Re: Gaps... ;^)

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:
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Between zero and any positive x there is a unit fraction small enough to fit in the ["]gap["].

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Right. This follows from the so called "Archimedean property" of the reals. From this property we get:
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For all x e IR, x > 0, there is an n e IN such that 1/n < x.
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See: https://en.wikipedia.org/wiki/Archimedean_property
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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), ...
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We can even refer to such unit fraction "in terms of x":
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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), ...
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Between x and any y that is different than it (x), there will be a unit fraction to fit into the gap. infinitely many.... :^)

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Nope. There is no unit fraction (strictly) between, say, 1/2 and 1/1.

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What about 1/4? Ahhhh! You mentioned the word _strictly_. Okay.
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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:

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Really?
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??? 1/2 < 1/4 < 1/1 ???
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Are you sure?
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0.5 < 0.25 < 1
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Hmmm...?
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In other words, there is no unit fraction u such that 1/2 < u < 1/1.

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Concerning 1/4, in my book (of numbers):
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     1/4 < 1/2 < 1/1. :-P
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It's clear that you have/had 3/4 in mind. (i.e. 1/2 + 1/4. :-)
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But 3/4 isn

 't a unit fraction. :-P

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DOH!!!! I fucked up.
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1/1----->(1/4*3)----->(1/2)
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1----->.75------>.5
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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?