[loarevalo]

Originally Posted by Southtown

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?

[Southtown]

Originally Posted by loarevalo

Is +0 different from -0?

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

[nkt]

Originally Posted by loarevalo

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.

[C1ay]

Originally Posted by nkt

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.

[loarevalo]

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.

[loarevalo]

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?

[nkt]

Originally Posted by C1ay

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.

[Southtown]

Originally Posted by loarevalo

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

[loarevalo]

Originally Posted by Southtown

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?

[geokker]

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.