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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 28, 2017 4:49 PM

Am Donnerstag, 28. Dezember 2017 21:09:48 UTC+1 schrieb konyberg:
> torsdag 28. desember 2017 08.28.04 UTC+1 skrev WM følgende:

> > > This is not a proof as such, but it's a reasonable construction.
> > > 1 = 9/9 = 1/9 + 8/9 = 0.111... + 0.888... = 0.999...
> > > Show me what's wrong with this.

> >
> > This is wrong: 1/9 + 8/9 = 0.111... + 0.888...
> >
> > Correct would be 1/9 + 8/9 = lim{n-->oo} 0.111...1_n + lim{n-->oo} 0.888...8_n

>
> Does it change the result?

Yes, in case of irrationals. A sum of fractions will not yield an irrational value. The belief that infinitely many fractions could yield an irrational is a silly as the belief that infinitely many red points would yield a green line.

A digit sequence does not define its limit. That can only be done by finite formula defining this digit sequence. A Cantor-diagonal is never a transcendental and cannot define it.

One of the wrong "results" is the idea that Cantor proved the existence of transcendental numbers and the idea that the measure of the reals is 1 and the measure of the rationals is 0 in the unit interval.

Regards, WM