Re: Gaps... ;^)

Am 10.09.2024 um 22:27 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 is't a unit fraction. :-P
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 Still the gap between 1/1 and 1/2 is equal to 1/2.

I guess you mean the LENGTH of the gap.
Your "gap" is automatically translated to "interval" by me.

There are infinite unit fractions that are smaller than the

length of the

gap [interval]?

RIGHT! :-)