Re: Does the number of nines increase?

Chris M. Thomasson was thinking very hard :

On 7/9/2024 3:11 AM, FromTheRafters wrote:

Chris M. Thomasson pretended :

On 7/8/2024 6:39 AM, WM wrote:

Le 07/07/2024 à 22:28, "Chris M. Thomasson" a écrit :

On 7/7/2024 1:27 PM, WM wrote:

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There are only countably many finite expression.

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What generated this?
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.(121232)
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?

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The finite formula ".(121232)" generated this.

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Did you know that a rational generated it?

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Every periodic decimal number has a rational limit.

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Can every real be represented by the convergents of infinite continued fractions?

 Those and the terminating ones.

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For some reason, this makes me think that if any real can be represented by infinite convergents of continued fractions comprised of rationals, well... Does that mean there are as many rationals as there are reals? Is this a wrong way to ponder on such things? Thanks.

Using various representations (CDE, CFE) might be too confusing. Pi has 'as many' rationals in its CFE makeup terms as the set of natural numbers has elements. Rational numbers 'terminate' irrationals don't.
I don't think that relying on any particular representation is going to help you explore this area.