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Groups > comp.theory > #51785 > unrolled thread
| Started by | olcott <NoOne@NoWhere.com> |
|---|---|
| First post | 2022-06-03 17:17 -0500 |
| Last post | 2022-06-04 00:36 +0100 |
| Articles | 20 on this page of 165 — 11 participants |
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Refuting the HP proofs (adapted for software engineers) olcott <NoOne@NoWhere.com> - 2022-06-03 17:17 -0500
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-03 18:50 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc.corp> - 2022-06-04 00:35 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-03 18:56 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-03 20:20 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-03 22:51 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2022-06-04 03:01 -0700
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-04 10:11 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 11:38 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ BRAIN DEAD MORON ] olcott <NoOne@NoWhere.com> - 2022-06-04 10:51 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ BRAIN DEAD MORON ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 12:11 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ BRAIN DEAD MORON ] olcott <NoOne@NoWhere.com> - 2022-06-04 11:25 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ BRAIN DEAD MORON ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 13:15 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ BRAIN DEAD MORON ] olcott <NoOne@NoWhere.com> - 2022-06-04 12:23 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ BRAIN DEAD MORON ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 14:09 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ BRAIN DEAD MORON ] olcott <NoOne@NoWhere.com> - 2022-06-04 13:14 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ BRAIN DEAD MORON ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 14:31 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ BRAIN DEAD MORON ] olcott <NoOne@NoWhere.com> - 2022-06-04 13:39 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ BRAIN DEAD MORON ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 14:49 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Alan Mackenzie <acm@muc.de> - 2022-06-04 18:17 +0000
Re: Refuting the HP proofs (adapted for software engineers)[ Alan Mackenzie ] olcott <NoOne@NoWhere.com> - 2022-06-04 13:37 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Alan Mackenzie ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 14:54 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Alan Mackenzie ] olcott <NoOne@NoWhere.com> - 2022-06-04 14:01 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Alan Mackenzie ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 15:57 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Alan Mackenzie ] Alan Mackenzie <acm@muc.de> - 2022-06-04 19:02 +0000
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-04 14:28 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 16:05 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] [OT] Jeff Barnett <jbb@notatt.com> - 2022-06-04 17:30 -0600
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Mikko <mikko.levanto@iki.fi> - 2022-06-05 13:14 +0300
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 05:34 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Alan Mackenzie <acm@muc.de> - 2022-06-05 11:12 +0000
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 06:21 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 07:58 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 14:47 +0100
Re: Refuting the HP proofs (adapted for software engineers) Andy Walker <anw@cuboid.co.uk> - 2022-06-05 16:28 +0100
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 16:34 +0100
Re: Refuting the HP proofs (adapted for software engineers) Alan Mackenzie <acm@muc.de> - 2022-06-05 15:44 +0000
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 16:49 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:22 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 17:28 +0100
Re: Refuting the HP proofs (adapted for software engineers) olcott <NoOne@NoWhere.com> - 2022-06-05 11:35 -0500
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:50 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 17:56 +0100
Re: Refuting the HP proofs (adapted for software engineers) olcott <NoOne@NoWhere.com> - 2022-06-05 12:01 -0500
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 18:19 +0100
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 18:27 +0100
Re: Refuting the HP proofs (adapted for software engineers) olcott <NoOne@NoWhere.com> - 2022-06-05 12:58 -0500
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 14:13 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 20:14 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 17:46 -0400
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 13:05 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 18:22 +0100
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 18:26 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 14:17 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 20:17 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 15:30 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 20:33 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 15:47 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 20:56 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 16:09 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 21:23 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 16:32 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mikko <mikko.levanto@iki.fi> - 2022-06-06 16:10 +0300
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-06 17:47 +0100
Re: Refuting the HP proofs (adapted for software engineers) Andy Walker <anw@cuboid.co.uk> - 2022-06-05 18:44 +0100
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 18:48 +0100
Re: Refuting the HP proofs (adapted for software engineers) olcott <NoOne@NoWhere.com> - 2022-06-05 11:29 -0500
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:53 -0400
Re: Refuting the HP proofs (adapted for software engineers) Alan Mackenzie <acm@muc.de> - 2022-06-05 16:34 +0000
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 17:38 +0100
Re: Refuting the HP proofs (adapted for software engineers) olcott <NoOne@NoWhere.com> - 2022-06-05 11:41 -0500
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 17:42 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:54 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 17:58 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 13:07 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 18:23 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 14:20 -0400
Re: Refuting the HP proofs (adapted for software engineers) Alan Mackenzie <acm@muc.de> - 2022-06-05 17:04 +0000
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:17 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 17:37 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:57 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 18:17 +0100
Re: Refuting the HP proofs (adapted for software engineers) Alan Mackenzie <acm@muc.de> - 2022-06-05 18:07 +0000
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 20:19 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 15:32 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 20:34 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 15:49 -0400
Re: Refuting the HP proofs (adapted for software engineers) Alan Mackenzie <acm@muc.de> - 2022-06-05 19:42 +0000
Re: Refuting the HP proofs (adapted for software engineers) Mikko <mikko.levanto@iki.fi> - 2022-06-06 16:03 +0300
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 14:24 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 20:18 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 15:38 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 20:44 +0100
Re: Refuting the HP proofs (adapted for software engineers) Richard Damon <Richard@Damon-Family.org> - 2022-06-05 15:54 -0400
Re: Refuting the HP proofs (adapted for software engineers) Ben <ben.usenet@bsb.me.uk> - 2022-06-05 18:56 +0100
Re: Refuting the HP proofs (adapted for software engineers) [ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 13:07 -0500
Re: Refuting the HP proofs (adapted for software engineers) [ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 14:29 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Alan Mackenzie <acm@muc.de> - 2022-06-05 12:14 +0000
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Ben <ben.usenet@bsb.me.uk> - 2022-06-05 13:38 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Ben <ben.usenet@bsb.me.uk> - 2022-06-05 16:17 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 10:59 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:29 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 10:57 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:31 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 11:39 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:59 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 12:02 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 14:31 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 13:35 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 14:54 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 13:57 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 14:09 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 15:25 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 14:33 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 15:43 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 11:24 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2022-06-05 15:46 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Alan Mackenzie <acm@muc.de> - 2022-06-05 15:16 +0000
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 11:10 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2022-06-05 21:07 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 15:15 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 21:28 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 15:36 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 16:44 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 16:38 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 15:41 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 16:57 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Jeff Barnett <jbb@notatt.com> - 2022-06-05 15:59 -0600
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2022-06-06 00:59 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Jeff Barnett <jbb@notatt.com> - 2022-06-05 18:24 -0600
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Ben <ben.usenet@bsb.me.uk> - 2022-06-06 01:40 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Jeff Barnett <jbb@notatt.com> - 2022-06-05 18:44 -0600
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 20:03 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 21:59 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 21:14 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 22:44 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2022-06-06 02:58 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 21:11 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 22:20 -0400
Re: Refuting the HP proofs (adapted for software engineers[ brand new computer science ] olcott <NoOne@NoWhere.com> - 2022-06-05 21:37 -0500
Re: Refuting the HP proofs (adapted for software engineers[ brand new computer science ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 22:52 -0400
Re: Refuting the HP proofs (adapted for software engineers[ brand new computer science ] olcott <NoOne@NoWhere.com> - 2022-06-05 22:03 -0500
Re: Refuting the HP proofs (adapted for software engineers[ brand new computer science ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 23:26 -0400
Re: Refuting the HP proofs (adapted for software engineers[ Ordinary software engineering ] olcott <NoOne@NoWhere.com> - 2022-06-05 22:41 -0500
Re: Refuting the HP proofs (adapted for software engineers[ Ordinary software engineering ] Richard Damon <Richard@Damon-Family.org> - 2022-06-06 00:17 -0400
Re: Refuting the HP proofs (adapted for software engineers[ Ordinary software engineering ] olcott <NoOne@NoWhere.com> - 2022-06-06 10:28 -0500
Re: Refuting the HP proofs (adapted for software engineers[ Ordinary software engineering ] Richard Damon <Richard@Damon-Family.org> - 2022-06-06 21:04 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 22:15 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 21:22 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 22:38 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Mike Terry ] olcott <NoOne@NoWhere.com> - 2022-06-05 19:27 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Mike Terry ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 20:56 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ members of c/c++ ] olcott <NoOne@NoWhere.com> - 2022-06-07 20:04 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ members of c/c++ ] Richard Damon <Richard@Damon-Family.org> - 2022-06-07 22:45 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Mike Terry ] Mr Flibble <flibble@reddwarf.jmc> - 2022-06-06 17:49 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Mike Terry ] olcott <NoOne@NoWhere.com> - 2022-06-06 11:59 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 11:07 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Mr Flibble <flibble@reddwarf.jmc> - 2022-06-05 17:12 +0100
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-05 11:15 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:45 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-05 12:41 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 06:27 -0400
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] olcott <NoOne@NoWhere.com> - 2022-06-04 10:28 -0500
Re: Refuting the HP proofs (adapted for software engineers)[ Andy Walker ] Richard Damon <Richard@Damon-Family.org> - 2022-06-04 11:51 -0400
Re: Refuting the HP proofs (adapted for software engineers) Mr Flibble <flibble@reddwarf.jmc.corp> - 2022-06-04 00:36 +0100
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| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 21:23 +0100 |
| Message-ID | <20220605212319.0000040a@reddwarf.jmc> |
| In reply to | #51944 |
On Sun, 5 Jun 2022 16:09:13 -0400
Richard Damon <Richard@Damon-Family.org> wrote:
> On 6/5/22 3:56 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 15:47:30 -0400
> > Richard Damon <Richard@Damon-Family.org> wrote:
> >
> >> On 6/5/22 3:33 PM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 15:30:45 -0400
> >>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>
> >>>> On 6/5/22 3:17 PM, Mr Flibble wrote:
> >>>>> On Sun, 5 Jun 2022 14:17:29 -0400
> >>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>
> >>>>>> On 6/5/22 1:22 PM, Mr Flibble wrote:
> >>>>>>> On Sun, 5 Jun 2022 13:05:09 -0400
> >>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>
> >>>>>>>> On 6/5/22 12:56 PM, Mr Flibble wrote:
> >>>>>>>>> On Sun, 5 Jun 2022 12:50:13 -0400
> >>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>
> >>>>>>>>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
> >>>>>>>>>>> On Sun, 5 Jun 2022 12:22:45 -0400
> >>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>
> >>>>>>>>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
> >>>>>>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> >>>>>>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>>>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>>>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>>>>>>>>>>>> exact value, but you can never actually express it
> >>>>>>>>>>>>>>>>>> (because it takes an infinite number of digits).
> >>>>>>>>>>>>>>>>> PI does not have an exact value; no irrational
> >>>>>>>>>>>>>>>>> number has an exact value.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> Of course "pi" has an exact value; as
> >>>>>>>>>>>>>>>> do [eg] "sqrt(2)", "e", and all the other computable
> >>>>>>>>>>>>>>>> real [and complex] numbers. Whether that value can be
> >>>>>>>>>>>>>>>> expressed in finite terms in some particular
> >>>>>>>>>>>>>>>> representation is quite another matter. That in turn
> >>>>>>>>>>>>>>>> depends on the representation; standard decimals is
> >>>>>>>>>>>>>>>> merely one [common] choice. Note that in symbolic
> >>>>>>>>>>>>>>>> computer systems, those computable reals are
> >>>>>>>>>>>>>>>> typically written "pi" [or whatever], and the
> >>>>>>>>>>>>>>>> computer works with that exactly, so that [eg]
> >>>>>>>>>>>>>>>> "sin^2 (pi/3) == 3/4", not 0.7499...; and also that
> >>>>>>>>>>>>>>>> in decimal-type notations most rationals equally
> >>>>>>>>>>>>>>>> have no terminating expansion. Numbers such as "pi"
> >>>>>>>>>>>>>>>> and "sqrt(2)" are not defined as decimal expansions
> >>>>>>>>>>>>>>>> but via their properties [eg that "sqrt(2)" is the
> >>>>>>>>>>>>>>>> unique positive real whose square is 2, or
> >>>>>>>>>>>>>>>> equivalently that it is the ratio of the diagonal of
> >>>>>>>>>>>>>>>> a square to its side, and "pi" is the least positive
> >>>>>>>>>>>>>>>> real whose sine is zero]. Those properties are
> >>>>>>>>>>>>>>>> exact, and tell you all you ever need to know about
> >>>>>>>>>>>>>>>> those numbers.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> [ .... ]
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> What has decimal (base 10) expansion got to do with
> >>>>>>>>>>>>>>> anything? An irrational number has a non-terminating
> >>>>>>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
> >>>>>>>>>>>>>>> mistaken: irrational numbers do NOT have an exact
> >>>>>>>>>>>>>>> value; this is obvious to anyone who understands
> >>>>>>>>>>>>>>> logic and uses a sane definition for infinity.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> That irrational numbers are exact values is clear to
> >>>>>>>>>>>>>> anybody with a degree in maths. Definitions of
> >>>>>>>>>>>>>> "infinity" (of which there are many) have nothing to do
> >>>>>>>>>>>>>> with this.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> You are wrong and fractally so so your degree in maths
> >>>>>>>>>>>>> appears to be worthless. An irrational number's
> >>>>>>>>>>>>> sequence is statistically random, has no fixed point on
> >>>>>>>>>>>>> the number line ergo has no exact representation. Any
> >>>>>>>>>>>>> number with no exact representation has, by definition,
> >>>>>>>>>>>>> no exact value, only an approximation. Infinity has
> >>>>>>>>>>>>> everything to do with this as an irrational's sequence
> >>>>>>>>>>>>> ("digits") never terminates (i.e. it is an INFINITELY
> >>>>>>>>>>>>> long sequence).
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> /Flibble
> >>>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>> Nope. Irrational numbers DO have exact points on the
> >>>>>>>>>>>> number line.
> >>>>>>>>>>>>
> >>>>>>>>>>>> And what does representation have to do with exact value?
