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On 3/30/2024 3:18 PM, Fred. Zwarts wrote:No, olcott is trying to change the meaning of the symbol '='. That *is* what the '=' means for real numbers, because 'exactly the same' is too vague. Is 1.0 exactly the same as 1/1? It contains different symbols, so why should they be exactly the same?Op 30.mrt.2024 om 20:57 schreef olcott:That is not what the "=" sign means. It means exactly the same as.On 3/30/2024 2:45 PM, Fred. Zwarts wrote:>Op 30.mrt.2024 om 14:56 schreef olcott:>On 3/30/2024 7:10 AM, Fred. Zwarts wrote:>Op 30.mrt.2024 om 02:31 schreef olcott:>On 3/29/2024 8:21 PM, Keith Thompson wrote:>olcott <polcott2@gmail.com> writes:>On 3/29/2024 7:25 PM, Keith Thompson wrote:[...]>What he either doesn't understand, or pretends not to understand, isIn other words when one gets to the end of a never ending sequence
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.
>
(a contradiction) thenn (then and only then) they reach 1.0.
No.
>
You either don't understand, or are pretending not to understand, what
the limit of sequence is. I'm not offering to explain it to you.
>
I know (or at least knew) what limits are from my college calculus 40
years ago. If anyone or anything in any way says that 0.999... equals
1.0 then they <are> saying what happens at the end of a never ending
sequence and this is a contradiction.
>
It is clear that olcott does not understand limits, because he is changing the meaning of the words and the symbols. Limits are not talking about what happens at the end of a sequence. It seems it has to be spelled out for him, otherwise he will not understand.
>
0.999... Limits basically pretend that we reach the end of this infinite sequence even though that it impossible, and says after we reach this
impossible end the value would be 1.0.
No, if olcott had paid attention to the text below, or the article I referenced:https://en.wikipedia.org/wiki/Construction_of_the_real_numbers>
he would have noted that limits do not pretend to reach the end. They
Other people were saying that math says 0.999... = 1.0
Indeed and they were right. Olcott's problem seems to be that he thinks that he has to go to the end to prove it, but that is not needed. We only have to go as far as needed for any given ε. Going to the end is his problem, not that of math in the real number system.
0.999... = 1.0 means that with this sequence we can come as close to 1.0 as needed.
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