Re: rational numbers

From	alessandro volturno <alessandro.volturno@libero.it>
Newsgroups	comp.lang.c++
Subject	Re: rational numbers
Date	2021-09-16 18:24 +0200
Organization	Aioe.org NNTP Server
Message-ID	<shvr5o$ovs$1@gioia.aioe.org> (permalink)
References	<shvf4c$ark$3@gioia.aioe.org> <shvhfq$mt2$1@dont-email.me>

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Il 16/09/2021 15:40, Paavo Helde ha scritto:
> 16.09.2021 15:59 alessandro volturno kirjutas:
>>
>> Maybe my question is stupid, but
>> why not to introduce a way to handle rational numbers inside the C++ 
>> standard or better make it a built in type?

I speak as a non-competent, hobbyist programmer and computer-user.

> What are the use cases for rationals? If something is not in the 
> standard, then often there are different use cases which cannot be 
> easily covered by a common "standard" implementation.

Computers were born to manipulate numbers and the languages for doing 
that, at that time, were mainly two: FORTRAN, and LISP. I have noticed 
that C++ (that I presume it could be thought of as a descendant of 
FORTRAN, is shifting towards Functional programming language facilities, 
like that offered by the LISP family. That's good, but numerical 
programming is still important nowadays.

> For rationals it feels like a fast implementation would have a pretty 
> limited numeric range, and unlimited range would require arbitrary 
> precision integers and would probably be many times slower even in case 
> of small numbers.

Mine uses long long int, that is the maximum offered by the language, 
but calculation speed is constantly increasing. I don't see that a as a 
limiting factor.

True that C++ applications can be built for every available hardware 
configuration (and so they can be run on older or slower cpus), but 
rational are an important class of numbers and its family is wider than 
integers'

> Nevertheless, there is at least one proposal to include rationals in the 
> standard, see 
> http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n3363.html

The proposal you presented is quite dated, so I presume many talkings 
have been made about it. I don't know the reason why, but it evidently 
came out not to add this feature to the language.

Many computer languages are evolving and new ones are born every one or 
two years. In my opinion, to keep a language competitive with respect to 
the others around, that language should also permit basic number 
manipulation as easier as possible.

I repeat myself, I'm a hobbyist, spare-time programmer, but as complex 
numbers have been added to C++, so there should be some mechanics to 
handle rationals.

> Meanwhile one can use the Boost Rational library.

I've heard about the Boost library but I've never made an attempt to 
study it nor to use it.

My projects are small ones and I write them by scratch just for the sake 
of knowing how I progress in this discipline (computer programming).