> >>>>>>>>>>>>
> >>>>>>>>>>>> Also, irrational numbers sequence of digits are not
> >>>>>>>>>>>> necessarily statistically random, in some
> >>>>>>>>>>>> representations, they can be VERY predictible for some
> >>>>>>>>>>>> numbers.
> >>>>>>>>>>>>
> >>>>>>>>>>>> One simple construction to show exact position, draw a
> >>>>>>>>>>>> box with sides exactly 1.
> >>>>>>>>>>>>
> >>>>>>>>>>>> Draw a line though opposite corners and make one point
> >>>>>>>>>>>> the value 0.
> >>>>>>>>>>>>
> >>>>>>>>>>>> The other corner will be EXACTLY at the point sqrt(2), so
> >>>>>>>>>>>> that irrational number has an exact point on the number
> >>>>>>>>>>>> line.
> >>>>>>>>>>>>
> >>>>>>>>>>>> You just don't understand what an exact value means,
> >>>>>>>>>>>> likely because you can't understand things that are
> >>>>>>>>>>>> somewhat abstract.
> >>>>>>>>>>>
> >>>>>>>>>>> An irrational number does not have an exact point on the
> >>>>>>>>>>> number line as it will move about as you "zoom in", you
> >>>>>>>>>>> can keep "zooming in" forever (i.e. infinitely) and it
> >>>>>>>>>>> will keep moving about because the number never
> >>>>>>>>>>> terminates.
> >>>>>>>>>>>
> >>>>>>>>>>> If I couldn't understand things that are somewhat abstract
> >>>>>>>>>>> then I wouldn't have a computer science degree (BSc Hons)
> >>>>>>>>>>> and 30 years of industry experience.
> >>>>>>>>>>>
> >>>>>>>>>>> /Flibble
> >>>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> Then why do you think irrational numbers don't have an
> >>>>>>>>>> exact location?
> >>>>>>>>>>
> >>>>>>>>>> I know people with degrees (even with honors) and industry
> >>>>>>>>>> experiance that still show that they don't really
> >>>>>>>>>> understand what they are talking about.
> >>>>>>>>>>
> >>>>>>>>>> The "width" of the point representing the location of an
> >>>>>>>>>> irrational number is just as much "0" as that of a rational
> >>>>>>>>>> number, so specifies just as exact of a location.
> >>>>>>>>>>
> >>>>>>>>>> The fact that we can't write it in a rational base with a
> >>>>>>>>>> finite number of digits doesn't actally mean anything.
> >>>>>>>>>
> >>>>>>>>> 3.1415xxxxxxxxx (a)
> >>>>>>>>> 3.14159xxxxxxxx (b)
> >>>>>>>>>
> >>>>>>>>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates
> >>>>>>>>> hence its value "moves about" as accuracy increases.
> >>>>>>>>>
> >>>>>>>>> /Flibble
> >>>>>>>>>
> >>>>>>>>
> >>>>>>>> But (a) and (b) aren't "pi"
> >>>>>>>
> >>>>>>> No, they are approximations of pi. The value of the
> >>>>>>> approximation changes up to a factor of 1/base as you evaluate
> >>>>>>> it at increasing accuracy.
> >>>>>>>
> >>>>>>>>
> >>>>>>>> All you are showing is that approximations to numbers get
> >>>>>>>> better as they get better, which is just a strange
> >>>>>>>> tautology.
> >>>>>>>
> >>>>>>> Yes and therefor they jump about on the number line as
> >>>>>>> accuracy increases.
> >>>>>>>
> >>>>>>>>
> >>>>>>>> The number PI, has only one precise value, the exact ratio of
> >>>>>>>> the circumference of a circle to its diameter on a flat plane
> >>>>>>>> (which will always be the same).
> >>>>>>>
> >>>>>>> pi cannot have a precise value as it neither terminates nor
> >>>>>>> has a repetend in any base.
> >>>>>>>
> >>>>>>>>
> >>>>>>>> The fact that it can't be expressed, isn't an issue on
> >>>>>>>> exactness, but of finite representation.
> >>>>>>>
> >>>>>>> We can only ever have a finite representation.
> >>>>>>>
> >>>>>>>>
> >>>>>>>> Note, that the set of numbers with finite representation is a
> >>>>>>>> countable set, so it isn't surprising that the uncountable
> >>>>>>>> infinity of the reals (that includes the irrationals) is not
> >>>>>>>> all finitely expressible.
> >>>>>>>
> >>>>>>> Stating the obvious.
> >>>>>>>
> >>>>>>> /Flibble
> >>>>>>>
> >>>>>>
> >>>>>> So you DO have problems with abstractions.
> >>>>>>
> >>>>>> The problem is, we don't actually need the finite numerical
> >>>>>> representation of a number if we have the definition of what
> >>>>>> the number is (which is another sort of abstract finite
> >>>>>> representation).
> >>>>>>
> >>>>>> The "finite representation" of pi is the ratio of the
> >>>>>> circumference to the diameter of a circle on a plane.
> >>>>>
> >>>>> No, instead if you use logic you must come to the conclusion
> >>>>> that there is no ratio of the circumference to the diameter of a
> >>>>> circle: rational numbers describe ratios, irrational numbers do
> >>>>> not. QED.
> >>>>>
> >>>>> /Flibble
> >>>>>
> >>>>
> >>>> Then you don't understand what is a ratio. Rational numbers are
> >>>> the ratio of INTEGERS, not all ratios. Ratios define the relative
> >>>> magnatude of one number to another, ANY number.
> >>>>
> >>>> Since it is clear you don't understand the meaning of the basic
> >>>> terms, your OPINION about the weightier things becomes suspect.
> >>>
> >>> Prove that there is a ratio between the circumference and diameter
> >>> of a circle. Hint: you can't.
> >>>
> >>> /Flibble
> >>>
> >>
> >> Ratio of numbers A and B is defined as A / B (assuming B is not 0)
> >>
> >> The circumference of the circle is a length of the arc all the way
> >> around the circle, so is a real number.
> >>
> >> The diameter of the circle is the length of the line from one side
> >> of the circle to the other through the center, so is a real number.
> >> Assuming the circle is not degenerate to a point, that number is
> >> not zero.
> >>
> >> Thus we have two real numbers, the second not zero, so by closure,
> >> the first divided by the second is a real number.
> >>
> >> Thus those numbers have a ratio.
> >
> > But at least one of those numbers must be irrational otherwise the
> > ratio would be a rational number ergo at least one of those numbers
> > is inexact ergo the resulting ratio between the circumference
> > and diameter of a circle must also be inexact which as an irrational
> > number it is. QED.
> >
> > /Flibble
> >
>
> You are ASSUMING that irrational numbers are inexact, and that is a
> false statement so you logic is unsound.
>
> Note, the square root of 2 can be proved exact by simple geometric
> construction, thus disproving your claim that irrational numbers are
> inexact.
>
Lets not change the subject and stick with pi shall we?
(1) You have defined a ratio to be A/B.
(2) You have defined a rational number to be the ratio
between two integers.
(3) You have defined pi as the ratio between circumference, C and
diameter, D, of a circle
(4) pi is defined to be an irrational number.
Given all of the above C, D, or both must be irrational for the
resultant ratio, pi, to also be irrational.
I cannot see a flaw in my logic. Either the circumference or the
diameter of a circle must ALWAYS be an inexact (viz. unmeasurable)
irrational number if we assert that the ratio is ALWAYS the irrational
number, pi.
/Flibble
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 16:32 -0400 |
| Message-ID | <Pp8nK.13629$Rvub.11404@fx35.iad> |
| In reply to | #51946 |
On 6/5/22 4:23 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 16:09:13 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 3:56 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 15:47:30 -0400
>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>
>>>> On 6/5/22 3:33 PM, Mr Flibble wrote:
>>>>> On Sun, 5 Jun 2022 15:30:45 -0400
>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>
>>>>>> On 6/5/22 3:17 PM, Mr Flibble wrote:
>>>>>>> On Sun, 5 Jun 2022 14:17:29 -0400
>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>
>>>>>>>> On 6/5/22 1:22 PM, Mr Flibble wrote:
>>>>>>>>> On Sun, 5 Jun 2022 13:05:09 -0400
>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>
>>>>>>>>>> On 6/5/22 12:56 PM, Mr Flibble wrote:
>>>>>>>>>>> On Sun, 5 Jun 2022 12:50:13 -0400
>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>
>>>>>>>>>>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
>>>>>>>>>>>>> On Sun, 5 Jun 2022 12:22:45 -0400
>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>>>>>>>>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>>>>>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>>>>>>>>>>>>>> exact value, but you can never actually express it
>>>>>>>>>>>>>>>>>>>> (because it takes an infinite number of digits).
>>>>>>>>>>>>>>>>>>> PI does not have an exact value; no irrational
>>>>>>>>>>>>>>>>>>> number has an exact value.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Of course "pi" has an exact value; as
>>>>>>>>>>>>>>>>>> do [eg] "sqrt(2)", "e", and all the other computable
>>>>>>>>>>>>>>>>>> real [and complex] numbers. Whether that value can be
>>>>>>>>>>>>>>>>>> expressed in finite terms in some particular
>>>>>>>>>>>>>>>>>> representation is quite another matter. That in turn
>>>>>>>>>>>>>>>>>> depends on the representation; standard decimals is
>>>>>>>>>>>>>>>>>> merely one [common] choice. Note that in symbolic
>>>>>>>>>>>>>>>>>> computer systems, those computable reals are
>>>>>>>>>>>>>>>>>> typically written "pi" [or whatever], and the
>>>>>>>>>>>>>>>>>> computer works with that exactly, so that [eg]
>>>>>>>>>>>>>>>>>> "sin^2 (pi/3) == 3/4", not 0.7499...; and also that
>>>>>>>>>>>>>>>>>> in decimal-type notations most rationals equally
>>>>>>>>>>>>>>>>>> have no terminating expansion. Numbers such as "pi"
>>>>>>>>>>>>>>>>>> and "sqrt(2)" are not defined as decimal expansions
>>>>>>>>>>>>>>>>>> but via their properties [eg that "sqrt(2)" is the
>>>>>>>>>>>>>>>>>> unique positive real whose square is 2, or
>>>>>>>>>>>>>>>>>> equivalently that it is the ratio of the diagonal of
>>>>>>>>>>>>>>>>>> a square to its side, and "pi" is the least positive
>>>>>>>>>>>>>>>>>> real whose sine is zero]. Those properties are
>>>>>>>>>>>>>>>>>> exact, and tell you all you ever need to know about
>>>>>>>>>>>>>>>>>> those numbers.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> [ .... ]
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> What has decimal (base 10) expansion got to do with
>>>>>>>>>>>>>>>>> anything? An irrational number has a non-terminating
>>>>>>>>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
>>>>>>>>>>>>>>>>> mistaken: irrational numbers do NOT have an exact
>>>>>>>>>>>>>>>>> value; this is obvious to anyone who understands
>>>>>>>>>>>>>>>>> logic and uses a sane definition for infinity.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> That irrational numbers are exact values is clear to
>>>>>>>>>>>>>>>> anybody with a degree in maths. Definitions of
>>>>>>>>>>>>>>>> "infinity" (of which there are many) have nothing to do
>>>>>>>>>>>>>>>> with this.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> You are wrong and fractally so so your degree in maths
>>>>>>>>>>>>>>> appears to be worthless. An irrational number's
>>>>>>>>>>>>>>> sequence is statistically random, has no fixed point on
>>>>>>>>>>>>>>> the number line ergo has no exact representation. Any
>>>>>>>>>>>>>>> number with no exact representation has, by definition,
>>>>>>>>>>>>>>> no exact value, only an approximation. Infinity has
>>>>>>>>>>>>>>> everything to do with this as an irrational's sequence
>>>>>>>>>>>>>>> ("digits") never terminates (i.e. it is an INFINITELY
>>>>>>>>>>>>>>> long sequence).
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> /Flibble
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Nope. Irrational numbers DO have exact points on the
>>>>>>>>>>>>>> number line.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> And what does representation have to do with exact value?
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Also, irrational numbers sequence of digits are not
>>>>>>>>>>>>>> necessarily statistically random, in some
>>>>>>>>>>>>>> representations, they can be VERY predictible for some
>>>>>>>>>>>>>> numbers.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> One simple construction to show exact position, draw a
>>>>>>>>>>>>>> box with sides exactly 1.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Draw a line though opposite corners and make one point
>>>>>>>>>>>>>> the value 0.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> The other corner will be EXACTLY at the point sqrt(2), so
>>>>>>>>>>>>>> that irrational number has an exact point on the number
>>>>>>>>>>>>>> line.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> You just don't understand what an exact value means,
>>>>>>>>>>>>>> likely because you can't understand things that are
>>>>>>>>>>>>>> somewhat abstract.
>>>>>>>>>>>>>
>>>>>>>>>>>>> An irrational number does not have an exact point on the
>>>>>>>>>>>>> number line as it will move about as you "zoom in", you
>>>>>>>>>>>>> can keep "zooming in" forever (i.e. infinitely) and it
>>>>>>>>>>>>> will keep moving about because the number never
>>>>>>>>>>>>> terminates.
>>>>>>>>>>>>>
>>>>>>>>>>>>> If I couldn't understand things that are somewhat abstract
>>>>>>>>>>>>> then I wouldn't have a computer science degree (BSc Hons)
>>>>>>>>>>>>> and 30 years of industry experience.
>>>>>>>>>>>>>
>>>>>>>>>>>>> /Flibble
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Then why do you think irrational numbers don't have an
>>>>>>>>>>>> exact location?
>>>>>>>>>>>>
>>>>>>>>>>>> I know people with degrees (even with honors) and industry
>>>>>>>>>>>> experiance that still show that they don't really
>>>>>>>>>>>> understand what they are talking about.
>>>>>>>>>>>>
>>>>>>>>>>>> The "width" of the point representing the location of an
>>>>>>>>>>>> irrational number is just as much "0" as that of a rational
>>>>>>>>>>>> number, so specifies just as exact of a location.
