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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 | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2022-06-05 11:35 -0500 |
| Message-ID | <2tidnRCUeI_NRgH_nZ2dnUU7_8zNnZ2d@giganews.com> |
| In reply to | #51873 |
On 6/5/2022 11:28 AM, 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
>
You haven't studied the specific matter at hand thoroughly enough.
With the same credentials as me you could learn these things.
It has taken me at least 15,000 hours since 2004.
You can find the 86612 postings of my prior work right here in this
forum. My work prior to 2004 has not yet been fully restored to the
USENET archives.
--
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 12:50 -0400 |
| Message-ID | <995nK.66928$GTEb.66655@fx48.iad> |
| In reply to | #51873 |
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.
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 17:56 +0100 |
| Message-ID | <20220605175617.00001647@reddwarf.jmc> |
| In reply to | #51886 |
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
[toc] | [prev] | [next] | [standalone]
| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2022-06-05 12:01 -0500 |
| Message-ID | <o_qdnWACaI3wfAH_nZ2dnUU7_8xh4p2d@giganews.com> |
| In reply to | #51889 |
On 6/5/2022 11:56 AM, 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
>
You are confusing the representation of the number in decimal digits
with the actual number itself.
--
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 18:19 +0100 |
| Message-ID | <20220605181947.000013f1@reddwarf.jmc> |
| In reply to | #51893 |
On Sun, 5 Jun 2022 12:01:32 -0500
olcott <NoOne@NoWhere.com> wrote:
> On 6/5/2022 11:56 AM, 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
> >
>
> You are confusing the representation of the number in decimal digits
> with the actual number itself.
No, you are. I am merely pointing out that the number changes up to a
factor of 1/base as you evaluate it at increasing accuracy.
/Flibble
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 18:27 +0100 |
| Message-ID | <20220605182700.00000804@reddwarf.jmc> |
| In reply to | #51899 |
On Sun, 5 Jun 2022 18:19:47 +0100
Mr Flibble <flibble@reddwarf.jmc> wrote:
> On Sun, 5 Jun 2022 12:01:32 -0500
> olcott <NoOne@NoWhere.com> wrote:
>
> > On 6/5/2022 11:56 AM, 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
> > >
> >
> > You are confusing the representation of the number in decimal
> > digits with the actual number itself.
>
> No, you are. I am merely pointing out that the number changes up to a
> factor of 1/base as you evaluate it at increasing accuracy.
I of course meant base^n not 1/base.
/Flibble
[toc] | [prev] | [next] | [standalone]
| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2022-06-05 12:58 -0500 |
| Message-ID | <Nc-dnf4gXMBbcwH_nZ2dnUU7_81g4p2d@giganews.com> |
| In reply to | #51899 |
On 6/5/2022 12:19 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 12:01:32 -0500
> olcott <NoOne@NoWhere.com> wrote:
>
>> On 6/5/2022 11:56 AM, 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
>>>
>>
>> You are confusing the representation of the number in decimal digits
>> with the actual number itself.
>
> No, you are. I am merely pointing out that the number changes up to a
> factor of 1/base as you evaluate it at increasing accuracy.
>
> /Flibble
>
Every real number has a unique point on the number line.
These points do not jump around.
--
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 14:13 -0400 |
| Message-ID | <En6nK.170698$zgr9.56621@fx13.iad> |
| In reply to | #51899 |
On 6/5/22 1:19 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 12:01:32 -0500
> olcott <NoOne@NoWhere.com> wrote:
>
>> On 6/5/2022 11:56 AM, 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
>>>
>>
>> You are confusing the representation of the number in decimal digits
>> with the actual number itself.
>
> No, you are. I am merely pointing out that the number changes up to a
> factor of 1/base as you evaluate it at increasing accuracy.
>
> /Flibble
>
But APPROXIMATIONS to the number are not the number itself.
The irrational number all have an EXACT location on the number line.
