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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Tue, 10 Sep 2024 13:48:16 -0700
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On 9/10/2024 1:38 PM, Moebius wrote:
> Am 10.09.2024 um 22:33 schrieb Chris M. Thomasson:
>> On 9/10/2024 12:30 PM, Moebius wrote:
>>> Am 10.09.2024 um 20:38 schrieb Chris M. Thomasson:
>>>> On 9/10/2024 11:35 AM, Chris M. Thomasson wrote:
>>>>> On 9/9/2024 5:32 PM, Moebius wrote:
>>>>>> Am 10.09.2024 um 01:08 schrieb Chris M. Thomasson:
>>>>>>
>>>>>>> Take the gap between:
>>>>>>>
>>>>>>> 1/1 and 1/2. There are infinitely many unit fractions that are 
>>>>>>> small enough to fit within that gap.
>>>>>>
>>>>>> Oh, really?! Could you name (just) o n e? :-)
>>>>>
>>>>> 1/4?
>>>>
>>>> 1/4 is between 1/1 and 1/2:
>>>>
>>>> 1/1---->1/4---->1/2
>>>
>>> lol. If you say so. :-)
>>>
>>>> So, it's not _strict_ 
>>>
>>> Actually, it's not between 1/2 and 1/1 at all (if we presuppose the 
>>> usual order < on the rational numbers).
>>
>> The LENGTH OF THE gap from 1/1 to 1/2 is 1/2 so there are infinity 
>> many unit fractions that are smaller than 1/2.
> 
> MAN, YOUR ORIGINAL CLAIM WAS: "1/4 is between 1/1 and 1/2."
> 
> THIS CLAIM IS WRONG. (WRONG! WORONG!)
> 
> FUCK YOUR "GAP".
> 
> Please don't do the Mückenheim her. One crank is enough.

1/4 is smaller than the gap. Think of a normalized gap:

To normalize the distance between two points in n-ary space. p0 and p1 
can be n-ary vectors.

p0 = 1/2
p1 = 1/1
pdif = p1 - p0

the mid point would use the unit fraction 1/2 at:

pmid = p0 + pdif * 1/2

right?

this means there are infinite unit fractions that can be used for the 
gap. The line from p0 to p1?