>>>>>>>>>>>>
>>>>>>>>>>>> The fact that we can't write it in a rational base with a
>>>>>>>>>>>> finite number of digits doesn't actally mean anything.
>>>>>>>>>>>
>>>>>>>>>>> 3.1415xxxxxxxxx (a)
>>>>>>>>>>> 3.14159xxxxxxxx (b)
>>>>>>>>>>>
>>>>>>>>>>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates
>>>>>>>>>>> hence its value "moves about" as accuracy increases.
>>>>>>>>>>>
>>>>>>>>>>> /Flibble
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> But (a) and (b) aren't "pi"
>>>>>>>>>
>>>>>>>>> No, they are approximations of pi. The value of the
>>>>>>>>> approximation changes up to a factor of 1/base as you evaluate
>>>>>>>>> it at increasing accuracy.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> All you are showing is that approximations to numbers get
>>>>>>>>>> better as they get better, which is just a strange
>>>>>>>>>> tautology.
>>>>>>>>>
>>>>>>>>> Yes and therefor they jump about on the number line as
>>>>>>>>> accuracy increases.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> The number PI, has only one precise value, the exact ratio of
>>>>>>>>>> the circumference of a circle to its diameter on a flat plane
>>>>>>>>>> (which will always be the same).
>>>>>>>>>
>>>>>>>>> pi cannot have a precise value as it neither terminates nor
>>>>>>>>> has a repetend in any base.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> The fact that it can't be expressed, isn't an issue on
>>>>>>>>>> exactness, but of finite representation.
>>>>>>>>>
>>>>>>>>> We can only ever have a finite representation.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Note, that the set of numbers with finite representation is a
>>>>>>>>>> countable set, so it isn't surprising that the uncountable
>>>>>>>>>> infinity of the reals (that includes the irrationals) is not
>>>>>>>>>> all finitely expressible.
>>>>>>>>>
>>>>>>>>> Stating the obvious.
>>>>>>>>>
>>>>>>>>> /Flibble
>>>>>>>>>
>>>>>>>>
>>>>>>>> So you DO have problems with abstractions.
>>>>>>>>
>>>>>>>> The problem is, we don't actually need the finite numerical
>>>>>>>> representation of a number if we have the definition of what
>>>>>>>> the number is (which is another sort of abstract finite
>>>>>>>> representation).
>>>>>>>>
>>>>>>>> The "finite representation" of pi is the ratio of the
>>>>>>>> circumference to the diameter of a circle on a plane.
>>>>>>>
>>>>>>> No, instead if you use logic you must come to the conclusion
>>>>>>> that there is no ratio of the circumference to the diameter of a
>>>>>>> circle: rational numbers describe ratios, irrational numbers do
>>>>>>> not. QED.
>>>>>>>
>>>>>>> /Flibble
>>>>>>>
>>>>>>
>>>>>> Then you don't understand what is a ratio. Rational numbers are
>>>>>> the ratio of INTEGERS, not all ratios. Ratios define the relative
>>>>>> magnatude of one number to another, ANY number.
>>>>>>
>>>>>> Since it is clear you don't understand the meaning of the basic
>>>>>> terms, your OPINION about the weightier things becomes suspect.
>>>>>
>>>>> Prove that there is a ratio between the circumference and diameter
>>>>> of a circle. Hint: you can't.
>>>>>
>>>>> /Flibble
>>>>>
>>>>
>>>> Ratio of numbers A and B is defined as A / B (assuming B is not 0)
>>>>
>>>> The circumference of the circle is a length of the arc all the way
>>>> around the circle, so is a real number.
>>>>
>>>> The diameter of the circle is the length of the line from one side
>>>> of the circle to the other through the center, so is a real number.
>>>> Assuming the circle is not degenerate to a point, that number is
>>>> not zero.
>>>>
>>>> Thus we have two real numbers, the second not zero, so by closure,
>>>> the first divided by the second is a real number.
>>>>
>>>> Thus those numbers have a ratio.
>>>
>>> But at least one of those numbers must be irrational otherwise the
>>> ratio would be a rational number ergo at least one of those numbers
>>> is inexact ergo the resulting ratio between the circumference
>>> and diameter of a circle must also be inexact which as an irrational
>>> number it is. QED.
>>>
>>> /Flibble
>>>
>>
>> You are ASSUMING that irrational numbers are inexact, and that is a
>> false statement so you logic is unsound.
>>
>> Note, the square root of 2 can be proved exact by simple geometric
>> construction, thus disproving your claim that irrational numbers are
>> inexact.
>>
>
> Lets not change the subject and stick with pi shall we?
>
> (1) You have defined a ratio to be A/B.
> (2) You have defined a rational number to be the ratio
> between two integers.
> (3) You have defined pi as the ratio between circumference, C and
> diameter, D, of a circle
> (4) pi is defined to be an irrational number.
>
> Given all of the above C, D, or both must be irrational for the
> resultant ratio, pi, to also be irrational.
>
> I cannot see a flaw in my logic. Either the circumference or the
> diameter of a circle must ALWAYS be an inexact (viz. unmeasurable)
> irrational number if we assert that the ratio is ALWAYS the irrational
> number, pi.
>
> /Flibble
>
Nothing wrong in deducing that at least one of them is irrational.
But that fact doesn't say they are inexact.
Being irrational doesn't make it "unmeasurable", just that its measure
is an irrational number.
Measureable doesn't mean the measure needs to be expressible as a finite
digit string, just as a real number on the number line.
(This is why [0, 1] and [0, 1) have the same measure of length, as the
value of the measure of length is 1.0, as there is no real number
exactly 1 point less.)
[toc] | [prev] | [next] | [standalone]
| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2022-06-06 16:10 +0300 |
| Message-ID | <t7kuc4$b5l$1@dont-email.me> |
| In reply to | #51946 |
On 2022-06-05 20:23:19 +0000, Mr Flibble said:
> On Sun, 5 Jun 2022 16:09:13 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 3:56 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 15:47:30 -0400
>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>
>>>> On 6/5/22 3:33 PM, Mr Flibble wrote:
>>>>> On Sun, 5 Jun 2022 15:30:45 -0400
>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>
>>>>>> On 6/5/22 3:17 PM, Mr Flibble wrote:
>>>>>>> On Sun, 5 Jun 2022 14:17:29 -0400
>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>
>>>>>>>> On 6/5/22 1:22 PM, Mr Flibble wrote:
>>>>>>>>> On Sun, 5 Jun 2022 13:05:09 -0400
>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>
>>>>>>>>>> On 6/5/22 12:56 PM, Mr Flibble wrote:
>>>>>>>>>>> On Sun, 5 Jun 2022 12:50:13 -0400
>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>
>>>>>>>>>>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
>>>>>>>>>>>>> On Sun, 5 Jun 2022 12:22:45 -0400
>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>>>>>>>>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>>>>>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>>>>>>>>>>>>>> exact value, but you can never actually express it
>>>>>>>>>>>>>>>>>>>> (because it takes an infinite number of digits).
>>>>>>>>>>>>>>>>>>> PI does not have an exact value; no irrational
>>>>>>>>>>>>>>>>>>> number has an exact value.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Of course "pi" has an exact value; as
>>>>>>>>>>>>>>>>>> do [eg] "sqrt(2)", "e", and all the other computable
>>>>>>>>>>>>>>>>>> real [and complex] numbers. Whether that value can be
>>>>>>>>>>>>>>>>>> expressed in finite terms in some particular
>>>>>>>>>>>>>>>>>> representation is quite another matter. That in turn
>>>>>>>>>>>>>>>>>> depends on the representation; standard decimals is
>>>>>>>>>>>>>>>>>> merely one [common] choice. Note that in symbolic
>>>>>>>>>>>>>>>>>> computer systems, those computable reals are
>>>>>>>>>>>>>>>>>> typically written "pi" [or whatever], and the
>>>>>>>>>>>>>>>>>> computer works with that exactly, so that [eg]
>>>>>>>>>>>>>>>>>> "sin^2 (pi/3) == 3/4", not 0.7499...; and also that
>>>>>>>>>>>>>>>>>> in decimal-type notations most rationals equally
>>>>>>>>>>>>>>>>>> have no terminating expansion. Numbers such as "pi"
>>>>>>>>>>>>>>>>>> and "sqrt(2)" are not defined as decimal expansions
>>>>>>>>>>>>>>>>>> but via their properties [eg that "sqrt(2)" is the
>>>>>>>>>>>>>>>>>> unique positive real whose square is 2, or
>>>>>>>>>>>>>>>>>> equivalently that it is the ratio of the diagonal of
>>>>>>>>>>>>>>>>>> a square to its side, and "pi" is the least positive
>>>>>>>>>>>>>>>>>> real whose sine is zero]. Those properties are
>>>>>>>>>>>>>>>>>> exact, and tell you all you ever need to know about
>>>>>>>>>>>>>>>>>> those numbers.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> [ .... ]
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> What has decimal (base 10) expansion got to do with
>>>>>>>>>>>>>>>>> anything? An irrational number has a non-terminating
>>>>>>>>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
>>>>>>>>>>>>>>>>> mistaken: irrational numbers do NOT have an exact
>>>>>>>>>>>>>>>>> value; this is obvious to anyone who understands
>>>>>>>>>>>>>>>>> logic and uses a sane definition for infinity.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> That irrational numbers are exact values is clear to
>>>>>>>>>>>>>>>> anybody with a degree in maths. Definitions of
>>>>>>>>>>>>>>>> "infinity" (of which there are many) have nothing to do
>>>>>>>>>>>>>>>> with this.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> You are wrong and fractally so so your degree in maths
>>>>>>>>>>>>>>> appears to be worthless. An irrational number's
>>>>>>>>>>>>>>> sequence is statistically random, has no fixed point on
>>>>>>>>>>>>>>> the number line ergo has no exact representation. Any
>>>>>>>>>>>>>>> number with no exact representation has, by definition,
>>>>>>>>>>>>>>> no exact value, only an approximation. Infinity has
>>>>>>>>>>>>>>> everything to do with this as an irrational's sequence
>>>>>>>>>>>>>>> ("digits") never terminates (i.e. it is an INFINITELY
>>>>>>>>>>>>>>> long sequence).
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> /Flibble
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Nope. Irrational numbers DO have exact points on the
>>>>>>>>>>>>>> number line.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> And what does representation have to do with exact value?
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Also, irrational numbers sequence of digits are not
>>>>>>>>>>>>>> necessarily statistically random, in some
>>>>>>>>>>>>>> representations, they can be VERY predictible for some
>>>>>>>>>>>>>> numbers.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> One simple construction to show exact position, draw a
>>>>>>>>>>>>>> box with sides exactly 1.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Draw a line though opposite corners and make one point
>>>>>>>>>>>>>> the value 0.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> The other corner will be EXACTLY at the point sqrt(2), so
>>>>>>>>>>>>>> that irrational number has an exact point on the number
>>>>>>>>>>>>>> line.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> You just don't understand what an exact value means,
>>>>>>>>>>>>>> likely because you can't understand things that are
>>>>>>>>>>>>>> somewhat abstract.
>>>>>>>>>>>>>
>>>>>>>>>>>>> An irrational number does not have an exact point on the
>>>>>>>>>>>>> number line as it will move about as you "zoom in", you
>>>>>>>>>>>>> can keep "zooming in" forever (i.e. infinitely) and it
>>>>>>>>>>>>> will keep moving about because the number never
>>>>>>>>>>>>> terminates.
>>>>>>>>>>>>>
>>>>>>>>>>>>> If I couldn't understand things that are somewhat abstract
>>>>>>>>>>>>> then I wouldn't have a computer science degree (BSc Hons)
>>>>>>>>>>>>> and 30 years of industry experience.
>>>>>>>>>>>>>
>>>>>>>>>>>>> /Flibble
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Then why do you think irrational numbers don't have an
>>>>>>>>>>>> exact location?
>>>>>>>>>>>>
>>>>>>>>>>>> I know people with degrees (even with honors) and industry
>>>>>>>>>>>> experiance that still show that they don't really
>>>>>>>>>>>> understand what they are talking about.
>>>>>>>>>>>>
>>>>>>>>>>>> The "width" of the point representing the location of an
>>>>>>>>>>>> irrational number is just as much "0" as that of a rational
>>>>>>>>>>>> number, so specifies just as exact of a location.
>>>>>>>>>>>>
>>>>>>>>>>>> The fact that we can't write it in a rational base with a
>>>>>>>>>>>> finite number of digits doesn't actally mean anything.
>>>>>>>>>>>
>>>>>>>>>>> 3.1415xxxxxxxxx (a)
>>>>>>>>>>> 3.14159xxxxxxxx (b)
>>>>>>>>>>>
>>>>>>>>>>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates
>>>>>>>>>>> hence its value "moves about" as accuracy increases.
>>>>>>>>>>>
>>>>>>>>>>> /Flibble
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> But (a) and (b) aren't "pi"
>>>>>>>>>
>>>>>>>>> No, they are approximations of pi. The value of the
>>>>>>>>> approximation changes up to a factor of 1/base as you evaluate
>>>>>>>>> it at increasing accuracy.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> All you are showing is that approximations to numbers get
>>>>>>>>>> better as they get better, which is just a strange
>>>>>>>>>> tautology.
>>>>>>>>>
>>>>>>>>> Yes and therefor they jump about on the number line as
>>>>>>>>> accuracy increases.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> The number PI, has only one precise value, the exact ratio of
>>>>>>>>>> the circumference of a circle to its diameter on a flat plane
>>>>>>>>>> (which will always be the same).
>>>>>>>>>
>>>>>>>>> pi cannot have a precise value as it neither terminates nor
>>>>>>>>> has a repetend in any base.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> The fact that it can't be expressed, isn't an issue on
>>>>>>>>>> exactness, but of finite representation.
>>>>>>>>>
>>>>>>>>> We can only ever have a finite representation.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Note, that the set of numbers with finite representation is a
>>>>>>>>>> countable set, so it isn't surprising that the uncountable
>>>>>>>>>> infinity of the reals (that includes the irrationals) is not
>>>>>>>>>> all finitely expressible.
