Re: How many different unit fractions are lessorequal than all unit fractions? (circle limits)

On 10/01/2024 06:37 PM, Chris M. Thomasson wrote:

On 10/1/2024 2:11 PM, FromTheRafters wrote:

Chris M. Thomasson wrote :

On 10/1/2024 6:28 AM, FromTheRafters wrote:

Chris M. Thomasson wrote :

On 9/30/2024 4:13 AM, Richard Damon wrote:

On 9/29/24 3:16 PM, WM wrote:

On 28.09.2024 14:58, Richard Damon wrote:

On 9/27/24 3:06 PM, WM wrote:

On 25.09.2024 19:12, Richard Damon wrote:
>

The problem is that it turns out the NUF(x) NEVER actually
"increments" by 0ne at any finite point, it jumps from 0 to
infinity (Aleph_0) in the unboundedly small gap between 0 and
all x

0,

>
How do you distinguish them?

>
They have different values, so why can't you?

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Then distinguish the first one.
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Regards, WM

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There isn't a first one.
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Show me a circle with 4 sides.

>
;^) Humm, an n-gon where n is taken to infinity is a circle?

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As n goes to infinity, the angle of the vertices goes to 180 degrees
-- is a straight line a circle?

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No. As n goes to infinity it makes a circle. Think of a finite view
of a "large" number for n:
_________________________
n = 696969
>
normal_base = 1.f / n;
>
for (i = 0; i < n; ++i)
{
     normal = normal_base * i;
     angle = pi2 * normal;
>
     p0 = { cos(angle), sin(angle) };
>
     plot(p0);
}
_________________________
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I typed this in the newsreader, so sorry for any typos! This a finite
view of a unit circle. Not a line.
>
Take n to infinity, well, its a circle...
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Taking an n-gon to infinity is a circle.

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It is 'never' a circle.
>
https://www.craig-wood.com/nick/articles/pi-archimedes/

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an n-gon as n goes to infinity approaches a circle? Fair enough?

Is the sum of its interior angles infinity, or, 2pi?
How about this one, in a circle, draw two circles whose
diameters are on the same line that is a diameter of
the outside circle. Now, the perimeter or length of the
circle is pi times diameter, so, the perimeter of the
internal circles is half each the outer. Now you can
notice that you can trace one then the other side making
a yin-yang the length of which curve, is the same as
half the perimeter. So, it's line three curves from
the upper point, to the lower point, each same length,
pi D / 2 or pi/2 D.
So anyways as you repeat on that one main diameter,
making the circles smaller yet having the same overall
perimeter, in the limit, the line goes flat, and the limit
of the curves on it, each with length pi D / 2, is D,
so that in the limit, it's pi/2 less than each case
of induction, yet, it's no different, for each case
of induction.
So anyways, it seems a sort of example that by
dividing the diameter into infinitely many pieces,
its measure is different than for any finitely many.
Similarly is that for blowing it up, is what results
where the zero solution is trivial instead of the envelope,
the length of the curve goes from D to pi/2 D while all
along that in induction, it looks constant, in the infinite
limit, asymptotically different.
Here pi/2 > 1.
Anyways the sum of the interior angles is a different
sort of thing, about the definition of curvature and so
on, as with regards to the triangle segments going
to tape segments in the infinite limit, for example,
where most people have them vanish, also about as:
they change shape as they vanish.