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From: Richard Heathfield <rjh@cpax.org.uk>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Mon, 7 Apr 2025 11:50:01 +0100
Organization: Fix this later
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On 07/04/2025 11:31, wij wrote:
> On Mon, 2025-04-07 at 11:28 +0300, Mikko wrote:
>> On 2025-04-06 10:42:05 +0000, wij said:
>>
>>> On Sun, 2025-04-06 at 13:35 +0300, Mikko wrote:
>>>> On 2025-04-06 07:15:51 +0000, wij said:
>>>>
>>>>> On Sun, 2025-04-06 at 06:43 +0000, Lawrence D'Oliveiro wrote:
>>>>>> On Sun, 6 Apr 2025 07:27:43 +0100, Richard Heathfield wrote:
>>>>>>
>>>>>>> On 06/04/2025 06:40, Lawrence D'Oliveiro wrote:
>>>>>>>
>>>>>>>> On Sat, 5 Apr 2025 09:07:22 +0100, Richard Heathfield wrote:
>>>>>>>>
>>>>>>>>> But to be computable, numbers must be computed in a finite number of
>>>>>>>>> steps.
>>>>>>>>
>>>>>>>> “Computable Number: A number which can be computed to any number of
>>>>>>>> digits desired by a Turing machine.”
>>>>>>>>
>>>>>>>> <https://mathworld.wolfram.com/ComputableNumber.html>
>>>>>>>
>>>>>>> "The “computable” numbers may be described briefly as the real numbers
>>>>>>> whose expressions as a decimal are calculable by finite means." - Alan
>>>>>>> Turing.
>>>>>>>
>>>>>>> And therefore, to be computable, numbers must be computed in a finite
>>>>>>> number of steps.
>>>>>>
>>>>>> I would say you are quoting Turing out of context. By your>> > >
>>>>>> (mis)interpretation of his words, even something like 1/3 is an>> > >
>>>>>> incomputable number, since its “expressions as a decimal are not>> > >
>>>>>> calculable by finite means”.
>>>>>
>>>>> Simply put, repeating decimals are irrational.
>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/download
>>>>>
>>>>>
>>>>
>>>> Repeating decimals are rational.
>>>
>>> Prove it (be sure not to make mistakes shown in the link above)
>>
>> See
>> https://math.stackexchange.com/questions/549254/why-is-a-repeating-decimal-a-rational-number
>>
> 
> Still can't prove, except posting a copy from the internet?

Posting a link to a proof /is/ proving it.

Let's work an example.

We take a repeating decimal such as r = 0.142857142857142857...

What's a million times that? Clearly it's 1000000r = 
142857.142857142857142857...

Subtracting:

1000000r = 142857.142857142857142857...
        r =      0.142857142857142857...

yields:

999999r = 142857

Dividing both sides by 142857:

7r = 1

Dividing both sides by 7:

r = 1/7

1 is an integer, 7 is an integer, so their ratio r is rational.

QED.

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
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