>>>>>>>>>
>>>>>>>>> Stating the obvious.
>>>>>>>>>
>>>>>>>>> /Flibble
>>>>>>>>>
>>>>>>>>
>>>>>>>> So you DO have problems with abstractions.
>>>>>>>>
>>>>>>>> The problem is, we don't actually need the finite numerical
>>>>>>>> representation of a number if we have the definition of what
>>>>>>>> the number is (which is another sort of abstract finite
>>>>>>>> representation).
>>>>>>>>
>>>>>>>> The "finite representation" of pi is the ratio of the
>>>>>>>> circumference to the diameter of a circle on a plane.
>>>>>>>
>>>>>>> No, instead if you use logic you must come to the conclusion
>>>>>>> that there is no ratio of the circumference to the diameter of a
>>>>>>> circle: rational numbers describe ratios, irrational numbers do
>>>>>>> not. QED.
>>>>>>>
>>>>>>> /Flibble
>>>>>>>
>>>>>>
>>>>>> Then you don't understand what is a ratio. Rational numbers are
>>>>>> the ratio of INTEGERS, not all ratios. Ratios define the relative
>>>>>> magnatude of one number to another, ANY number.
>>>>>>
>>>>>> Since it is clear you don't understand the meaning of the basic
>>>>>> terms, your OPINION about the weightier things becomes suspect.
>>>>>
>>>>> Prove that there is a ratio between the circumference and diameter
>>>>> of a circle. Hint: you can't.
>>>>>
>>>>> /Flibble
>>>>>
>>>>
>>>> Ratio of numbers A and B is defined as A / B (assuming B is not 0)
>>>>
>>>> The circumference of the circle is a length of the arc all the way
>>>> around the circle, so is a real number.
>>>>
>>>> The diameter of the circle is the length of the line from one side
>>>> of the circle to the other through the center, so is a real number.
>>>> Assuming the circle is not degenerate to a point, that number is
>>>> not zero.
>>>>
>>>> Thus we have two real numbers, the second not zero, so by closure,
>>>> the first divided by the second is a real number.
>>>>
>>>> Thus those numbers have a ratio.
>>>
>>> But at least one of those numbers must be irrational otherwise the
>>> ratio would be a rational number ergo at least one of those numbers
>>> is inexact ergo the resulting ratio between the circumference
>>> and diameter of a circle must also be inexact which as an irrational
>>> number it is. QED.
>>>
>>> /Flibble
>>>
>>
>> You are ASSUMING that irrational numbers are inexact, and that is a
>> false statement so you logic is unsound.
>>
>> Note, the square root of 2 can be proved exact by simple geometric
>> construction, thus disproving your claim that irrational numbers are
>> inexact.
>>
>
> Lets not change the subject and stick with pi shall we?
>
> (1) You have defined a ratio to be A/B.
> (2) You have defined a rational number to be the ratio
> between two integers.
> (3) You have defined pi as the ratio between circumference, C and
> diameter, D, of a circle
> (4) pi is defined to be an irrational number.
No, pi is not defined to be an irrational number. It is proven to be
irrational, i.e., it is proven that every rational number is either
less than or greater than pi.
> Given all of the above C, D, or both must be irrational for the
> resultant ratio, pi, to also be irrational.
If C and D are numbers. Not if they are just lengths. The question
"is the distance from P to Q rational" is meaningless unless one
postulates that distances are numbers.
Mikko
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| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-06 17:47 +0100 |
| Message-ID | <20220606174712.000039f0@reddwarf.jmc> |
| In reply to | #51990 |
On Mon, 6 Jun 2022 16:10:28 +0300
Mikko <mikko.levanto@iki.fi> wrote:
> On 2022-06-05 20:23:19 +0000, Mr Flibble said:
>
> > On Sun, 5 Jun 2022 16:09:13 -0400
> > Richard Damon <Richard@Damon-Family.org> wrote:
> >
> >> On 6/5/22 3:56 PM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 15:47:30 -0400
> >>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>
> >>>> On 6/5/22 3:33 PM, Mr Flibble wrote:
> >>>>> On Sun, 5 Jun 2022 15:30:45 -0400
> >>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>
> >>>>>> On 6/5/22 3:17 PM, Mr Flibble wrote:
> >>>>>>> On Sun, 5 Jun 2022 14:17:29 -0400
> >>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>
> >>>>>>>> On 6/5/22 1:22 PM, Mr Flibble wrote:
> >>>>>>>>> On Sun, 5 Jun 2022 13:05:09 -0400
> >>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>
> >>>>>>>>>> On 6/5/22 12:56 PM, Mr Flibble wrote:
> >>>>>>>>>>> On Sun, 5 Jun 2022 12:50:13 -0400
> >>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>
> >>>>>>>>>>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
> >>>>>>>>>>>>> On Sun, 5 Jun 2022 12:22:45 -0400
> >>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
> >>>>>>>>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> >>>>>>>>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>>>>>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>>>>>>>>>>>>>> exact value, but you can never actually express
> >>>>>>>>>>>>>>>>>>>> it (because it takes an infinite number of
> >>>>>>>>>>>>>>>>>>>> digits).
> >>>>>>>>>>>>>>>>>>> PI does not have an exact value; no irrational
> >>>>>>>>>>>>>>>>>>> number has an exact value.
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> Of course "pi" has an exact value; as
> >>>>>>>>>>>>>>>>>> do [eg] "sqrt(2)", "e", and all the other
> >>>>>>>>>>>>>>>>>> computable real [and complex] numbers. Whether
> >>>>>>>>>>>>>>>>>> that value can be expressed in finite terms in
> >>>>>>>>>>>>>>>>>> some particular representation is quite another
> >>>>>>>>>>>>>>>>>> matter. That in turn depends on the
> >>>>>>>>>>>>>>>>>> representation; standard decimals is merely one
> >>>>>>>>>>>>>>>>>> [common] choice. Note that in symbolic computer
> >>>>>>>>>>>>>>>>>> systems, those computable reals are typically
> >>>>>>>>>>>>>>>>>> written "pi" [or whatever], and the computer works
> >>>>>>>>>>>>>>>>>> with that exactly, so that [eg] "sin^2 (pi/3) ==
> >>>>>>>>>>>>>>>>>> 3/4", not 0.7499...; and also that in decimal-type
> >>>>>>>>>>>>>>>>>> notations most rationals equally have no
> >>>>>>>>>>>>>>>>>> terminating expansion. Numbers such as "pi" and
> >>>>>>>>>>>>>>>>>> "sqrt(2)" are not defined as decimal expansions
> >>>>>>>>>>>>>>>>>> but via their properties [eg that "sqrt(2)" is the
> >>>>>>>>>>>>>>>>>> unique positive real whose square is 2, or
> >>>>>>>>>>>>>>>>>> equivalently that it is the ratio of the diagonal
> >>>>>>>>>>>>>>>>>> of a square to its side, and "pi" is the least
> >>>>>>>>>>>>>>>>>> positive real whose sine is zero]. Those
> >>>>>>>>>>>>>>>>>> properties are exact, and tell you all you ever
> >>>>>>>>>>>>>>>>>> need to know about those numbers.
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> [ .... ]
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> What has decimal (base 10) expansion got to do with
> >>>>>>>>>>>>>>>>> anything? An irrational number has a non-terminating
> >>>>>>>>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
> >>>>>>>>>>>>>>>>> mistaken: irrational numbers do NOT have an exact
> >>>>>>>>>>>>>>>>> value; this is obvious to anyone who understands
> >>>>>>>>>>>>>>>>> logic and uses a sane definition for infinity.
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> That irrational numbers are exact values is clear to
> >>>>>>>>>>>>>>>> anybody with a degree in maths. Definitions of
> >>>>>>>>>>>>>>>> "infinity" (of which there are many) have nothing to
> >>>>>>>>>>>>>>>> do with this.
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> You are wrong and fractally so so your degree in maths
> >>>>>>>>>>>>>>> appears to be worthless. An irrational number's
> >>>>>>>>>>>>>>> sequence is statistically random, has no fixed point
> >>>>>>>>>>>>>>> on the number line ergo has no exact representation.
> >>>>>>>>>>>>>>> Any number with no exact representation has, by
> >>>>>>>>>>>>>>> definition, no exact value, only an approximation.
> >>>>>>>>>>>>>>> Infinity has everything to do with this as an
> >>>>>>>>>>>>>>> irrational's sequence ("digits") never terminates
> >>>>>>>>>>>>>>> (i.e. it is an INFINITELY long sequence).
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> /Flibble
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Nope. Irrational numbers DO have exact points on the
> >>>>>>>>>>>>>> number line.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> And what does representation have to do with exact
> >>>>>>>>>>>>>> value?
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Also, irrational numbers sequence of digits are not
> >>>>>>>>>>>>>> necessarily statistically random, in some
> >>>>>>>>>>>>>> representations, they can be VERY predictible for some
> >>>>>>>>>>>>>> numbers.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> One simple construction to show exact position, draw a
> >>>>>>>>>>>>>> box with sides exactly 1.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Draw a line though opposite corners and make one point
> >>>>>>>>>>>>>> the value 0.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> The other corner will be EXACTLY at the point sqrt(2),
> >>>>>>>>>>>>>> so that irrational number has an exact point on the
> >>>>>>>>>>>>>> number line.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> You just don't understand what an exact value means,
> >>>>>>>>>>>>>> likely because you can't understand things that are
> >>>>>>>>>>>>>> somewhat abstract.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> An irrational number does not have an exact point on the
> >>>>>>>>>>>>> number line as it will move about as you "zoom in", you
> >>>>>>>>>>>>> can keep "zooming in" forever (i.e. infinitely) and it
> >>>>>>>>>>>>> will keep moving about because the number never
> >>>>>>>>>>>>> terminates.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> If I couldn't understand things that are somewhat
> >>>>>>>>>>>>> abstract then I wouldn't have a computer science degree
> >>>>>>>>>>>>> (BSc Hons) and 30 years of industry experience.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> /Flibble
> >>>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>> Then why do you think irrational numbers don't have an
> >>>>>>>>>>>> exact location?
> >>>>>>>>>>>>
> >>>>>>>>>>>> I know people with degrees (even with honors) and
> >>>>>>>>>>>> industry experiance that still show that they don't
> >>>>>>>>>>>> really understand what they are talking about.
> >>>>>>>>>>>>
> >>>>>>>>>>>> The "width" of the point representing the location of an
> >>>>>>>>>>>> irrational number is just as much "0" as that of a
> >>>>>>>>>>>> rational number, so specifies just as exact of a
> >>>>>>>>>>>> location.
> >>>>>>>>>>>>
> >>>>>>>>>>>> The fact that we can't write it in a rational base with a
> >>>>>>>>>>>> finite number of digits doesn't actally mean anything.
> >>>>>>>>>>>
> >>>>>>>>>>> 3.1415xxxxxxxxx (a)
> >>>>>>>>>>> 3.14159xxxxxxxx (b)
> >>>>>>>>>>>
> >>>>>>>>>>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates
> >>>>>>>>>>> hence its value "moves about" as accuracy increases.
> >>>>>>>>>>>
> >>>>>>>>>>> /Flibble
> >>>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> But (a) and (b) aren't "pi"
> >>>>>>>>>
> >>>>>>>>> No, they are approximations of pi. The value of the
> >>>>>>>>> approximation changes up to a factor of 1/base as you
> >>>>>>>>> evaluate it at increasing accuracy.
> >>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> All you are showing is that approximations to numbers get
> >>>>>>>>>> better as they get better, which is just a strange
> >>>>>>>>>> tautology.
> >>>>>>>>>
> >>>>>>>>> Yes and therefor they jump about on the number line as
> >>>>>>>>> accuracy increases.
> >>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> The number PI, has only one precise value, the exact ratio
> >>>>>>>>>> of the circumference of a circle to its diameter on a flat
> >>>>>>>>>> plane (which will always be the same).
> >>>>>>>>>
> >>>>>>>>> pi cannot have a precise value as it neither terminates nor
> >>>>>>>>> has a repetend in any base.
> >>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> The fact that it can't be expressed, isn't an issue on
> >>>>>>>>>> exactness, but of finite representation.
> >>>>>>>>>
> >>>>>>>>> We can only ever have a finite representation.
> >>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> Note, that the set of numbers with finite representation
> >>>>>>>>>> is a countable set, so it isn't surprising that the
> >>>>>>>>>> uncountable infinity of the reals (that includes the
> >>>>>>>>>> irrationals) is not all finitely expressible.
> >>>>>>>>>
> >>>>>>>>> Stating the obvious.
> >>>>>>>>>
> >>>>>>>>> /Flibble
> >>>>>>>>>
> >>>>>>>>
> >>>>>>>> So you DO have problems with abstractions.
> >>>>>>>>
> >>>>>>>> The problem is, we don't actually need the finite numerical
> >>>>>>>> representation of a number if we have the definition of what
> >>>>>>>> the number is (which is another sort of abstract finite
> >>>>>>>> representation).
> >>>>>>>>
> >>>>>>>> The "finite representation" of pi is the ratio of the
> >>>>>>>> circumference to the diameter of a circle on a plane.
> >>>>>>>
> >>>>>>> No, instead if you use logic you must come to the conclusion
> >>>>>>> that there is no ratio of the circumference to the diameter
> >>>>>>> of a circle: rational numbers describe ratios, irrational
> >>>>>>> numbers do not. QED.
> >>>>>>>
> >>>>>>> /Flibble
> >>>>>>>
> >>>>>>
> >>>>>> Then you don't understand what is a ratio. Rational numbers are
> >>>>>> the ratio of INTEGERS, not all ratios. Ratios define the
> >>>>>> relative magnatude of one number to another, ANY number.
> >>>>>>
> >>>>>> Since it is clear you don't understand the meaning of the basic
> >>>>>> terms, your OPINION about the weightier things becomes
> >>>>>> suspect.
> >>>>>
> >>>>> Prove that there is a ratio between the circumference and
> >>>>> diameter of a circle. Hint: you can't.
