From: antispam@fricas.org

David Brown  wrote:
> On 26/01/2026 21:34, wij wrote:
>> On Mon, 2026-01-26 at 21:07 +0100, David Brown wrote:
>>> On 26/01/2026 16:51, wij wrote:
>>>> On Mon, 2026-01-26 at 01:25 +0100, Janis Papanagnou wrote:
>>>>> (I probably regret answering to your post.)
>>>>>
>>>>> On 2026-01-25 18:20, wij wrote:
>>>>>>
>>>>>> You need to prove 4/33 exactly equal to 0.1212..., not approximation.
>>>>>
>>>>> Is that all you want proven; a specific example?
>>>>>
>>>>> This appears to be as trivial as the more general approach that James
>>>>> gave and that you (for reasons beyond me) don't accept (or don't see).
>>>>>
>>>>> First
>>>>>        __
>>>>>      0.12   or   0.1212...
>>>>>
>>>>> are just finite representations of real numbers; conventions. And 4/33
>>>>> is an expression representing an operation, the division. You can just
>>>>> do that computation (as you've certainly learned at school decades ago)
>>>>> in individual steps, continuing each step with the remainder
>>>>>
>>>>>      4/33 = 0     => 0
>>>>>      40/33 = 1    => 0.1
>>>>>      remainder 7
>>>>>      70/33 = 2    => 0.12
>>>>>      remainder 4
>>>>>      40/33 = 1    and at this point you see that the _operations_
*repeat*
>>>>>
>>>>> so the calculated decimals (1 and 2) will also repeat. And sensibly we
>>>>> need a finite representation (see above) to express that.
>>>>>
>>>>>      Albert Einstein (for example) said: „Die Definition von
Wahnsinn ist,
>>>>>      immer wieder das Gleiche zu tun und andere Ergebnisse zu
erwarten“.
>>>>>
>>>>> Are you expecting the sequence of decimals differing at some point?
>>>>>
>>>>> If not you see that the number represented by the convention "0.1212..."
>>>>> equals to the number calculated or expressed by "4/33".
>>>>>
>>>>> Janis
>>>> Not quite sure what you mean.
>>>>
>>>> https://sourceforge.net/projects/cscall/files/MisFiles/Real
umber2-en.txt/download
>>>>            3. 1/3 = 0.333... + non-zero-remainder (True
identity. How to deny?)
>>>>
>>>> How would you deny it, and call the cut-off 'equation' identity?
>>>
>>> Have you ever heard of the concept of "limits" ?  You might want to
>>> learn something about them before embarrassing yourself.
>>
>> What do you know about the concept of "limits"? (You invented? Don't try to
be
>> the next one, again. I remember the other expert in this forum has
humiliated himself
>> once, not sure which one, if I can safely predict. And I ignored the other
reply,
>> because it is too obvious, I leave as record)
>>
>
> No, I did not invent the concept of limits.  Newton and Leibnitz were
> probably the first to use them, then Cauchy formalized them (if I
> remember my history correctly).  But I /learned/ about them - understood
> them, understood proofs about them, understood how to use them.

Actually, part that is needed here is ancient, due to Eudoksos.  Namely,
real numebers a and b are equal if and only if comparing them with
rational numbers gives the same result.  In other words, a is different
from b if and only if there is a rational number c between a and b,
but not equal to either a or b.  When computing something this
principle is clumsy, but for proofs it works quit well.  Thanks
to this ancients we able compute (and prove) a bunch of transcendental
equalities.  Theory was finished by Dededking and Cantor who proved
that number produced by limiting process exist (earlier this was
consider true "by faith" or by invoking geometric intuition).

--
                              Waldek Hebisch

--- SoupGate-Win32 v1.05
 * Origin: you cannot sedate... all the things you hate (1:229/2)