Re: Does the number of nines increase?

On 7/11/2024 3:59 AM, FromTheRafters wrote:

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

Agreed. It's automatic, so to speak. sqrt(2) is all of its infinite glory, all at once.