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Topic: If you claim 0.999... is a rational number, then you must find
p/q such that 0.999... = p/q. 12/26/2017

Replies: 40   Last Post: Jan 4, 2018 7:37 AM

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 wolfgang.mueckenheim@hs-augsburg.de Posts: 3,394 Registered: 10/18/08
Re: If you claim 0.999... is a rational number, then you must find
p/q such that 0.999... = p/q. 12/26/2017

Posted: Dec 31, 2017 8:36 AM
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Am Sonntag, 31. Dezember 2017 11:00:37 UTC+1 schrieb Zelos Malum:
> >Yes. Nobody has denied that. But since oo is no natural number you have not a sum over terms with natural indices but the limit.
>
> Again, oo is a shorthand for the limit and the infinite sum is defiend as the limit.

I agree. But the infinite sum, whether or not defined as the limit, *is* not a limit. A sum is the addition of summands --- and that is rational for every summand added. The adding does never stop, and the rational character of the result does never stop.

Why should the one persist but the other cease?
>
> >The last does not follow. Correct is: The limit of the infinite sum of rationals is irrational.
>
> Which is the definition of an infinite sum?

It is an infinite adding of terms which never yields a final result but approaches the limit better and better.
>
> >If there are already *all* finite numbers of nines then your claim is wrong. More than all is not possible.
>
> More than any finite number.

Of course, an infinite number of terms is more than any finite number. But that should not allow you to drop logic.
>
> >It does not. Logic clearly states: If all infinitely many terms fail, then all infinitely many terms fail. This is even simplest kind of logic: a tautology.
>
> Yes but infinitely many terms does not fail.

The word "infinite" seems to paralyse the common sense of many mathematicians.

If all terms fail and if there are infinitely many, then infinitely many fail.
>
> >Difficult to understand? Or only difficult to answer?
>
> Asking for clearification for what you are exactly asking for.

Why don't you comprehend this simple case?: If all terms fail and if there are infinitely many, then infinitely many fail.

Regards, WM

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