From: david.brown@hesbynett.no
On 27/01/2026 17:31, Waldek Hebisch wrote:
> 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/Rea
Number2-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.
Thanks for that - I was not familiar with his work. (My knowledge of
the history of maths is mostly from whatever books, articles, Youtube
videos, Wikipedia pages, etc., that I have looked at - so it is quite
sporadic, and is often missing many of the great names. I have never
done any kind of methodical study.) But of course mathematics is built
up in stages - it's very rare that new concepts appear completely out of
the blue. So it is natural that Newton and Leibnitz built on earlier
work, through many other mathematicians, back to Eudoksos. And
Eudoksos' work was an extension of earlier work on approximating pi, and
so on.
I always think it is fascinating that between any two distinct rationals
there is at least one real number, and between any two distinct real
numbers there is at least one ration number - and yet the cardinality of
the reals is the power-set of the cardinality of the rational numbers.
> 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).
>
Dedekind cuts are one of the ways to construct the real numbers -
basically, filling in the gaps in the rationals by including the numbers
produced by limiting processes in sets of rationals. So he did not
really prove that these limits exist - he defined the reals to be these
limits, and then proved that this formal definition of the real numbers
worked the way everyone had assumed they worked.
--- SoupGate-Win32 v1.05
* Origin: you cannot sedate... all the things you hate (1:229/2)
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