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 20:14 +0100 |
| Message-ID | <20220605201453.000072f4@reddwarf.jmc> |
| In reply to | #51911 |
On Sun, 5 Jun 2022 14:13:55 -0400
Richard Damon <Richard@Damon-Family.org> wrote:
> On 6/5/22 1:19 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 12:01:32 -0500
> > olcott <NoOne@NoWhere.com> wrote:
> >
> >> On 6/5/2022 11:56 AM, 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
> >>>
> >>
> >> You are confusing the representation of the number in decimal
> >> digits with the actual number itself.
> >
> > No, you are. I am merely pointing out that the number changes up to
> > a factor of 1/base as you evaluate it at increasing accuracy.
> >
> > /Flibble
> >
>
> But APPROXIMATIONS to the number are not the number itself.
>
> The irrational number all have an EXACT location on the number line.
Prove it.
/Flibble
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 17:46 -0400 |
| Message-ID | <vu9nK.33050$qt97.2485@fx97.iad> |
| In reply to | #51922 |
On 6/5/22 3:14 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 14:13:55 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 1:19 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 12:01:32 -0500
>>> olcott <NoOne@NoWhere.com> wrote:
>>>
>>>> On 6/5/2022 11:56 AM, 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
>>>>>
>>>>
>>>> You are confusing the representation of the number in decimal
>>>> digits with the actual number itself.
>>>
>>> No, you are. I am merely pointing out that the number changes up to
>>> a factor of 1/base as you evaluate it at increasing accuracy.
>>>
>>> /Flibble
>>>
>>
>> But APPROXIMATIONS to the number are not the number itself.
>>
>> The irrational number all have an EXACT location on the number line.
>
> Prove it.
>
> /Flibble
>
The DEFINITION?
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 13:05 -0400 |
| Message-ID | <bn5nK.14468$xZtb.13326@fx41.iad> |
| In reply to | #51889 |
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"
All you are showing is that approximations to numbers get better as they
get better, which is just a strange tautology.
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).
The fact that it can't be expressed, isn't an issue on exactness, but of
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.
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 18:22 +0100 |
| Message-ID | <20220605182217.00006176@reddwarf.jmc> |
| In reply to | #51896 |
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
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 18:26 +0100 |
| Message-ID | <20220605182643.00005c70@reddwarf.jmc> |
| In reply to | #51900 |
On Sun, 5 Jun 2022 18:22:17 +0100
Mr Flibble <flibble@reddwarf.jmc> 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.
I of course meant base^n not 1/base.
/Flibble
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 14:17 -0400 |
| Message-ID | <_q6nK.40212$ssF.1607@fx14.iad> |
| In reply to | #51900 |
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.
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 20:17 +0100 |
| Message-ID | <20220605201701.00005880@reddwarf.jmc> |
| In reply to | #51912 |
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
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 15:30 -0400 |
| Message-ID | <Gv7nK.42946$elob.22159@fx43.iad> |
| In reply to | #51923 |
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.
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 20:33 +0100 |
| Message-ID | <20220605203320.000053a1@reddwarf.jmc> |
| In reply to | #51928 |
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
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 15:47 -0400 |
| Message-ID | <nL7nK.65115$ntj.27275@fx15.iad> |
| In reply to | #51930 |
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.
[toc] | [prev] | [next] | [standalone]
| From | Mr Flibble <flibble@reddwarf.jmc> |
|---|---|
| Date | 2022-06-05 20:56 +0100 |
| Message-ID | <20220605205620.00006729@reddwarf.jmc> |
| In reply to | #51939 |
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
[toc] | [prev] | [next] | [standalone]
| From | Richard Damon <Richard@Damon-Family.org> |
|---|---|
| Date | 2022-06-05 16:09 -0400 |
| Message-ID | <K38nK.47082$X_i.4514@fx18.iad> |
| In reply to | #51942 |
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.
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