Thank you for your kind reply,

alessandro

Thread

rational numbers alessandro volturno <alessandro.volturno@libero.it> - 2021-09-16 14:59 +0200
  Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-16 16:40 +0300
    Re: rational numbers alessandro volturno <alessandro.volturno@libero.it> - 2021-09-16 18:24 +0200
      Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-16 17:05 +0000
        Re: rational numbers Manfred <noname@add.invalid> - 2021-09-16 19:58 +0200
        Re: rational numbers Christian Gollwitzer <auriocus@gmx.de> - 2021-09-17 09:21 +0200
          Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-17 10:34 +0200
            Re: rational numbers Christian Gollwitzer <auriocus@gmx.de> - 2021-09-17 15:55 +0200
              Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-17 17:49 +0200
                Re: rational numbers Christian Gollwitzer <auriocus@gmx.de> - 2021-09-17 18:22 +0200
          Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-17 22:07 +1200
            Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-17 13:32 +0200
              Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-18 09:15 +1200
            Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-17 14:06 +0000
      Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-16 20:41 +0200
        Re: rational numbers Bart <bc@freeuk.com> - 2021-09-16 21:11 +0100
          Re: rational numbers Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-16 21:26 +0100
            Re: rational numbers Bart <bc@freeuk.com> - 2021-09-16 21:54 +0100
            Re: rational numbers "Alf P. Steinbach" <alf.p.steinbach@gmail.com> - 2021-09-16 23:07 +0200
              Re: rational numbers "Alf P. Steinbach" <alf.p.steinbach@gmail.com> - 2021-09-16 23:27 +0200
      Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-16 23:12 +0300
        Re: rational numbers alessandro volturno <alessandro.volturno@libero.it> - 2021-09-17 09:50 +0200
          Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-17 11:15 +0200
            Re: rational numbers alessandro volturno <alessandro.volturno@libero.it> - 2021-09-17 19:19 +0200
              Re: rational numbers James Kuyper <jameskuyper@alumni.caltech.edu> - 2021-09-17 19:00 -0400
              Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-18 11:39 +0200
                Re: rational numbers alessandro volturno <alessandro.volturno@libero.it> - 2021-09-18 11:56 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-18 11:01 +0100
  Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-17 05:36 +0000
    Re: rational numbers alessandro volturno <alessandro.volturno@libero.it> - 2021-09-17 09:55 +0200
      Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-17 11:16 +0000
        Re: rational numbers alessandro volturno <alessandro.volturno@libero.it> - 2021-09-17 18:33 +0200
        Re: rational numbers Tim Rentsch <tr.17687@z991.linuxsc.com> - 2021-10-06 13:06 -0700
    Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-17 14:03 +0000
      Re: rational numbers Christian Gollwitzer <auriocus@gmx.de> - 2021-09-17 17:12 +0200
        Re: rational numbers Christian Gollwitzer <auriocus@gmx.de> - 2021-09-17 17:16 +0200
          Re: rational numbers Manfred <noname@add.invalid> - 2021-09-17 18:59 +0200
            Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-17 17:13 +0000
              Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-19 10:17 +0000
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-19 14:07 +0000
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-19 15:58 +0000
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-19 21:53 +0000
      Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-18 09:34 +1200
        Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-18 12:00 +0200
          Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-18 14:54 +0000
            Re: rational numbers Bart <bc@freeuk.com> - 2021-09-18 17:36 +0100
              Re: rational numbers Christian Gollwitzer <auriocus@gmx.de> - 2021-09-18 23:29 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-19 01:09 +0100
                Re: rational numbers red floyd <no.spam.here@its.invalid> - 2021-09-18 23:45 -0700
              Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-18 22:48 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-19 00:46 +0100
                Re: rational numbers Christian Gollwitzer <auriocus@gmx.de> - 2021-09-19 08:06 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-19 10:26 +0100
              Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-19 10:23 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-19 12:04 +0100
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-19 16:01 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-19 17:39 +0100
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-20 05:25 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-20 10:37 +0100
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-22 04:58 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-22 11:26 +0100
                Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-22 23:13 +1200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-22 13:45 +0100
                Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-24 21:37 +1200
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-22 15:29 +0300
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-23 07:26 +0000
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-19 21:54 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-20 00:31 +0100
                Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-20 08:33 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-20 16:11 +0100
                Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-20 18:08 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-20 18:39 +0100
                Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-20 19:49 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-20 19:04 +0100
                Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-20 21:35 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-20 21:38 +0100
                Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-21 10:28 +1200
                Re: rational numbers Bo Persson <bo@bo-persson.se> - 2021-09-21 09:08 +0200
                Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-21 09:39 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 11:12 +0100
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-21 15:45 +0300
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 14:37 +0100
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-21 17:09 +0300
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 16:26 +0100
                Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-21 18:04 +0200
                Re: rational numbers Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2021-09-21 11:51 -0700
                Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-22 11:09 +0200
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-21 15:38 +0000
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-21 15:36 +0000
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-21 15:49 +0000
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-21 16:21 +0000
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-22 09:21 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 17:45 +0100
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-21 17:02 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 18:16 +0100
                Re: rational numbers Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2021-09-21 11:53 -0700
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 20:48 +0100
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-21 20:23 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 23:29 +0100
                Re: rational numbers Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2021-09-21 13:47 -0700
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-21 20:11 +0300
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 18:33 +0100
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-21 20:43 +0300
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 18:59 +0100
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-21 21:20 +0300
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 20:32 +0100
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-21 20:24 +0000
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-21 18:52 +0000
                Re: rational numbers Bo Persson <bo@bo-persson.se> - 2021-09-21 18:34 +0200
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-21 01:16 +0300
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-21 00:39 +0100
                Re: rational numbers Manfred <noname@add.invalid> - 2021-09-22 23:45 +0200
                Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-22 23:51 +0200
                Re: rational numbers Manfred <noname@add.invalid> - 2021-09-23 18:42 +0200
                Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-23 21:52 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-23 00:17 +0100
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-23 02:03 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-23 11:18 +0100
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-23 10:26 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-23 12:21 +0100
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-23 12:40 +0100
                Re: rational numbers Richard Damon <Richard@Damon-Family.org> - 2021-09-23 08:05 -0400
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-23 15:02 +0100
                Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-24 08:48 +1200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-23 23:05 +0100
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-23 23:47 +0000
                Re: rational numbers Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2021-09-23 16:56 -0700
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-24 01:58 +0100
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-24 01:40 +0000
                Re: rational numbers Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2021-09-23 19:35 -0700
                Re: rational numbers "daniel...@gmail.com" <danielaparker@gmail.com> - 2021-09-24 09:07 -0700
                Re: rational numbers "daniel...@gmail.com" <danielaparker@gmail.com> - 2021-09-24 09:21 -0700
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-23 14:58 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-23 17:02 +0100
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-23 16:21 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-23 18:31 +0100
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-24 09:12 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-24 12:10 +0100
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-24 11:23 +0000
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-24 15:19 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-24 17:25 +0100
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-25 09:27 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-25 11:07 +0100
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-25 10:14 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-25 11:53 +0100
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-25 11:23 +0000
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-25 12:34 +0000
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-25 15:37 +0300
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-25 15:46 +0100
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-25 15:19 +0000
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-25 17:16 +0000
                Re: rational numbers Christian Gollwitzer <auriocus@gmx.de> - 2021-09-27 23:51 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-27 23:41 +0100
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-23 10:26 +0000
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-23 14:04 +0000
                Re: rational numbers Manfred <noname@add.invalid> - 2021-09-23 17:47 +0200
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-23 17:07 +0000
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-23 17:08 +0000
                Re: rational numbers Manfred <noname@add.invalid> - 2021-09-23 19:30 +0200
                Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-24 10:06 +1200
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-22 05:06 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-22 10:51 +0100
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-22 10:54 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-22 11:59 +0100
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-22 11:04 +0000
                Re: rational numbers "Alf P. Steinbach" <alf.p.steinbach@gmail.com> - 2021-09-22 14:17 +0200
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-22 14:32 +0100
                Re: rational numbers "Alf P. Steinbach" <alf.p.steinbach@gmail.com> - 2021-09-22 16:23 +0200
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-22 15:18 +0300
                Re: rational numbers "Alf P. Steinbach" <alf.p.steinbach@gmail.com> - 2021-09-22 16:16 +0200
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-22 15:15 +0000
                Re: rational numbers "Alf P. Steinbach" <alf.p.steinbach@gmail.com> - 2021-09-22 17:57 +0200
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-22 16:05 +0000
                Re: rational numbers Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2021-09-22 14:13 -0700
                Re: rational numbers HorseyWorsey@the_stables.com - 2021-09-23 09:01 +0000
                Re: rational numbers James Kuyper <jameskuyper@alumni.caltech.edu> - 2021-09-23 12:08 -0400
                Re: rational numbers Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2021-09-23 10:55 -0700
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-22 20:46 +0300
                Re: rational numbers "daniel...@gmail.com" <danielaparker@gmail.com> - 2021-09-22 14:42 -0700
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-23 09:15 +0300
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-22 14:04 +0000
                Re: rational numbers Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2021-09-22 07:18 -0700
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-23 07:31 +0000
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-23 11:05 +0300
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-23 08:28 +0000
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-24 00:51 +0300
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-24 04:37 +0000
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-24 07:46 +0000
                Re: rational numbers "daniel...@gmail.com" <danielaparker@gmail.com> - 2021-09-24 08:59 -0700
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-24 22:49 +0300
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-25 00:55 +0000
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-24 12:17 +0300
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-27 05:25 +0000
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-27 11:37 +0300
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-28 04:49 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-23 11:29 +0100
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-24 04:38 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-24 11:15 +0100
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-24 10:19 +0000
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-27 05:34 +0000
                Re: rational numbers scott@slp53.sl.home (Scott Lurndal) - 2021-09-23 14:07 +0000
                Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-23 14:50 +0000
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-20 05:20 +0000
                Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-24 21:45 +1200
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-25 16:15 +0300
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-27 05:36 +0000
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-27 11:19 +0300
                Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-27 21:27 +1300
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-27 15:49 +0300
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-28 04:53 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-28 10:16 +0100
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-29 12:10 +0000
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-29 14:34 +0100
                Re: rational numbers Bart <bc@freeuk.com> - 2021-09-29 15:54 +0100
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-28 13:23 +0300
                Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-29 10:34 +1300
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-29 12:18 +0300
                Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-29 22:46 +1300
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-29 13:07 +0300
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-10-02 22:57 +0300
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-29 12:16 +0000
                Re: rational numbers Paavo Helde <myfirstname@osa.pri.ee> - 2021-09-29 20:15 +0300
                Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-30 04:46 +0000
            Re: rational numbers Ian Collins <ian-news@hotmail.com> - 2021-09-19 11:03 +1200
      Re: rational numbers Juha Nieminen <nospam@thanks.invalid> - 2021-09-19 10:12 +0000
  Re: rational numbers Siri Cruise <chine.bleu@yahoo.com> - 2021-09-23 07:38 -0700
    Re: rational numbers alessandro volturno <alessandro.volturno@libero.it> - 2021-09-24 09:45 +0200
      Re: rational numbers Branimir Maksimovic <branimir.maksimovic@gmail.com> - 2021-09-24 07:50 +0000
      Re: rational numbers David Brown <david.brown@hesbynett.no> - 2021-09-24 10:58 +0200
        Re: rational numbers Siri Cruise <chine.bleu@yahoo.com> - 2021-09-24 02:48 -0700

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