From jcowan@reutershealth.com Mon Jul 10 11:20:47 2000 Return-Path: Received: (qmail 18602 invoked from network); 10 Jul 2000 18:17:15 -0000 Received: from unknown (10.1.10.142) by m2.onelist.org with QMQP; 10 Jul 2000 18:17:15 -0000 Received: from unknown (HELO mail.reutershealth.com) (204.243.9.36) by mta1 with SMTP; 10 Jul 2000 18:17:14 -0000 Received: from reutershealth.com (IDENT:cowan@skunk.reutershealth.com [204.243.9.153]) by mail.reutershealth.com (Pro-8.9.3/8.9.3) with ESMTP id OAA27179; Mon, 10 Jul 2000 14:17:09 -0400 (EDT) Sender: cowan@mail.reutershealth.com Message-ID: <396A12E3.82125F2@reutershealth.com> Date: Mon, 10 Jul 2000 14:16:03 -0400 Organization: Reuters Health Information X-Mailer: Mozilla 4.7 [en] (X11; I; Linux 2.2.5-15 i686) X-Accept-Language: en MIME-Version: 1.0 To: And Rosta , "lojban@onelist.com" Subject: Re: [lojban] 2 maths questions References: Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: John Cowan And Rosta wrote: > What is density? [Give me dimbo's answer only.] No problem. A sequence is dense if between any two members of it you can always find another member. The whole numbers are not dense: there is no whole number between 3 and 4. The rational numbers (fractions) are dense: between 3/1 and 4/1 there is 355/113, among many others. In general, between fraction a / b and fraction c / d the fraction (ad+cb)/2 / bd exists. Density is a polar property: a set is either dense or not. > Anyway, I originally was trying to ask (i) whether "thickness" is a > recognized notion, Seemingly not. > I don't at all understand pc's or C.D.Wright's replies, I'm afraid. Okay. The idea is that we take our two sets, the set of evens and the set of integers, and truncate them to some finite range, say 0 to 1,000,000. In this range, we find that the ratio of the truncated evens to the truncated integers is 1 to 2. As we take larger and larger finite ranges, the ratio remains 1 to 2, no matter how large they get. So we can say that if the integers are assigned "Wright measure" 1, then the evens have Wright measure 1/2, which is what we want. What is the Wright measure of the prime numbers? For small ranges, there are many prime numbers, e.g. in the range 0-5, we have six integers and three primes, for a Wright measure of 1/2. But as the ranges get larger and larger, the primes get rarer and rarer, and it can be shown that the limiting value is zero. So the whole infinite set of primes has Wright measure 0. So does the set of powers of 2 (or any integer). > All replies the set of whose addressees includes me should be > expressed in a maximally elementary [...] It's an interesting sociolinguistic fact that all mathematical treatises, however esoteric, are invariably titled something like "An Introduction to the Elementary Theory of X", even if the beginner will be lost after the first five sentences. The physicist and wit Richard Feynman went so far as to define "trivial theorem" as "theorem whose proof is known." > [The set of primes] is of uneven thickness. Fairly thick in some areas and fairly > thin in others. Like trains in peak and offpeak hours. And, like > buses, they often come in pairs. Yes, but unlike the trains, and like the atmosphere, it becomes thinner and thinner as you ascend Mount Integer, never actually disappearing but becoming as rarefied as you like. This may be easier to think about if you think about the number of unbroken integers in sequence *without* a prime. The lists of these get longer and longer: do you want a million, billion, trillion, ... integers in sequence, none of which are prime? Just keep looking! In the end, the ratio of primes to nonprimes becomes as small as you like, so that we say that "in the limit" it reaches zero, even though there are always more primes (as good old Euclid proved some 2500 years ago). ObJoke: This allows me to tell my favorite anecdote on the subject, which is about a mathematics teacher (British, obviously), who accidentally gave a test whose questions were worth, in aggregate, only 99 marks (points, in American). He noticed this only after grading it, and conscientiously converted all the scores to percentages, thus: 75 marks got a grade of 75.75757575.... and 58 marks a grade of 58.5858585..... But there was a student, Smith by name, who had a perfect score of 99 marks, and was duly assigned the grade 99.99999..... "And what say you to this, Smith minor, that although your paper is quite perfect, it falls short of 100 percent?" "Sir," said Smith minor, moved to anger, "I call that the limit." -- Schlingt dreifach einen Kreis um dies! || John Cowan Schliesst euer Aug vor heiliger Schau, || http://www.reutershealth.com Denn er genoss vom Honig-Tau, || http://www.ccil.org/~cowan Und trank die Milch vom Paradies. -- Coleridge (tr. Politzer)