Re: Does the number of nines increase?

Chris M. Thomasson explained :

On 7/10/2024 3:26 PM, Chris M. Thomasson wrote:

On 7/10/2024 3:07 PM, Moebius wrote:

Am 10.07.2024 um 23:56 schrieb Chris M. Thomasson:

On 7/9/2024 2:49 PM, Moebius wrote:

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I guess you might have a _sequence_ of rational numbers in mind, say,
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(1, 1.4, 1.41, 1.414, ...).
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So we might say that this SEQUENCE represents the real number sqrt(2) - in a certain sense. :-P
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Actually, its limit is sqrt(2).
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Well, basically, I was thinking that for any element of:
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(1, 1.4, 1.41, 1.414, ...)
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there is a rational that can represent it.

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lol. (Sorry!)
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Which one, if I may ask? :-P
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So, it kind of makes my brain want to bleed from time to time, shit happens! Uggg.

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lol. (Sorry again!)
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Imho you are "on a good way"!
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Just take your time, and don't do the Mückenheim! :-)
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Taken to infinity, there are rationals that can represent [...] sqrt 2:
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(1, 1.4, 1.41, 1.414, ...)

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Yeah, but when speaking of a mathematical objekts (in this connection) we (usually) refer to the SEQUENCE (1, 1.4, 1.41, 1.414, ...)
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Set theory allows to refer to such objekts (sets).
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However, there is no single rational that equals sqrt 2.

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Exactly! :-)
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A thing even the ancient greeks new! :-P
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Humm... Fair enough?

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Absolutely!
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 (1, 1.4, 1.41, 1.414, ...)
 For each element there is a rational that can represent it.
 No single rational can represent the whole...
 However, a real can represent it the whole... Fair enough? Or am I drifting off deeper into WM land? Oh shit.
 ;^o
  

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Think of the whole as simply, sqrt(2)
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The finite parts are the step wise construction of the whole, can be something akin to:
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(1, 1.4, 1.41, 1.414, ...)
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Is this a decent line of thought?

Just be careful not to assume that such a step by step method requires time for each step or that it is the (only) way it is constructed. It is one way to think about its construction.