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"Fred. Zwarts" <F.Zwarts@HetNet.nl> writes:In other words when one gets to the end of a never ending sequenceOp 29.mrt.2024 om 16:46 schreef olcott:olcott almost has a point, in that the sequence of values 0.9, 0.99,On 3/29/2024 8:13 AM, Richard Damon wrote:>On 3/28/24 11:50 PM, olcott wrote:That is not true. 0.999... means never reaches 1.0On 3/28/2024 10:36 PM, Keith Thompson wrote:>olcott <polcott2@gmail.com> writes:>
[...]It seems dead obvious that 0.999... is infinitesimally less than 1.0.>
Yes, it *seems* dead obvious. That doesn't make it true, and in
fact it
isn't.
>
0.999... means that is never reaches 1.0.
and math simply stipulates that it does even though it does not.
>
0.999... isn't a "number" in the Real Number system, just an
alternate representation for the number 1.
>
Maybe for olcott's unspecified olcott numbers. For real numbers
0.999... equals 1.0. There are many proofs. See e.g.https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
0.999, 0.9999, ... (continuing in the obvious manner) never reaches
1.0. No element of that unending sequence of real numbers is exactly
equal to 1.0.
What he either doesn't understand, or pretends not to understand, is
that the notation "0.999..." does not refer either to any element of
that sequence or to the entire sequence. It refers to the *limit* of
the sequence. The limit of the sequence happens not to be an element of
the sequence, and it's exactly equal to 1.0.
This is all stated in terms of the real numbers, which are a wellPi exists at a single geometric point on the number line.
defined set. There are other systems with different properties. If we
were talking about the hyperreals, for example, olcott's statement might
be correct (though I'm not sure of that). But olcott seems to be
insisting, quite incorrectly, that his statements apply to the reals.
If he's talking about the reals, he's wrong. If he's talking about--
something other than the reals, he's boring. Either way, he will not
change his mind. Attempts to explain limits and real numbers to him
will fail.
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