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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Gaps... ;^)
Date: Tue, 10 Sep 2024 22:30:53 +0200
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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.