T20:42:23.717-07:002010-04-25T20:42:23.717-07:00Algebraic can be complex, and thus are not a subse...Algebraic can be complex, and thus are not a subset of reals. Are you only considering real algebraic numbers?Jaredhttp://www.blogger.com/profile/11392214290089134699noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-19432068216170547782010-04-25T16:27:01.184-07:002010-04-25T16:27:01.184-07:00Interesting article. This leaves one wondering abo...Interesting article. This leaves one wondering about the numbers in R\D, although it would be hard to talk about them :)<br /><br />In "every algebraic number can be described by a finite string of symbols, and there are only finitely many such strings."<br />I think that should be "there are only *countably* many such strings".Twan van Laarhovenhttp://www.blogger.com/profile/18138442561179666544noreply@blogger.com