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Date: Sun, 30 Oct 2022 09:56:00 -0700 (PDT)
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Subject: Re: Three awkward and helpless answers. Who knows a better one?
From: "Ross A. Finlayson" <ross.finlayson@gmail.com>
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On Sunday, October 30, 2022 at 2:16:56 AM UTC-7, WM wrote:
> FromTheRafters schrieb am Samstag, 29. Oktober 2022 um 14:34:26 UTC+2:=20
> > WM has brought this to us :=20
> > > FromTheRafters schrieb am Samstag, 29. Oktober 2022 um 00:27:55 UTC+2=
:=20
> > >> Chris M. Thomasson has brought this to us :=20
> > >=20
> > >> The set of reals from zero to one inclusive is a larger cardinality =
set=20
> > >> than the set of natural number indices.=20
> > >=20
> > > Something claimed cranks worldwide.=20
> > >=20
> > > Proof: It is impossible to define a real number by an infinite sequen=
ce of=20
> > > digits=20
> > Representations are not definitions.
> The number pi is defined by pi as well as by 3.1415... . It would be defi=
ned by the complete digit sequence too, if this could be given. But if you =
dislike the word you can also say: It is impossible to mention a real numbe=
r by an infinite sequence of digits=20
>=20
> Regards, WM


Noticed this Nilakantha identity on the Pascal's triangle the other day.

pi =3D 3 + Sigma_n=3D1^oo (-1)^(n+1) =20

(2n+1
   1 )
--------
(2n+1) (2n+2)
    2             2

https://en.wikipedia.org/wiki/Pascal%27s_triangle#Rows

It amuses me because 3 <  pi < 4, that the non-integer part is defined by s=
o simple a series.

I.e., one would usually expect that the integer and non-integer part togeth=
er only would be so defined.

Not sure about it.

Reminds me of Apery's.



3.1415926, ....