Gaps... ;^)

[Chris M. Thomasson]

Between zero and any positive non-zero x there is a unit fraction small
enough to fit in the gap. The x can even be a real that is not a unit
fraction.

Between x and any y that is different than it (x), there will be a unit
fraction to fit into the gap. infinitely many.... :^)

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.

Any thoughts? Did I miss something? Thanks.

[Moebius]

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.

[Moebius]

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.

[Chris M. Thomasson]

Post by Moebius

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
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), ...
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:

p0 = 1/2
p1 = 1/1
dif = p1 - p0
unit_mid = dif / 2

So:

p0 = 1/2 = .5
p1 = 1/1 = 1
dif = p1 - p0 = .5
unit_mid = dif / 2 = .5 / 2 = .25 = 1/4

?

Or is this just crank bullshit that removes the strict factor?

Post by Moebius
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.

thanks.

[Moebius]

Post by Chris M. Thomasson

Post by Moebius

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
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), ...
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

Really?

??? 1/2 < 1/4 < 1/1 ???

Are you sure?

0.5 < 0.25 < 1

Hmmm...?

Post by Chris M. Thomasson

Post by Moebius
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

[Chris M. Thomasson]

Post by Moebius

Post by Chris M. Thomasson

Post by Moebius

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
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), ...
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

Really?
??? 1/2 < 1/4 < 1/1 ???
Are you sure?
0.5 < 0.25 < 1
Hmmm...?

Post by Chris M. Thomasson

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

    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

DOH!!!! I fucked up.

1/1----->(1/4*3)----->(1/2)

1----->.75------>.5

YIKES!!!!

[Moebius]

Post by Chris M. Thomasson

Post by Moebius

Post by Chris M. Thomasson

Post by Moebius

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
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), ...
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

Really?
??? 1/2 < 1/4 < 1/1 ???
Are you sure?
0.5 < 0.25 < 1
Hmmm...?

Post by Chris M. Thomasson

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

     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

Post by Chris M. Thomasson
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.

[Chris M. Thomasson]

Post by Moebius

Post by Chris M. Thomasson

Post by Moebius

Post by Chris M. Thomasson

Post by Moebius

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
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), ...
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

Really?
??? 1/2 < 1/4 < 1/1 ???
Are you sure?
0.5 < 0.25 < 1
Hmmm...?

Post by Chris M. Thomasson

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

     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

Post by Chris M. Thomasson
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?

[Chris M. Thomasson]

Post by Moebius

Post by Chris M. Thomasson

Post by Moebius

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
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), ...
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

Really?
??? 1/2 < 1/4 < 1/1 ???
Are you sure?
0.5 < 0.25 < 1
Hmmm...?

Post by Chris M. Thomasson

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

    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. There are infinite
unit fractions that are smaller than the gap?

[Moebius]

Post by Chris M. Thomasson

Post by Moebius

Post by Chris M. Thomasson

Post by Moebius

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
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), ...
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

Really?
??? 1/2 < 1/4 < 1/1 ???
Are you sure?
0.5 < 0.25 < 1
Hmmm...?

Post by Chris M. Thomasson

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

     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.

Post by Chris M. Thomasson
There are infinite unit fractions that are smaller than the

length of the

Post by Chris M. Thomasson
gap [interval]?

RIGHT! :-)

[Chris M. Thomasson]

Post by Moebius

Post by Chris M. Thomasson

Post by Moebius

Post by Chris M. Thomasson

Post by Moebius

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
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), ...
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

Really?
??? 1/2 < 1/4 < 1/1 ???
Are you sure?
0.5 < 0.25 < 1
Hmmm...?

Post by Chris M. Thomasson

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

     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.

Post by Chris M. Thomasson
There are infinite unit fractions that are smaller than the

length of the

Post by Chris M. Thomasson
gap [interval]?

RIGHT! :-)

Also, think about normalizing the gaps as in:

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?