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# What's half of forever?

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eternity

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I think it's important to remember just how big forever is. The factorial of a trillion trillion years is not even the smallest fraction of the span of forever. In fact - it isn't and cannot be. Forever is impregnable by definition. Philosophically, the 80 year old man has lived a meaningless and pointless existence in the face of an eternal life. Those who believe in oblivion as a flavour, may have a point when they maintain that their lives on Earth mean more.

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It's still a very long time.

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Indeed, is a totally different concept than .

It is arguable whether -INF and +INF are different or if there really is only a signless INF - I personally think that both perspectives are correct and should be used together to form a more complete view of numbers.

Check mathworld for these two perspectives on infinity:

Projectively Extended Real Numbers (signless INF)

Affinely Extended Real Numbers (-INF and +INF)

Is +0 different from -0?

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Is +0 different from -0?

Not in the real world, unless you need them to make an equation work...

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Is +0 different from -0?
I ponder this every time I use my digital vernier-style callipers. They don't have the vernier scale markings, but a digital unit that gives me millimeters or decimal inches at the press of a button. It also floats between 0 and -0 after I hit the zero reset button! :)

There should be no such reading, but as an artifact of the digital readings, it claims that there is. At least Vernier callipers don't drift in strong magnetic fields...

I think negative infinity is as valid a concept as infinity, but it is less useful generally.

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I ponder this every time I use my digital vernier-style callipers.

That's really an oxymoron. I have digital calipers and vernier calipers but I've never seen any digital vernier calipers.

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Is +0 and -0 different?

I answer: Yes and No, both approaches are true and complete our understanding of zero, infinitesimals, and an absolutely smallest quantity. SOUTHTOWN, you should check a thread I started about that issue - I'm sure it will make sense to you once you get its point.

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Let's say we adopt the Projectively Extended Real Numbers (Real numbers + {∞}).

When we graph in regular Cartesian XY coordinates we place 0 at the origin, and count from it 1,2,3...

I wondered how the graph would look if , using the same plane, only we change and place ∞ at the origin, and count from it replacing x with 1/x: 1, 1/2 , 1/3 ... The graph y=f(x), would look as if graphed like y=1/f(1/x). Of course, the scale would be distorted:

(The marks "|" should be evenly spaced)

********`*****1/3 -

********`*****1/2 -

*********`**.***1 -

-----|-------|------|-------+-------|------|-------|-------

...-1/3..-1/2....-1......∞......1.....1/2....1/3 ...

******.****.****-1 -

******.*******-1/2 -

******.*******-1/3 -

Can anyone think of way of graphing the ENTIRE real numbers? That is, so not only some x are plotted, but every x (every real number) is plotted?

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That's really an oxymoron. I have digital calipers and vernier calipers but I've never seen any digital vernier calipers.

To remain off-point for a moment, that's why I said "digital Vernier style callipers" rather than "Vernier callipers". They look like Vernier callipers, but have an electronic unit where the Vernier scale normally is.

Either way, they still read -0 and 0 quite randomly when reset.

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Let's say we adopt the Projectively Extended Real Numbers (Real numbers + {∞}).

When we graph in regular Cartesian XY coordinates we place 0 at the origin, and count from it 1,2,3...

I wondered how the graph would look if , using the same plane, only we change and place ∞ at the origin, and count from it replacing x with 1/x: 1, 1/2 , 1/3 ... The graph y=f(x), would look as if graphed like y=1/f(1/x). Of course, the scale would be distorted:

(The marks "|" should be evenly spaced)

********`*****1/3 -

********`*****1/2 -

*********`**.***1 -

-----|-------|------|-------+-------|------|-------|-------

...-1/3..-1/2....-1......∞......1.....1/2....1/3 ...

******.****.****-1 -

******.*******-1/2 -

******.*******-1/3 -

Can anyone think of way of graphing the ENTIRE real numbers? That is, so not only some x are plotted, but every x (every real number) is plotted?