> >>>>>
> >>>>> /Flibble
> >>>>>
> >>>>
> >>>> Ratio of numbers A and B is defined as A / B (assuming B is not
> >>>> 0)
> >>>>
> >>>> The circumference of the circle is a length of the arc all the
> >>>> way around the circle, so is a real number.
> >>>>
> >>>> The diameter of the circle is the length of the line from one
> >>>> side of the circle to the other through the center, so is a real
> >>>> number. Assuming the circle is not degenerate to a point, that
> >>>> number is not zero.
> >>>>
> >>>> Thus we have two real numbers, the second not zero, so by
> >>>> closure, the first divided by the second is a real number.
> >>>>
> >>>> Thus those numbers have a ratio.
> >>>
> >>> But at least one of those numbers must be irrational otherwise the
> >>> ratio would be a rational number ergo at least one of those
> >>> numbers is inexact ergo the resulting ratio between the
> >>> circumference and diameter of a circle must also be inexact which
> >>> as an irrational number it is. QED.
> >>>
> >>> /Flibble
> >>>
> >>
> >> You are ASSUMING that irrational numbers are inexact, and that is a
> >> false statement so you logic is unsound.
> >>
> >> Note, the square root of 2 can be proved exact by simple geometric
> >> construction, thus disproving your claim that irrational numbers
> >> are inexact.
> >>
> >
> > Lets not change the subject and stick with pi shall we?
> >
> > (1) You have defined a ratio to be A/B.
> > (2) You have defined a rational number to be the ratio
> > between two integers.
> > (3) You have defined pi as the ratio between circumference, C and
> > diameter, D, of a circle
> > (4) pi is defined to be an irrational number.
>
> No, pi is not defined to be an irrational number. It is proven to be
> irrational, i.e., it is proven that every rational number is either
> less than or greater than pi.
>
> > Given all of the above C, D, or both must be irrational for the
> > resultant ratio, pi, to also be irrational.
>
> If C and D are numbers. Not if they are just lengths. The question
> "is the distance from P to Q rational" is meaningless unless one
> postulates that distances are numbers.
Nonsense.
/Flibble
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| From | Andy Walker <anw@cuboid.co.uk> |
|---|---|
| Date | 2022-06-05 18:44 +0100 |
| Message-ID | <t7iq18$329$1@gioia.aioe.org> |
| In reply to | #51896 |
On 05/06/2022 18:05, Richard Damon wrote:
> Note, that the set of numbers with finite representation is a
> countable set, so it isn't surprising that the uncountable infinity
> of the reals (that includes the irrationals) is not all finitely
> expressible.
Perhaps worth noting that "almost all" real numbers have no form
of expression ["are indescribable"], because sentences, whether in English
or in the form of a computer program or any similar form, are themselves
countable. Every computable number [those for which a computer program
exists], a subset of those with descriptions, has a finite representation
[eg as that computer program]; that includes irrationals such as "pi",
"sqrt(2)" and "e", and /every/ number of practical interest in physics,
applied mathematics and other similar sciences.
FWIW, Mr Flibble perpetrated his same nonsense quite recently.
Whether he is a troll or simply ill-educated in mathematics can safely
be left to the reader's imagination.
--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Palmgren
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 18:48 +0100 |
| Message-ID | <20220605184839.00002db6@reddwarf.jmc> |
| In reply to | #51904 |
On Sun, 5 Jun 2022 18:44:08 +0100 Andy Walker <anw@cuboid.co.uk> wrote: > On 05/06/2022 18:05, Richard Damon wrote: > > Note, that the set of numbers with finite representation is a > > countable set, so it isn't surprising that the uncountable infinity > > of the reals (that includes the irrationals) is not all finitely > > expressible. > > Perhaps worth noting that "almost all" real numbers have no > form of expression ["are indescribable"], because sentences, whether > in English or in the form of a computer program or any similar form, > are themselves countable. Every computable number [those for which a > computer program exists], a subset of those with descriptions, has a > finite representation [eg as that computer program]; that includes > irrationals such as "pi", "sqrt(2)" and "e", and /every/ number of > practical interest in physics, applied mathematics and other similar > sciences. > > FWIW, Mr Flibble perpetrated his same nonsense quite recently. > Whether he is a troll or simply ill-educated in mathematics can safely > be left to the reader's imagination. FWIW, your mathematics degree is worthless as you don't understand the basics; your mind has been clouded by the manipulation of symbols at the expense of logic. I wouldn't like to see your code. /Flibble
[toc] | [prev] | [next] | [standalone]
| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2022-06-05 11:29 -0500 |
| Message-ID | <JKGdnXC3u9lhRAH_nZ2dnUU7_8zNnZ2d@giganews.com> |
| In reply to | #51871 |
On 6/5/2022 11:22 AM, Richard Damon wrote:
> On 6/5/22 11:49 AM, Mr Flibble wrote:
>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>> Alan Mackenzie <acm@muc.de> wrote:
>>
>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>
>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>> exact value, but you can never actually express it (because it
>>>>>>> takes an infinite number of digits).
>>>>>> PI does not have an exact value; no irrational number has an
>>>>>> exact value.
>>>
>>>>> Â Â Â Â Â Â Of course "pi" has an exact value;Â as do [eg] "sqrt(2)",
>>>>> "e", and all the other computable real [and complex] numbers.
>>>>> Whether that value can be expressed in finite terms in some
>>>>> particular representation is quite another matter. That in turn
>>>>> depends on the representation;Â standard decimals is merely one
>>>>> [common] choice. Note that in symbolic computer systems, those
>>>>> computable reals are typically written "pi" [or whatever], and the
>>>>> computer works with that exactly, so that [eg] "sin^2 (pi/3) ==
>>>>> 3/4", not 0.7499...; and also that in decimal-type notations most
>>>>> rationals equally have no terminating expansion. Numbers such as
>>>>> "pi" and "sqrt(2)" are not defined as decimal expansions but via
>>>>> their properties [eg that "sqrt(2)" is the unique positive real
>>>>> whose square is 2, or equivalently that it is the ratio of the
>>>>> diagonal of a square to its side, and "pi" is the least positive
>>>>> real whose sine is zero]. Those properties are exact, and tell
>>>>> you all you ever need to know about those numbers.
>>>
>>> [ .... ]
>>>
>>>> What has decimal (base 10) expansion got to do with anything? An
>>>> irrational number has a non-terminating sequence in ANY base. I am
>>>> sorry but you are simply mistaken: irrational numbers do NOT have an
>>>> exact value; this is obvious to anyone who understands logic and
>>>> uses a sane definition for infinity.
>>>
>>> That irrational numbers are exact values is clear to anybody with a
>>> degree in maths. Definitions of "infinity" (of which there are many)
>>> have nothing to do with this.
>>
>> You are wrong and fractally so so your degree in maths appears to be
>> worthless. An irrational number's sequence is statistically random,
>> has no fixed point on the number line ergo has no exact representation.
>> Any number with no exact representation has, by definition, no exact
>> value, only an approximation. Infinity has everything to do with this
>> as an irrational's sequence ("digits") never terminates (i.e. it is an
>> INFINITELY long sequence).
>>
>> /Flibble
>>
>
> Nope. Irrational numbers DO have exact points on the number line.
In this we agree. Also every identifiable point on a number line has
another identifiable point that is immediately adjacent to it with no
other points inbetween.
[0,1] is exactly one geometric point longer than [0,1)
--
Copyright 2022 Pete Olcott
"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 12:53 -0400 |
| Message-ID | <Nb5nK.66929$GTEb.20082@fx48.iad> |
| In reply to | #51874 |
On 6/5/22 12:29 PM, olcott wrote:
> On 6/5/2022 11:22 AM, Richard Damon wrote:
>> On 6/5/22 11:49 AM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>>> Alan Mackenzie <acm@muc.de> wrote:
>>>
>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>
>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>> exact value, but you can never actually express it (because it
>>>>>>>> takes an infinite number of digits).
>>>>>>> PI does not have an exact value; no irrational number has an
>>>>>>> exact value.
>>>>
>>>>>> Â Â Â Â Â Â Of course "pi" has an exact value;Â as do [eg] "sqrt(2)",
>>>>>> "e", and all the other computable real [and complex] numbers.
>>>>>> Whether that value can be expressed in finite terms in some
>>>>>> particular representation is quite another matter. That in turn
>>>>>> depends on the representation;Â standard decimals is merely one
>>>>>> [common] choice. Note that in symbolic computer systems, those
>>>>>> computable reals are typically written "pi" [or whatever], and the
>>>>>> computer works with that exactly, so that [eg] "sin^2 (pi/3) ==
>>>>>> 3/4", not 0.7499...; and also that in decimal-type notations most
>>>>>> rationals equally have no terminating expansion. Numbers such as
>>>>>> "pi" and "sqrt(2)" are not defined as decimal expansions but via
>>>>>> their properties [eg that "sqrt(2)" is the unique positive real
>>>>>> whose square is 2, or equivalently that it is the ratio of the
>>>>>> diagonal of a square to its side, and "pi" is the least positive
>>>>>> real whose sine is zero]. Those properties are exact, and tell
>>>>>> you all you ever need to know about those numbers.
>>>>
>>>> [ .... ]
>>>>
>>>>> What has decimal (base 10) expansion got to do with anything? An
>>>>> irrational number has a non-terminating sequence in ANY base. I am
>>>>> sorry but you are simply mistaken: irrational numbers do NOT have an
>>>>> exact value; this is obvious to anyone who understands logic and
>>>>> uses a sane definition for infinity.
>>>>
>>>> That irrational numbers are exact values is clear to anybody with a
>>>> degree in maths. Definitions of "infinity" (of which there are many)
>>>> have nothing to do with this.
>>>
>>> You are wrong and fractally so so your degree in maths appears to be
>>> worthless. An irrational number's sequence is statistically random,
>>> has no fixed point on the number line ergo has no exact representation.
>>> Any number with no exact representation has, by definition, no exact
>>> value, only an approximation. Infinity has everything to do with this
>>> as an irrational's sequence ("digits") never terminates (i.e. it is an
>>> INFINITELY long sequence).
>>>
>>> /Flibble
>>>
>>
>> Nope. Irrational numbers DO have exact points on the number line.
>
> In this we agree. Also every identifiable point on a number line has
> another identifiable point that is immediately adjacent to it with no
> other points inbetween.
>
> [0,1] is exactly one geometric point longer than [0,1)
>
>
Well yes, there is one more point (the point 1.0000) is [0, 1] then in
[0, 1), but that doesn't make it "longer" as both lines have a
measurement lenght of "1.00" (the missing point is immeasurable in the
Reals).
[toc] | [prev] | [next] | [standalone]
| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2022-06-05 16:34 +0000 |
| Message-ID | <t7ilua$1qaq$3@news.muc.de> |
| In reply to | #51863 |
Mr Flibble <flibble@reddwarf.jmc> wrote:
> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> Alan Mackenzie <acm@muc.de> wrote:
>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>> > On Sun, 5 Jun 2022 16:28:05 +0100
>> > Andy Walker <anw@cuboid.co.uk> wrote:
>> >> On 05/06/2022 14:47, Mr Flibble wrote:
>> >> > On Sun, 5 Jun 2022 07:58:42 -0400
>> >> > Richard Damon <Richard@Damon-Family.org> wrote:
>> >> >> [...] Sort of like how the number Pi has an
>> >> >> exact value, but you can never actually express it (because it
>> >> >> takes an infinite number of digits).
>> >> > PI does not have an exact value; no irrational number has an
>> >> > exact value.
>> >> Of course "pi" has an exact value; as do [eg] "sqrt(2)",
>> >> "e", and all the other computable real [and complex] numbers.
>> >> Whether that value can be expressed in finite terms in some
>> >> particular representation is quite another matter. That in turn
>> >> depends on the representation; standard decimals is merely one
>> >> [common] choice. Note that in symbolic computer systems, those
>> >> computable reals are typically written "pi" [or whatever], and the
>> >> computer works with that exactly, so that [eg] "sin^2 (pi/3) ==
>> >> 3/4", not 0.7499...; and also that in decimal-type notations most
>> >> rationals equally have no terminating expansion. Numbers such as
>> >> "pi" and "sqrt(2)" are not defined as decimal expansions but via
>> >> their properties [eg that "sqrt(2)" is the unique positive real
>> >> whose square is 2, or equivalently that it is the ratio of the
>> >> diagonal of a square to its side, and "pi" is the least positive
>> >> real whose sine is zero]. Those properties are exact, and tell
>> >> you all you ever need to know about those numbers.
>> [ .... ]
>> > What has decimal (base 10) expansion got to do with anything? An
>> > irrational number has a non-terminating sequence in ANY base. I am
>> > sorry but you are simply mistaken: irrational numbers do NOT have
>> > an exact value; this is obvious to anyone who understands logic and
>> > uses a sane definition for infinity.
>> That irrational numbers are exact values is clear to anybody with a
>> degree in maths. Definitions of "infinity" (of which there are many)
>> have nothing to do with this.
> You are wrong and fractally so so your degree in maths appears to be
> worthless.
No, I am right, along with the world's other mathematics graduates. You
are stuck in the distant (100s of years) past, when mathematicians were
still puzzling over what you're puzzling over. The fundamentals of
maths have been worked out, and you are in the position of an alchemist
faced with modern chemistry.
> An irrational number's sequence is statistically random, has no fixed
> point on the number line ergo has no exact representation. Any number
> with no exact representation has, by definition, no exact value, only
> an approximation. Infinity has everything to do with this as an
> irrational's sequence ("digits") never terminates (i.e. it is an
> INFINITELY long sequence).
> /Flibble
--
Alan Mackenzie (Nuremberg, Germany).
