Re: Does the number of nines increase?

Am 16.07.2024 um 04:55 schrieb Moebius:

Repost:
 Am 16.07.2024 um 04:02 schrieb Chris M. Thomasson:

On 7/15/2024 6:57 PM, Moebius wrote:

 

See? 😛

 

I see that you increased the granularity from natural numbers into the unit fractions...

Yeah, the real numbers comprise the naturals, the integers, the unit fractons, the rational numbers, ..., you know. 🙂
 

Wrt enumeration unit fractions I like to go from 1/1, to 1/2, to 1/3, ect...

  You may like to do that, still:
 0 < ... < 1/3 < 1/2 < 1/1.
 Meaning: Concening the < relation as _defined on the reals_ (as well on the rationals) 1/3 is SMALLER than 1/2 and 1/2 is smaller than 1/1. In general: 1/(n+1) is smaller than 1/n.
 

Is that wrong?

 Nope. You may define a SEQUENCE (of unit fractions):
 (1/1, 1/2, 1/3, ...)
 Here (referring to these sequence) 1/1 is "before", say, 1/2. 🙂
 

we have to think of a a smallest unit fraction, WM world, right?

 Right. There simply is no such unit fraction because for each and every unit fraction u: 1/(1/u + 1) is a unit fraction that is smaller than u.

Of course, we may consider the set {1/1, 1/2, 1/3, ...} of all unit fractions and define a certain order << on it, such that
1/1 << 1/2 << 1/3 << ... .
But this is NOT the order WM is referring to. WM is referring to the usual order < as defined on the reals (or rationals). There
0 < ... < 1/3 < 1/2 < 1/1.
Nuff said. (Ufff...)