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.

In other words, there is no unit fraction u such that 1/2 < u < 1/1.

Say the gap is abs(x - y) where x and y can be real. If they are
different (aka abs(x - y) does not equal zero), then there are
infinitely many unit fractions that sit between them.

Nope. See counter example above.

Any thoughts? Did I miss something? Thanks.

Yes. It works for any (0, x) where x e IR, x > 0.

But it does not work "in general" for (x, y) where x,y e IR, x,y > 0 and
x < y (and hence abs(x - y) > 0).

If you'd consider _rational numbers_ (or fractions) instead of unit
fractions, your intuition would be right, though.

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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 unit fractions are smaller than x: 1/ceil(1/x + 1), 1/ceil(1/x + 2), 1/ceil(1/x + 2),

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.

In other words, there is no unit fraction u such that 1/2 < u < 1/1.

Say the gap is abs(x - y) where x and y can be real. If they are
different (aka abs(x - y) does not equal zero), then there are
infinitely many unit fractions that sit between them.

Nope. See counter example above.

Any thoughts? Did I miss something? Thanks.

Yes. It works for any (0, x) where x e IR, x > 0.

But it does not work "in general" for (x, y) where x,y e IR, x,y > 0 and
x < y (and hence abs(x - y) > 0).

If you'd consider _rational numbers_ (or fractions) instead of unit
fractions, your intuition would be right, though.

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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 is't a unit fraction. :-P

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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.

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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:

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 is't a unit fraction. :-P

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! :-)

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