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 17:38 +0100 |
| Message-ID | <20220605173844.00007fbd@reddwarf.jmc> |
| In reply to | #51877 |
On Sun, 5 Jun 2022 16:34:18 -0000 (UTC)
Alan Mackenzie <acm@muc.de> wrote:
> Mr Flibble <flibble@reddwarf.jmc> wrote:
> > On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> > Alan Mackenzie <acm@muc.de> wrote:
>
> >> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >> > On Sun, 5 Jun 2022 16:28:05 +0100
> >> > Andy Walker <anw@cuboid.co.uk> wrote:
>
> >> >> On 05/06/2022 14:47, Mr Flibble wrote:
> >> >> > On Sun, 5 Jun 2022 07:58:42 -0400
> >> >> > Richard Damon <Richard@Damon-Family.org> wrote:
> >> >> >> [...] Sort of like how the number Pi has an
> >> >> >> exact value, but you can never actually express it (because
> >> >> >> it takes an infinite number of digits).
> >> >> > PI does not have an exact value; no irrational number has an
> >> >> > exact value.
>
> >> >> Of course "pi" has an exact value; as do [eg] "sqrt(2)",
> >> >> "e", and all the other computable real [and complex] numbers.
> >> >> Whether that value can be expressed in finite terms in some
> >> >> particular representation is quite another matter. That in turn
> >> >> depends on the representation; standard decimals is merely one
> >> >> [common] choice. Note that in symbolic computer systems, those
> >> >> computable reals are typically written "pi" [or whatever], and
> >> >> the computer works with that exactly, so that [eg] "sin^2
> >> >> (pi/3) == 3/4", not 0.7499...; and also that in decimal-type
> >> >> notations most rationals equally have no terminating expansion.
> >> >> Numbers such as "pi" and "sqrt(2)" are not defined as decimal
> >> >> expansions but via their properties [eg that "sqrt(2)" is the
> >> >> unique positive real whose square is 2, or equivalently that it
> >> >> is the ratio of the diagonal of a square to its side, and "pi"
> >> >> is the least positive real whose sine is zero]. Those
> >> >> properties are exact, and tell you all you ever need to know
> >> >> about those numbers.
>
> >> [ .... ]
>
> >> > What has decimal (base 10) expansion got to do with anything? An
> >> > irrational number has a non-terminating sequence in ANY base. I
> >> > am sorry but you are simply mistaken: irrational numbers do NOT
> >> > have an exact value; this is obvious to anyone who understands
> >> > logic and uses a sane definition for infinity.
>
> >> That irrational numbers are exact values is clear to anybody with a
> >> degree in maths. Definitions of "infinity" (of which there are
> >> many) have nothing to do with this.
>
> > You are wrong and fractally so so your degree in maths appears to be
> > worthless.
>
> No, I am right, along with the world's other mathematics graduates.
> You are stuck in the distant (100s of years) past, when
> mathematicians were still puzzling over what you're puzzling over.
> The fundamentals of maths have been worked out, and you are in the
> position of an alchemist faced with modern chemistry.
Pure assertion with NOTHING to back it up.
>
> > An irrational number's sequence is statistically random, has no
> > fixed point on the number line ergo has no exact representation.
> > Any number with no exact representation has, by definition, no
> > exact value, only an approximation. Infinity has everything to do
> > with this as an irrational's sequence ("digits") never terminates
> > (i.e. it is an INFINITELY long sequence).
Ignored this part I see.
/Flibble
[toc] | [prev] | [next] | [standalone]
| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2022-06-05 11:41 -0500 |
| Message-ID | <2tidnRKUeI8-QQH_nZ2dnUU7_8xh4p2d@giganews.com> |
| In reply to | #51880 |
On 6/5/2022 11:38 AM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 16:34:18 -0000 (UTC)
> Alan Mackenzie <acm@muc.de> wrote:
>
>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>>> Alan Mackenzie <acm@muc.de> wrote:
>>
>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>
>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>> exact value, but you can never actually express it (because
>>>>>>>> it takes an infinite number of digits).
>>>>>>> PI does not have an exact value; no irrational number has an
>>>>>>> exact value.
>>
>>>>>> Of course "pi" has an exact value; as do [eg] "sqrt(2)",
>>>>>> "e", and all the other computable real [and complex] numbers.
>>>>>> Whether that value can be expressed in finite terms in some
>>>>>> particular representation is quite another matter. That in turn
>>>>>> depends on the representation; standard decimals is merely one
>>>>>> [common] choice. Note that in symbolic computer systems, those
>>>>>> computable reals are typically written "pi" [or whatever], and
>>>>>> the computer works with that exactly, so that [eg] "sin^2
>>>>>> (pi/3) == 3/4", not 0.7499...; and also that in decimal-type
>>>>>> notations most rationals equally have no terminating expansion.
>>>>>> Numbers such as "pi" and "sqrt(2)" are not defined as decimal
>>>>>> expansions but via their properties [eg that "sqrt(2)" is the
>>>>>> unique positive real whose square is 2, or equivalently that it
>>>>>> is the ratio of the diagonal of a square to its side, and "pi"
>>>>>> is the least positive real whose sine is zero]. Those
>>>>>> properties are exact, and tell you all you ever need to know
>>>>>> about those numbers.
>>
>>>> [ .... ]
>>
>>>>> What has decimal (base 10) expansion got to do with anything? An
>>>>> irrational number has a non-terminating sequence in ANY base. I
>>>>> am sorry but you are simply mistaken: irrational numbers do NOT
>>>>> have an exact value; this is obvious to anyone who understands
>>>>> logic and uses a sane definition for infinity.
>>
>>>> That irrational numbers are exact values is clear to anybody with a
>>>> degree in maths. Definitions of "infinity" (of which there are
>>>> many) have nothing to do with this.
>>
>>> You are wrong and fractally so so your degree in maths appears to be
>>> worthless.
>>
>> No, I am right, along with the world's other mathematics graduates.
>> You are stuck in the distant (100s of years) past, when
>> mathematicians were still puzzling over what you're puzzling over.
>> The fundamentals of maths have been worked out, and you are in the
>> position of an alchemist faced with modern chemistry.
>
> Pure assertion with NOTHING to back it up.
>
>>
>>> An irrational number's sequence is statistically random, has no
>>> fixed point on the number line ergo has no exact representation.
>>> Any number with no exact representation has, by definition, no
>>> exact value, only an approximation. Infinity has everything to do
>>> with this as an irrational's sequence ("digits") never terminates
>>> (i.e. it is an INFINITELY long sequence).
>
> Ignored this part I see.
>
> /Flibble
>
You are utterly clueless on these things.
The square-root of 2 does not jump around on the number line.
--
Copyright 2022 Pete Olcott
"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 17:42 +0100 |
| Message-ID | <20220605174252.0000475c@reddwarf.jmc> |
| In reply to | #51882 |
On Sun, 5 Jun 2022 11:41:05 -0500
olcott <NoOne@NoWhere.com> wrote:
> On 6/5/2022 11:38 AM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 16:34:18 -0000 (UTC)
> > Alan Mackenzie <acm@muc.de> wrote:
> >
> >> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> >>> Alan Mackenzie <acm@muc.de> wrote:
> >>
> >>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>
> >>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>> exact value, but you can never actually express it (because
> >>>>>>>> it takes an infinite number of digits).
> >>>>>>> PI does not have an exact value; no irrational number has an
> >>>>>>> exact value.
> >>
> >>>>>> Of course "pi" has an exact value; as do [eg]
> >>>>>> "sqrt(2)", "e", and all the other computable real [and
> >>>>>> complex] numbers. Whether that value can be expressed in
> >>>>>> finite terms in some particular representation is quite
> >>>>>> another matter. That in turn depends on the representation;
> >>>>>> standard decimals is merely one [common] choice. Note that in
> >>>>>> symbolic computer systems, those computable reals are
> >>>>>> typically written "pi" [or whatever], and the computer works
> >>>>>> with that exactly, so that [eg] "sin^2 (pi/3) == 3/4", not
> >>>>>> 0.7499...; and also that in decimal-type notations most
> >>>>>> rationals equally have no terminating expansion. Numbers such
> >>>>>> as "pi" and "sqrt(2)" are not defined as decimal expansions
> >>>>>> but via their properties [eg that "sqrt(2)" is the unique
> >>>>>> positive real whose square is 2, or equivalently that it is
> >>>>>> the ratio of the diagonal of a square to its side, and "pi" is
> >>>>>> the least positive real whose sine is zero]. Those properties
> >>>>>> are exact, and tell you all you ever need to know about those
> >>>>>> numbers.
> >>
> >>>> [ .... ]
> >>
> >>>>> What has decimal (base 10) expansion got to do with anything? An
> >>>>> irrational number has a non-terminating sequence in ANY base. I
> >>>>> am sorry but you are simply mistaken: irrational numbers do NOT
> >>>>> have an exact value; this is obvious to anyone who understands
> >>>>> logic and uses a sane definition for infinity.
> >>
> >>>> That irrational numbers are exact values is clear to anybody
> >>>> with a degree in maths. Definitions of "infinity" (of which
> >>>> there are many) have nothing to do with this.
> >>
> >>> You are wrong and fractally so so your degree in maths appears to
> >>> be worthless.
> >>
> >> No, I am right, along with the world's other mathematics graduates.
> >> You are stuck in the distant (100s of years) past, when
> >> mathematicians were still puzzling over what you're puzzling over.
> >> The fundamentals of maths have been worked out, and you are in the
> >> position of an alchemist faced with modern chemistry.
> >
> > Pure assertion with NOTHING to back it up.
> >
> >>
> >>> An irrational number's sequence is statistically random, has no
> >>> fixed point on the number line ergo has no exact representation.
> >>> Any number with no exact representation has, by definition, no
> >>> exact value, only an approximation. Infinity has everything to do
> >>> with this as an irrational's sequence ("digits") never terminates
> >>> (i.e. it is an INFINITELY long sequence).
> >
> > Ignored this part I see.
> >
> > /Flibble
> >
>
> You are utterly clueless on these things.
Projection.
> The square-root of 2 does not jump around on the number line.
Yes it does, all irrational numbers do.
/Flibble
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 12:54 -0400 |
| Message-ID | <3d5nK.66930$GTEb.20472@fx48.iad> |
| In reply to | #51884 |
On 6/5/22 12:42 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 11:41:05 -0500
> olcott <NoOne@NoWhere.com> wrote:
>
>> On 6/5/2022 11:38 AM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 16:34:18 -0000 (UTC)
>>> Alan Mackenzie <acm@muc.de> wrote:
>>>
>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>>>>> Alan Mackenzie <acm@muc.de> wrote:
>>>>
>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>
>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>>>> exact value, but you can never actually express it (because
>>>>>>>>>> it takes an infinite number of digits).
>>>>>>>>> PI does not have an exact value; no irrational number has an
>>>>>>>>> exact value.
>>>>
>>>>>>>> Of course "pi" has an exact value; as do [eg]
>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
>>>>>>>> complex] numbers. Whether that value can be expressed in
>>>>>>>> finite terms in some particular representation is quite
>>>>>>>> another matter. That in turn depends on the representation;
>>>>>>>> standard decimals is merely one [common] choice. Note that in
>>>>>>>> symbolic computer systems, those computable reals are
>>>>>>>> typically written "pi" [or whatever], and the computer works
>>>>>>>> with that exactly, so that [eg] "sin^2 (pi/3) == 3/4", not
>>>>>>>> 0.7499...; and also that in decimal-type notations most
>>>>>>>> rationals equally have no terminating expansion. Numbers such
>>>>>>>> as "pi" and "sqrt(2)" are not defined as decimal expansions
>>>>>>>> but via their properties [eg that "sqrt(2)" is the unique
>>>>>>>> positive real whose square is 2, or equivalently that it is
>>>>>>>> the ratio of the diagonal of a square to its side, and "pi" is
>>>>>>>> the least positive real whose sine is zero]. Those properties
>>>>>>>> are exact, and tell you all you ever need to know about those
>>>>>>>> numbers.
>>>>
>>>>>> [ .... ]
>>>>
>>>>>>> What has decimal (base 10) expansion got to do with anything? An
>>>>>>> irrational number has a non-terminating sequence in ANY base. I
>>>>>>> am sorry but you are simply mistaken: irrational numbers do NOT
>>>>>>> have an exact value; this is obvious to anyone who understands
>>>>>>> logic and uses a sane definition for infinity.
>>>>
>>>>>> That irrational numbers are exact values is clear to anybody
>>>>>> with a degree in maths. Definitions of "infinity" (of which
>>>>>> there are many) have nothing to do with this.
>>>>
>>>>> You are wrong and fractally so so your degree in maths appears to
>>>>> be worthless.
>>>>
>>>> No, I am right, along with the world's other mathematics graduates.
>>>> You are stuck in the distant (100s of years) past, when
>>>> mathematicians were still puzzling over what you're puzzling over.
>>>> The fundamentals of maths have been worked out, and you are in the
>>>> position of an alchemist faced with modern chemistry.
>>>
>>> Pure assertion with NOTHING to back it up.
>>>
>>>>
>>>>> An irrational number's sequence is statistically random, has no
>>>>> fixed point on the number line ergo has no exact representation.
>>>>> Any number with no exact representation has, by definition, no
>>>>> exact value, only an approximation. Infinity has everything to do
>>>>> with this as an irrational's sequence ("digits") never terminates
>>>>> (i.e. it is an INFINITELY long sequence).
>>>
>>> Ignored this part I see.
>>>
>>> /Flibble
>>>
>>
>> You are utterly clueless on these things.
>
> Projection.
>
>> The square-root of 2 does not jump around on the number line.
>
> Yes it does, all irrational numbers do.
>
> /Flibble
>
So, what other values does it jump to besides the actual value of sqrt(2)?
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 17:58 +0100 |
| Message-ID | <20220605175820.00002bfe@reddwarf.jmc> |
| In reply to | #51888 |
On Sun, 5 Jun 2022 12:54:22 -0400
Richard Damon <Richard@Damon-Family.org> wrote:
> On 6/5/22 12:42 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 11:41:05 -0500
> > olcott <NoOne@NoWhere.com> wrote:
> >
> >> On 6/5/2022 11:38 AM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 16:34:18 -0000 (UTC)
> >>> Alan Mackenzie <acm@muc.de> wrote:
> >>>
> >>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> >>>>> Alan Mackenzie <acm@muc.de> wrote:
> >>>>
> >>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>
> >>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>>>> exact value, but you can never actually express it (because
> >>>>>>>>>> it takes an infinite number of digits).
