Sujet : Re: Cantor Diagonal Proof
De : rjh (at) *nospam* cpax.org.uk (Richard Heathfield)
Groupes : comp.theoryDate : 07. Apr 2025, 11:50:01
Autres entêtes
Organisation : Fix this later
Message-ID : <vt0akp$38f07$2@dont-email.me>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
User-Agent : Mozilla Thunderbird
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:
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On Sun, 2025-04-06 at 13:35 +0300, Mikko wrote:
On 2025-04-06 07:15:51 +0000, wij said:
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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:
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On 06/04/2025 06:40, Lawrence D'Oliveiro wrote:
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On Sat, 5 Apr 2025 09:07:22 +0100, Richard Heathfield wrote:
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But to be computable, numbers must be computed in a finite number of
steps.
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“Computable Number: A number which can be computed to any number of
digits desired by a Turing machine.”
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<https://mathworld.wolfram.com/ComputableNumber.html>
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"The “computable” numbers may be described briefly as the real numbers
whose expressions as a decimal are calculable by finite means." - Alan
Turing.
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And therefore, to be computable, numbers must be computed in a finite
number of steps.
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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”.
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Simply put, repeating decimals are irrational.
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/download
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Repeating decimals are rational.
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Prove it (be sure not to make mistakes shown in the link above)
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See
https://math.stackexchange.com/questions/549254/why-is-a-repeating-decimal-a-rational-number
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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 HeathfieldEmail: rjh at cpax dot org dot uk"Usenet is a strange place" - dmr 29 July 1999Sig line 4 vacant - apply within
Cantor Diagonal Proof
Re: Cantor Diagonal Proof