Nice artwork! HAHA Next time try: [ COLOR=#F6F8FA ] for the astericks. But, I don't understand what you're saying. Infinitesimals are just reciprocated infinity. ( 1 / ∞ ) And it's the same as applying any other kind of arithmetic to the concept.

Ohhh... wait. I get ya. +/-0 Man, that's deep. Enter Keanu Reeves, "There is no zero." LOL

Kinda like infinity is wondering what the universe looks like from the outside, while 1/infinity is kinda like wondering what subatomic particles are made, and what that is made of, etc., etc. If infinity is possible, there can be neither a zero nor a whole... HAHAHA

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Ohhh... wait. I get ya. +/-0 Man, that's deep. Enter Keanu Reeves, "There is no zero." LOL

Kinda like infinity is wondering what the universe looks like from the outside, while 1/infinity is kinda like wondering what subatomic particles are made, and what that is made of, etc., etc. If infinity is possible, there can be neither a zero nor a whole... HAHAHA

That's it! :) Someone finally got it!

Just for fun, I envision the number line as a circle: here on one end is 0, and 180 deg. apart is ∞. See Projectively Extended Real Numbers for an illustration (Riemann's innovation).

So I thought "Hey, when we graph we have 0 at the center; why not go around to infinity, and graph with ∞ at the center?"

Thinking in this way, one envisions the XY plane as really the surface of an infinite sphere; so one realizes that a parabola closes as it gets to ∞, and the line y=x really is a circle of infinite diameter; the hyperbola y=1/x is two disjoint closed loops - they touch neither 0 nor ∞. That's my interpretation, please indicate if I am incorrect. Still, under this way of "seeing" the entire XY surface, there are graphs I can't fathom, like y=2; can anyone "see" how this graph would look in the entire XY surface?

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the XY plane as really the surface of an infinite sphere;

This is the crux of the issue. If, by this you mean a sphere with infinite volume, then it cannot be a sphere by the very definition of infinity. A sphere does not have a flat surface regardless of its diameter. This is as true as parallel lines not crossing, the internal angle of triangles etc.

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If infinity is possible, there can be neither a zero nor a whole... HAHAHA

I believe your speaking in more terms of exsistance than of numbers...but i guess it can apply to both...Maybe I sound stupid saying it like that...O well

Off my chest..LOL

Op5

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I believe your speaking in more terms of exsistance than of numbers...but i guess it can apply to both...Maybe I sound stupid saying it like that...O well

Off my chest..LOL

Op5

No, that's exactly right. Good call. Math is conceptual so everything is "possible" — like the whole complex number crap.

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No, that's exactly right. Good call. Math is conceptual so everything is "possible" — like the whole complex number crap.

Well...not EVERYTHING is possible. If it were, there would be no truth or falsehood in an equivalence relation. However, I do agree in that every equation has a solution - with few exceptions including:

THERE IS (x) FOR ALL (y) (x + y = x)

I think this x doesn't exist, which x would necesarily be The Absolute Infinity - according to Set Theory.

However, Zermelo-Fraenkel Set Theory (in spite of the previous) holds as an axiom:

THERE IS (x) FOR ALL (y) (y + x = y)

This x they call the Empy Set.

So, in Set Theory, there isn't a set containing all sets, but there is a set that contains no set; I think this is erronous - there should be symmetry.

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This is the crux of the issue. If, by this you mean a sphere with infinite volume, then it cannot be a sphere by the very definition of infinity. A sphere does not have a flat surface regardless of its diameter. This is as true as parallel lines not crossing, the internal angle of triangles etc.

I should've been clearer: With this perspective, we would recognize that the XY plane is not a plane, but the surface of an infinite sphere.

You are mistaken; a sphere does have a flat surface - in the infinitesimal level. That's what calculus is based on - the flatness of curves at infinitesimal level. The Universe is curved, but locally (as in here on Earth) space is fairly flat or Eucledian - that is, we appreciate the curvature of the Universe only in grand scale.

As a finite sphere's surface is flat at the infinitesimal level, so is an infinite sphere's surface flat at the finite level.

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