> >>>>>>>>> PI does not have an exact value; no irrational number has an
> >>>>>>>>> exact value.
> >>>>
> >>>>>>>> Of course "pi" has an exact value; as do [eg]
> >>>>>>>> "sqrt(2)", "e", and all the other computable real [and
> >>>>>>>> complex] numbers. Whether that value can be expressed in
> >>>>>>>> finite terms in some particular representation is quite
> >>>>>>>> another matter. That in turn depends on the representation;
> >>>>>>>> standard decimals is merely one [common] choice. Note that
> >>>>>>>> in symbolic computer systems, those computable reals are
> >>>>>>>> typically written "pi" [or whatever], and the computer works
> >>>>>>>> with that exactly, so that [eg] "sin^2 (pi/3) == 3/4", not
> >>>>>>>> 0.7499...; and also that in decimal-type notations most
> >>>>>>>> rationals equally have no terminating expansion. Numbers such
> >>>>>>>> as "pi" and "sqrt(2)" are not defined as decimal expansions
> >>>>>>>> but via their properties [eg that "sqrt(2)" is the unique
> >>>>>>>> positive real whose square is 2, or equivalently that it is
> >>>>>>>> the ratio of the diagonal of a square to its side, and "pi"
> >>>>>>>> is the least positive real whose sine is zero]. Those
> >>>>>>>> properties are exact, and tell you all you ever need to know
> >>>>>>>> about those numbers.
> >>>>
> >>>>>> [ .... ]
> >>>>
> >>>>>>> What has decimal (base 10) expansion got to do with anything?
> >>>>>>> An irrational number has a non-terminating sequence in ANY
> >>>>>>> base. I am sorry but you are simply mistaken: irrational
> >>>>>>> numbers do NOT have an exact value; this is obvious to anyone
> >>>>>>> who understands logic and uses a sane definition for
> >>>>>>> infinity.
> >>>>
> >>>>>> That irrational numbers are exact values is clear to anybody
> >>>>>> with a degree in maths. Definitions of "infinity" (of which
> >>>>>> there are many) have nothing to do with this.
> >>>>
> >>>>> You are wrong and fractally so so your degree in maths appears
> >>>>> to be worthless.
> >>>>
> >>>> No, I am right, along with the world's other mathematics
> >>>> graduates. You are stuck in the distant (100s of years) past,
> >>>> when mathematicians were still puzzling over what you're
> >>>> puzzling over. The fundamentals of maths have been worked out,
> >>>> and you are in the position of an alchemist faced with modern
> >>>> chemistry.
> >>>
> >>> Pure assertion with NOTHING to back it up.
> >>>
> >>>>
> >>>>> An irrational number's sequence is statistically random, has no
> >>>>> fixed point on the number line ergo has no exact representation.
> >>>>> Any number with no exact representation has, by definition, no
> >>>>> exact value, only an approximation. Infinity has everything to
> >>>>> do with this as an irrational's sequence ("digits") never
> >>>>> terminates (i.e. it is an INFINITELY long sequence).
> >>>
> >>> Ignored this part I see.
> >>>
> >>> /Flibble
> >>>
> >>
> >> You are utterly clueless on these things.
> >
> > Projection.
> >
> >> The square-root of 2 does not jump around on the number line.
> >
> > Yes it does, all irrational numbers do.
> >
> > /Flibble
> >
>
> So, what other values does it jump to besides the actual value of
> sqrt(2)?
It jumps to the next slightly more accurate value as the approximation
increases in accuracy.
/Flibble
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 13:07 -0400 |
| Message-ID | <cp5nK.14469$xZtb.4197@fx41.iad> |
| In reply to | #51891 |
On 6/5/22 12:58 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 12:54:22 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 12:42 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 11:41:05 -0500
>>> olcott <NoOne@NoWhere.com> wrote:
>>>
>>>> On 6/5/2022 11:38 AM, Mr Flibble wrote:
>>>>> On Sun, 5 Jun 2022 16:34:18 -0000 (UTC)
>>>>> Alan Mackenzie <acm@muc.de> wrote:
>>>>>
>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
>>>>>>
>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>>>
>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>>>>>> exact value, but you can never actually express it (because
>>>>>>>>>>>> it takes an infinite number of digits).
>>>>>>>>>>> PI does not have an exact value; no irrational number has an
>>>>>>>>>>> exact value.
>>>>>>
>>>>>>>>>> Of course "pi" has an exact value; as do [eg]
>>>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
>>>>>>>>>> complex] numbers. Whether that value can be expressed in
>>>>>>>>>> finite terms in some particular representation is quite
>>>>>>>>>> another matter. That in turn depends on the representation;
>>>>>>>>>> standard decimals is merely one [common] choice. Note that
>>>>>>>>>> in symbolic computer systems, those computable reals are
>>>>>>>>>> typically written "pi" [or whatever], and the computer works
>>>>>>>>>> with that exactly, so that [eg] "sin^2 (pi/3) == 3/4", not
>>>>>>>>>> 0.7499...; and also that in decimal-type notations most
>>>>>>>>>> rationals equally have no terminating expansion. Numbers such
>>>>>>>>>> as "pi" and "sqrt(2)" are not defined as decimal expansions
>>>>>>>>>> but via their properties [eg that "sqrt(2)" is the unique
>>>>>>>>>> positive real whose square is 2, or equivalently that it is
>>>>>>>>>> the ratio of the diagonal of a square to its side, and "pi"
>>>>>>>>>> is the least positive real whose sine is zero]. Those
>>>>>>>>>> properties are exact, and tell you all you ever need to know
>>>>>>>>>> about those numbers.
>>>>>>
>>>>>>>> [ .... ]
>>>>>>
>>>>>>>>> What has decimal (base 10) expansion got to do with anything?
>>>>>>>>> An irrational number has a non-terminating sequence in ANY
>>>>>>>>> base. I am sorry but you are simply mistaken: irrational
>>>>>>>>> numbers do NOT have an exact value; this is obvious to anyone
>>>>>>>>> who understands logic and uses a sane definition for
>>>>>>>>> infinity.
>>>>>>
>>>>>>>> That irrational numbers are exact values is clear to anybody
>>>>>>>> with a degree in maths. Definitions of "infinity" (of which
>>>>>>>> there are many) have nothing to do with this.
>>>>>>
>>>>>>> You are wrong and fractally so so your degree in maths appears
>>>>>>> to be worthless.
>>>>>>
>>>>>> No, I am right, along with the world's other mathematics
>>>>>> graduates. You are stuck in the distant (100s of years) past,
>>>>>> when mathematicians were still puzzling over what you're
>>>>>> puzzling over. The fundamentals of maths have been worked out,
>>>>>> and you are in the position of an alchemist faced with modern
>>>>>> chemistry.
>>>>>
>>>>> Pure assertion with NOTHING to back it up.
>>>>>
>>>>>>
>>>>>>> An irrational number's sequence is statistically random, has no
>>>>>>> fixed point on the number line ergo has no exact representation.
>>>>>>> Any number with no exact representation has, by definition, no
>>>>>>> exact value, only an approximation. Infinity has everything to
>>>>>>> do with this as an irrational's sequence ("digits") never
>>>>>>> terminates (i.e. it is an INFINITELY long sequence).
>>>>>
>>>>> Ignored this part I see.
>>>>>
>>>>> /Flibble
>>>>>
>>>>
>>>> You are utterly clueless on these things.
>>>
>>> Projection.
>>>
>>>> The square-root of 2 does not jump around on the number line.
>>>
>>> Yes it does, all irrational numbers do.
>>>
>>> /Flibble
>>>
>>
>> So, what other values does it jump to besides the actual value of
>> sqrt(2)?
>
> It jumps to the next slightly more accurate value as the approximation
> increases in accuracy.
>
> /Flibble
>
So, you are just claiming that APPROXIMATIONS aren't exact, that again
seems to be just a silly tautology.
Yes, approximations to a number aren't exact, but that doesn't say
anything about the number itself.
Approximations to the number 1/3 (as a decimal) aren't exact, are you
saying that 1/3 isn't an exact number?
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 18:23 +0100 |
| Message-ID | <20220605182310.00000c2c@reddwarf.jmc> |
| In reply to | #51897 |
On Sun, 5 Jun 2022 13:07:19 -0400
Richard Damon <Richard@Damon-Family.org> wrote:
> On 6/5/22 12:58 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 12:54:22 -0400
> > Richard Damon <Richard@Damon-Family.org> wrote:
> >
> >> On 6/5/22 12:42 PM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 11:41:05 -0500
> >>> olcott <NoOne@NoWhere.com> wrote:
> >>>
> >>>> On 6/5/2022 11:38 AM, Mr Flibble wrote:
> >>>>> On Sun, 5 Jun 2022 16:34:18 -0000 (UTC)
> >>>>> Alan Mackenzie <acm@muc.de> wrote:
> >>>>>
> >>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> >>>>>>> Alan Mackenzie <acm@muc.de> wrote:
> >>>>>>
> >>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>>>
> >>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>>>>>> exact value, but you can never actually express it
> >>>>>>>>>>>> (because it takes an infinite number of digits).
> >>>>>>>>>>> PI does not have an exact value; no irrational number has
> >>>>>>>>>>> an exact value.
> >>>>>>
> >>>>>>>>>> Of course "pi" has an exact value; as do [eg]
> >>>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
> >>>>>>>>>> complex] numbers. Whether that value can be expressed in
> >>>>>>>>>> finite terms in some particular representation is quite
> >>>>>>>>>> another matter. That in turn depends on the
> >>>>>>>>>> representation; standard decimals is merely one [common]
> >>>>>>>>>> choice. Note that in symbolic computer systems, those
> >>>>>>>>>> computable reals are typically written "pi" [or whatever],
> >>>>>>>>>> and the computer works with that exactly, so that [eg]
> >>>>>>>>>> "sin^2 (pi/3) == 3/4", not 0.7499...; and also that in
> >>>>>>>>>> decimal-type notations most rationals equally have no
> >>>>>>>>>> terminating expansion. Numbers such as "pi" and "sqrt(2)"
> >>>>>>>>>> are not defined as decimal expansions but via their
> >>>>>>>>>> properties [eg that "sqrt(2)" is the unique positive real
> >>>>>>>>>> whose square is 2, or equivalently that it is the ratio of
> >>>>>>>>>> the diagonal of a square to its side, and "pi" is the
> >>>>>>>>>> least positive real whose sine is zero]. Those properties
> >>>>>>>>>> are exact, and tell you all you ever need to know about
> >>>>>>>>>> those numbers.
> >>>>>>
> >>>>>>>> [ .... ]
> >>>>>>
> >>>>>>>>> What has decimal (base 10) expansion got to do with
> >>>>>>>>> anything? An irrational number has a non-terminating
> >>>>>>>>> sequence in ANY base. I am sorry but you are simply
> >>>>>>>>> mistaken: irrational numbers do NOT have an exact value;
> >>>>>>>>> this is obvious to anyone who understands logic and uses a
> >>>>>>>>> sane definition for infinity.
> >>>>>>
> >>>>>>>> That irrational numbers are exact values is clear to anybody
> >>>>>>>> with a degree in maths. Definitions of "infinity" (of which
> >>>>>>>> there are many) have nothing to do with this.
> >>>>>>
> >>>>>>> You are wrong and fractally so so your degree in maths appears
> >>>>>>> to be worthless.
> >>>>>>
> >>>>>> No, I am right, along with the world's other mathematics
> >>>>>> graduates. You are stuck in the distant (100s of years) past,
> >>>>>> when mathematicians were still puzzling over what you're
> >>>>>> puzzling over. The fundamentals of maths have been worked out,
> >>>>>> and you are in the position of an alchemist faced with modern
> >>>>>> chemistry.
> >>>>>
> >>>>> Pure assertion with NOTHING to back it up.
> >>>>>
> >>>>>>
> >>>>>>> An irrational number's sequence is statistically random, has
> >>>>>>> no fixed point on the number line ergo has no exact
> >>>>>>> representation. Any number with no exact representation has,
> >>>>>>> by definition, no exact value, only an approximation.
> >>>>>>> Infinity has everything to do with this as an irrational's
> >>>>>>> sequence ("digits") never terminates (i.e. it is an
> >>>>>>> INFINITELY long sequence).
> >>>>>
> >>>>> Ignored this part I see.
> >>>>>
> >>>>> /Flibble
> >>>>>
> >>>>
> >>>> You are utterly clueless on these things.
> >>>
> >>> Projection.
> >>>
> >>>> The square-root of 2 does not jump around on the number line.
> >>>
> >>> Yes it does, all irrational numbers do.
> >>>
> >>> /Flibble
> >>>
> >>
> >> So, what other values does it jump to besides the actual value of
> >> sqrt(2)?
> >
> > It jumps to the next slightly more accurate value as the
> > approximation increases in accuracy.
> >
> > /Flibble
> >
>
> So, you are just claiming that APPROXIMATIONS aren't exact, that
> again seems to be just a silly tautology.
>
> Yes, approximations to a number aren't exact, but that doesn't say
> anything about the number itself.
>
> Approximations to the number 1/3 (as a decimal) aren't exact, are you
> saying that 1/3 isn't an exact number?
1/3 is an exact number as it either terminates or has a repetend in
some base.
/Flibble
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 14:20 -0400 |
| Message-ID | <_t6nK.40213$ssF.4365@fx14.iad> |
| In reply to | #51901 |
On 6/5/22 1:23 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 13:07:19 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 12:58 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 12:54:22 -0400
>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>
>>>> On 6/5/22 12:42 PM, Mr Flibble wrote:
>>>>> On Sun, 5 Jun 2022 11:41:05 -0500
>>>>> olcott <NoOne@NoWhere.com> wrote:
>>>>>
>>>>>> On 6/5/2022 11:38 AM, Mr Flibble wrote:
>>>>>>> On Sun, 5 Jun 2022 16:34:18 -0000 (UTC)
>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
>>>>>>>
>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
>>>>>>>>
>>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>>>>>
>>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>>>>>>>> exact value, but you can never actually express it
>>>>>>>>>>>>>> (because it takes an infinite number of digits).
