Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" Newsgroups: sci.math Subject: Re: How many different unit fractions are lessorequal than all unit fractions? Date: Fri, 20 Sep 2024 18:36:18 -0700 Organization: A noiseless patient Spider Lines: 70 Message-ID: References: <405557f7289631d63264c712d137244c940b9926@i2pn2.org> <13c08e96ad635f8142b38d89863a80caf17a32a8@i2pn2.org> <4faa63d0ff8c163f01a38736aeb5732184218a29@i2pn2.org> <87jzfchnxc.fsf@bsb.me.uk> <4b617ec8573746a45483060acdd082653f14fda5@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Injection-Date: Sat, 21 Sep 2024 03:36:19 +0200 (CEST) Injection-Info: dont-email.me; posting-host="9fabd7f027d8f87c91e752d958126e44"; logging-data="1394268"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+R7VVvmlRdzEYeCJjCFoCc9V5NoO/14Us=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:EzafVx27bA0LFfAF8EwEF/BmZww= Content-Language: en-US In-Reply-To: Bytes: 4643 On 9/20/2024 6:03 PM, Moebius wrote: > Am 21.09.2024 um 01:23 schrieb Chris M. Thomasson: >> On 9/19/2024 2:13 PM, Moebius wrote: >>> Am 19.09.2024 um 21:13 schrieb Chris M. Thomasson: >>>> On 9/19/2024 5:55 AM, WM wrote: >>>>> On 18.09.2024 22:49, Moebius wrote: >>>>>> Am 18.09.2024 um 21:35 schrieb Chris M. Thomasson: >>>>>>> On 9/18/2024 5:44 AM, WM wrote: >>>>>>>> On 16.09.2024 03:16, Richard Damon wrote: >>>>>>>>> On 9/15/24 3:39 PM, WM wrote: >>>>>>>>>> On 15.09.2024 18:38, Ben Bacarisse wrote: >>>>>>>>>> >>>>>>>>>>> It might be worth pointing out that any non-trivial interval >>>>>>>>>>> [a, b] on >>>>>>>>>>> the real line (i.e. with b > a) contains an uncountable >>>>>>>>>>> number of >>>>>>>>>>> points. >>>>>>>>>> >>>>>>>>>> That proves that small intervals cannot be defined (they are >>>>>>>>>> dark). >>>>>>>>> >>>>>>>>> But arbitrary small intervals CAN be defined. >>> >>> For any eps e IR, eps > 0: [0, eps] is an interval (@WM: you see, I >>> just defined it) of length eps and it countains an uncountable number >>> of points. Hint: eps may be arbitrarily small, as long as it is > 0. >>> >>>>>>>>> Try to name one that can't. >>>>>>>> >>>>>>>> Define an interval comprising 9182024 points, starting at zero. >>>>>> >>>>>> There IS NO "interval comprising 9182024 points", hence NOTHING TO >>>>>> DEFINE HERE, you fucking asshole full of shit. >>>>> >>>>> How can infinitely many points be accumulated without a first one? >>> >>> @WM: There's no need for them "to be accumulated" (whatever this may >>> mean), you fucking asshole full of shit. >>> >>>> The real line is infinitely long >>> >>> WM is talking about some interval of finite length here, it seems. >>> >>>> and infinitely dense, or granular if you will... >>> >>> Sorta. >> >> Well, its infinitely long... >> >> ...(-1)------(0)--------(+1)... >> >> >> It has no end just like there is no end to the signed integers. Also, >> its infinitely dense due to the nature of the reals. >> >> ? > > Yes. > > Here's a mind bending fact concerning the reals and rational numbers. > > Between any two real numbers there is a rational number and between any > two rational number there is a real number. But there are only countably > many rational numbers while there are uncountably many real numbers. :-) > > Math lingo: "The rationals are dense in the reals". :-) As in there is a rational that can be generated by the infinite convergents of continued fractions that can be used to represent the target real up to an infinite desired accuracy? Fair enough?