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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Gaps... ;^)
Date: Tue, 10 Sep 2024 13:37:12 -0700
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On 9/10/2024 1:30 PM, Moebius wrote:
> Am 10.09.2024 um 22:24 schrieb Chris M. Thomasson:
>> On 9/10/2024 12:23 PM, Moebius wrote:
>>> Am 10.09.2024 um 20:30 schrieb Chris M. Thomasson:
>>>> On 9/9/2024 5:28 PM, Moebius wrote:
>>>>> Am 10.09.2024 um 00:59 schrieb Chris M. Thomasson:
>>>>>
>>>>>> Between zero and any positive x there is a unit fraction small 
>>>>>> enough to fit in the ["]gap["].
>>>>>
>>>>> Right. This follows from the so called "Archimedean property" of 
>>>>> the reals. From this property we get:
>>>>>
>>>>> For all x e IR, x > 0, there is an n e IN such that 1/n < x.
>>>>>
>>>>> See: https://en.wikipedia.org/wiki/Archimedean_property
>>>>>
>>>>> Of course, from this we get that there are infinitely many unit 
>>>>> fractions smaller than x, say, 1/n, 1/(n + 1), 1/(n + 2), 1/(n + 
>>>>> 3), ...
>>>>>
>>>>> We can even refer to such unit fraction "in terms of x":
>>>>>
>>>>> All of the following (infinitely many) unit fractions are smaller 
>>>>> than x: 1/ceil(1/x + 1), 1/ceil(1/x + 2),  1/ceil(1/x + 3), ...
>>>>>
>>>>>> Between x and any y that is different than it (x), there will be a 
>>>>>> unit fraction to fit into the gap. infinitely many.... :^)
>>>>>
>>>>> Nope. There is no unit fraction (strictly) between, say, 1/2 and 1/1.
>>>>
>>>> What about 1/4? Ahhhh! You mentioned the word _strictly_. Okay.
>>>>
>>>> Humm... Well, if we play some "games" ;^), then 1/4 would sit in the 
>>>> center of the gap between 1/2 and 1/1 where:
>>>
>>> Really?
>>>
>>> ??? 1/2 < 1/4 < 1/1 ???
>>>
>>> Are you sure?
>>>
>>> 0.5 < 0.25 < 1
>>>
>>> Hmmm...?
>>>
>>>>> In other words, there is no unit fraction u such that 1/2 < u < 1/1.
>>>
>>> Concerning 1/4, in my book (of numbers):
>>>
>>>      1/4 < 1/2 < 1/1. :-P
>>>
>>> It's clear that you have/had 3/4 in mind. (i.e. 1/2 + 1/4. :-)
>>>
>>> But 3/4 isn
> 
> 't a unit fraction. :-P
>>
>> DOH!!!! I fucked up.
>>
>> 1/1----->(1/4*3)----->(1/2)
>>
>> 1----->.75------>.5
>>
>> YIKES!!!!
> 
> N/p.
> 
> Of course you had
> 
> 1/2 ---> 1/2 + 1/4 ---> 1/1
> 
> in mind.
> 
> The __distance__ between the mid point (between 1/2 and 1/2) to 1/2 
> and/or 1/1 is 1/4. That tripped you up.
> 
> 
> 

Right. Now what about normalize the distance between any two points? Say 
p0 and p1. Where 0 maps to p0 and 1 maps to p1? This can be used to fill 
any gap with the unit fractions. Not nearly as dense as the reals, but 
the will get arbitrarily close to 0. A normalization between two points 
can be as simple as:

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?