>>>>>>>>>>>>> PI does not have an exact value; no irrational number has
>>>>>>>>>>>>> an exact value.
>>>>>>>>
>>>>>>>>>>>> Of course "pi" has an exact value; as do [eg]
>>>>>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
>>>>>>>>>>>> complex] numbers. Whether that value can be expressed in
>>>>>>>>>>>> finite terms in some particular representation is quite
>>>>>>>>>>>> another matter. That in turn depends on the
>>>>>>>>>>>> representation; standard decimals is merely one [common]
>>>>>>>>>>>> choice. Note that in symbolic computer systems, those
>>>>>>>>>>>> computable reals are typically written "pi" [or whatever],
>>>>>>>>>>>> and the computer works with that exactly, so that [eg]
>>>>>>>>>>>> "sin^2 (pi/3) == 3/4", not 0.7499...; and also that in
>>>>>>>>>>>> decimal-type notations most rationals equally have no
>>>>>>>>>>>> terminating expansion. Numbers such as "pi" and "sqrt(2)"
>>>>>>>>>>>> are not defined as decimal expansions but via their
>>>>>>>>>>>> properties [eg that "sqrt(2)" is the unique positive real
>>>>>>>>>>>> whose square is 2, or equivalently that it is the ratio of
>>>>>>>>>>>> the diagonal of a square to its side, and "pi" is the
>>>>>>>>>>>> least positive real whose sine is zero]. Those properties
>>>>>>>>>>>> are exact, and tell you all you ever need to know about
>>>>>>>>>>>> those numbers.
>>>>>>>>
>>>>>>>>>> [ .... ]
>>>>>>>>
>>>>>>>>>>> What has decimal (base 10) expansion got to do with
>>>>>>>>>>> anything? An irrational number has a non-terminating
>>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
>>>>>>>>>>> mistaken: irrational numbers do NOT have an exact value;
>>>>>>>>>>> this is obvious to anyone who understands logic and uses a
>>>>>>>>>>> sane definition for infinity.
>>>>>>>>
>>>>>>>>>> That irrational numbers are exact values is clear to anybody
>>>>>>>>>> with a degree in maths. Definitions of "infinity" (of which
>>>>>>>>>> there are many) have nothing to do with this.
>>>>>>>>
>>>>>>>>> You are wrong and fractally so so your degree in maths appears
>>>>>>>>> to be worthless.
>>>>>>>>
>>>>>>>> No, I am right, along with the world's other mathematics
>>>>>>>> graduates. You are stuck in the distant (100s of years) past,
>>>>>>>> when mathematicians were still puzzling over what you're
>>>>>>>> puzzling over. The fundamentals of maths have been worked out,
>>>>>>>> and you are in the position of an alchemist faced with modern
>>>>>>>> chemistry.
>>>>>>>
>>>>>>> Pure assertion with NOTHING to back it up.
>>>>>>>
>>>>>>>>
>>>>>>>>> An irrational number's sequence is statistically random, has
>>>>>>>>> no fixed point on the number line ergo has no exact
>>>>>>>>> representation. Any number with no exact representation has,
>>>>>>>>> by definition, no exact value, only an approximation.
>>>>>>>>> Infinity has everything to do with this as an irrational's
>>>>>>>>> sequence ("digits") never terminates (i.e. it is an
>>>>>>>>> INFINITELY long sequence).
>>>>>>>
>>>>>>> Ignored this part I see.
>>>>>>>
>>>>>>> /Flibble
>>>>>>>
>>>>>>
>>>>>> You are utterly clueless on these things.
>>>>>
>>>>> Projection.
>>>>>
>>>>>> The square-root of 2 does not jump around on the number line.
>>>>>
>>>>> Yes it does, all irrational numbers do.
>>>>>
>>>>> /Flibble
>>>>>
>>>>
>>>> So, what other values does it jump to besides the actual value of
>>>> sqrt(2)?
>>>
>>> It jumps to the next slightly more accurate value as the
>>> approximation increases in accuracy.
>>>
>>> /Flibble
>>>
>>
>> So, you are just claiming that APPROXIMATIONS aren't exact, that
>> again seems to be just a silly tautology.
>>
>> Yes, approximations to a number aren't exact, but that doesn't say
>> anything about the number itself.
>>
>> Approximations to the number 1/3 (as a decimal) aren't exact, are you
>> saying that 1/3 isn't an exact number?
>
> 1/3 is an exact number as it either terminates or has a repetend in
> some base.
>
> /Flibble
>
But "repeating" wasn't factor. You were looking at sequnces of numbers
to N digits, the fact that for 1/3 the pattern is predictable doesn't
change the fact that the approximations are of different values.
[toc] | [prev] | [next] | [standalone]
| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2022-06-05 17:04 +0000 |
| Message-ID | <t7inmq$1qaq$4@news.muc.de> |
| In reply to | #51880 |
Mr Flibble <flibble@reddwarf.jmc> wrote:
> On Sun, 5 Jun 2022 16:34:18 -0000 (UTC)
> Alan Mackenzie <acm@muc.de> wrote:
>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>> > On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>> > Alan Mackenzie <acm@muc.de> wrote:
>> >> Mr Flibble <flibble@reddwarf.jmc> wrote:
>> >> > On Sun, 5 Jun 2022 16:28:05 +0100
>> >> > Andy Walker <anw@cuboid.co.uk> wrote:
>> >> >> On 05/06/2022 14:47, Mr Flibble wrote:
>> >> >> > On Sun, 5 Jun 2022 07:58:42 -0400
>> >> >> > Richard Damon <Richard@Damon-Family.org> wrote:
>> >> >> >> [...] Sort of like how the number Pi has an
>> >> >> >> exact value, but you can never actually express it (because
>> >> >> >> it takes an infinite number of digits).
>> >> >> > PI does not have an exact value; no irrational number has an
>> >> >> > exact value.
>> >> >> Of course "pi" has an exact value; as do [eg] "sqrt(2)", "e",
>> >> >> and all the other computable real [and complex] numbers.
>> >> >> Whether that value can be expressed in finite terms in some
>> >> >> particular representation is quite another matter. That in turn
>> >> >> depends on the representation; standard decimals is merely one
>> >> >> [common] choice. Note that in symbolic computer systems, those
>> >> >> computable reals are typically written "pi" [or whatever], and
>> >> >> the computer works with that exactly, so that [eg] "sin^2 (pi/3)
>> >> >> == 3/4", not 0.7499...; and also that in decimal-type notations
>> >> >> most rationals equally have no terminating expansion. Numbers
>> >> >> such as "pi" and "sqrt(2)" are not defined as decimal expansions
>> >> >> but via their properties [eg that "sqrt(2)" is the unique
>> >> >> positive real whose square is 2, or equivalently that it is the
>> >> >> ratio of the diagonal of a square to its side, and "pi" is the
>> >> >> least positive real whose sine is zero]. Those properties are
>> >> >> exact, and tell you all you ever need to know about those
>> >> >> numbers.
>> >> [ .... ]
>> >> > What has decimal (base 10) expansion got to do with anything? An
>> >> > irrational number has a non-terminating sequence in ANY base. I
>> >> > am sorry but you are simply mistaken: irrational numbers do NOT
>> >> > have an exact value; this is obvious to anyone who understands
>> >> > logic and uses a sane definition for infinity.
>> >> That irrational numbers are exact values is clear to anybody with a
>> >> degree in maths. Definitions of "infinity" (of which there are
>> >> many) have nothing to do with this.
>> > You are wrong and fractally so so your degree in maths appears to be
>> > worthless.
>> No, I am right, along with the world's other mathematics graduates.
>> You are stuck in the distant (100s of years) past, when mathematicians
>> were still puzzling over what you're puzzling over. The fundamentals
>> of maths have been worked out, and you are in the position of an
>> alchemist faced with modern chemistry.
> Pure assertion with NOTHING to back it up.
I have several years of hard study to back it up, combined with the
authority of the world's mathematicians. If you were to insist the world
were flat rather than roughly a sphere, and I were to correct you, I
would likewise have "nothing to back it up". But you would still be
wrong. Why do you have so little respect for education?
You make the mistake of thinking that because everybody is free to hold
opinions, everybody's opinion is equally valid. This is untrue -
experts' expertise is worth far more than the opinionated peoples'
opinions.
>> > An irrational number's sequence is statistically random, has no
>> > fixed point on the number line ergo has no exact representation.
>> > Any number with no exact representation has, by definition, no
>> > exact value, only an approximation. Infinity has everything to do
>> > with this as an irrational's sequence ("digits") never terminates
>> > (i.e. it is an INFINITELY long sequence).
> Ignored this part I see.
Yes. It is so full of errors, misunderstandings of definitions, and
general lack of education, it is hardly worth countering. You don't know
what a real number is, for example. You don't know the axioms by which
the properties of numbers have been established. You don't know how
these numbers are constructed. And were I to spend several weeks and
months trying to get these points across to you, in the end you might
well just chose to stay ignorant. I've got better things to do with my
time.
> /Flibble
--
Alan Mackenzie (Nuremberg, Germany).
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 12:17 -0400 |
| Message-ID | <MG4nK.40209$ssF.1755@fx14.iad> |
| In reply to | #51861 |
On 6/5/22 11:34 AM, Mr Flibble wrote: > On Sun, 5 Jun 2022 16:28:05 +0100 > Andy Walker <anw@cuboid.co.uk> wrote: > >> On 05/06/2022 14:47, Mr Flibble wrote: >>> On Sun, 5 Jun 2022 07:58:42 -0400 >>> Richard Damon <Richard@Damon-Family.org> wrote: >>>> [...] Sort of like how the number Pi has an >>>> exact value, but you can never actually express it (because it >>>> takes an infinite number of digits). >>> PI does not have an exact value; no irrational number has an exact >>> value. >> >> Of course "pi" has an exact value; as do [eg] "sqrt(2)", >> "e", and all the other computable real [and complex] numbers. >> Whether that value can be expressed in finite terms in some >> particular representation is quite another matter. That in turn >> depends on the representation; standard decimals is merely one >> [common] choice. Note that in symbolic computer systems, those >> computable reals are typically written "pi" [or whatever], and the >> computer works with that exactly, so that [eg] "sin^2 (pi/3) == 3/4", >> not 0.7499...; and also that in decimal-type notations most rationals >> equally have no terminating expansion. Numbers such as "pi" and >> "sqrt(2)" are not defined as decimal expansions but via their >> properties [eg that "sqrt(2)" is the unique positive real whose >> square is 2, or equivalently that it is the ratio of the diagonal of >> a square to its side, and "pi" is the least positive real whose sine >> is zero]. Those properties are exact, and tell you all you ever need >> to know about those numbers. >> >> [I have removed my name from the "Subject:"; I don't know why >> anyone saw fit to attach it to this debate, such as it is, on the HP.] > > What has decimal (base 10) expansion got to do with anything? An > irrational number has a non-terminating sequence in ANY base. I am > sorry but you are simply mistaken: irrational numbers do NOT have an > exact value; this is obvious to anyone who understands logic and uses a > sane definition for infinity. > > /Flibble > How about in base pi? then it is the number 10 Base pi is an interesting base for some problems. What is your definition of "an exact value"? Maybe the problem is you don't quite understand the meaning of that term.
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 17:37 +0100 |
| Message-ID | <20220605173716.0000358e@reddwarf.jmc> |
| In reply to | #51870 |
On Sun, 5 Jun 2022 12:17:48 -0400 Richard Damon <Richard@Damon-Family.org> wrote: > On 6/5/22 11:34 AM, Mr Flibble wrote: > > On Sun, 5 Jun 2022 16:28:05 +0100 > > Andy Walker <anw@cuboid.co.uk> wrote: > > > >> On 05/06/2022 14:47, Mr Flibble wrote: > >>> On Sun, 5 Jun 2022 07:58:42 -0400 > >>> Richard Damon <Richard@Damon-Family.org> wrote: > >>>> [...] Sort of like how the number Pi has an > >>>> exact value, but you can never actually express it (because it > >>>> takes an infinite number of digits). > >>> PI does not have an exact value; no irrational number has an exact > >>> value. > >> > >> Of course "pi" has an exact value; as do [eg] "sqrt(2)", > >> "e", and all the other computable real [and complex] numbers. > >> Whether that value can be expressed in finite terms in some > >> particular representation is quite another matter. That in turn > >> depends on the representation; standard decimals is merely one > >> [common] choice. Note that in symbolic computer systems, those > >> computable reals are typically written "pi" [or whatever], and the > >> computer works with that exactly, so that [eg] "sin^2 (pi/3) == > >> 3/4", not 0.7499...; and also that in decimal-type notations most > >> rationals equally have no terminating expansion. Numbers such as > >> "pi" and "sqrt(2)" are not defined as decimal expansions but via > >> their properties [eg that "sqrt(2)" is the unique positive real > >> whose square is 2, or equivalently that it is the ratio of the > >> diagonal of a square to its side, and "pi" is the least positive > >> real whose sine is zero]. Those properties are exact, and tell > >> you all you ever need to know about those numbers. > >> > >> [I have removed my name from the "Subject:"; I don't know > >> why anyone saw fit to attach it to this debate, such as it is, on > >> the HP.] > > > > What has decimal (base 10) expansion got to do with anything? An > > irrational number has a non-terminating sequence in ANY base. I am > > sorry but you are simply mistaken: irrational numbers do NOT have an > > exact value; this is obvious to anyone who understands logic and > > uses a sane definition for infinity. > > > > /Flibble > > > > How about in base pi? then it is the number 10 how about base banana? then it is the number 10. PI, like banana, is just a symbol representing an irrational number that has no exact value. To use it here is circular and therefor erroneous. > > Base pi is an interesting base for some problems. > > What is your definition of "an exact value"? > > Maybe the problem is you don't quite understand the meaning of that > term. Of course I understand the fucking term. For the purposes of this discussion an exact value is a real number (non-integer) that terminates in a base that is not a multiple of itself. /Flibble
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