Does X/0 = Infinity? Results of My Calculus Class Discussion

[Hagie190]

I had an elightening conversation with my calculus teach this past friday we were arguing whether X/0= infinity. we ran out of time, but i wasn't satisfied that when i came up this:

Assuming that ALL lines are in fact circles (Bye Bye Euclid)
and Infinity is an equal distance from all numbers.

Imagine a number line since All lines are circles the number line becomes a circle containing all numbers. Now knowing that infinity is an equal distance from a numbers, infinity can be expressed as the center of this circle.

The Problem I face is using this can you at least difine zero?

 

[Hurkyl]

There is no argument: In the real numbers, not only is x/0 undefined for any x, but infinity is not a number, thus it cannot be the result of a division.


But there are ways to extend the real numbers (in other words, consider new systems) that do have one or more points at infinity, and some even permit arithmetic to be extended.



However, with your idea, I see this problem first:

If "infinity" is an equal distance from every point, then shouldn't the circle be the entire plane and not just a line?

 

[Hagie190]

If "infinity" is an equal distance from every point, then shouldn't the circle be the entire plane and not just a line?

No,You misunderstood infinity is an equal distance from every number and every number is contain on the afore mentioned circle. So the only way for that to work is for infinity to be a point at the center of the circle.

 

[quartodeciman]

Just pick some point on the real number "circle" to represent number 0. It really doesn't matter which one you pick, because all the points (numbers) on that "circle" have the same fixed distance from your "circle" center (infinity). A fundamental problem with this idea is that you must be able to go from "zero" to "one" and on to very large numbers (without bound) by traveling around that "circle". Somehow you must end up back at 0 again after going around the "circle". You even have to magically pass through negative numbers before returning to 0. Where do you change from positive numbers to negative numbers? It isn't at infinity, because infinity is not on the "circle" of numbers but in the center. Give this idea up!

A better scheme is called stereographic projection. It is usually expressed using a plane mapped onto the surface of a sphere, but it can also be applied to a line and an ordinary circle. Let some unbounded line represent the real numbers. Pick some point on the line to be 0. Pick a point directly above the 0 point at some distance( for example, 1). Let this be the center of a regular circle that touches the line only at the 0 point. OK, you now have a circle of radius 1 sitting on the point 0 of the line. The stereographic map is made by drawing a line from any selected point of the real number line to the top point of the circle. This line must cross the circle of radius 1 at one other point. This is the stereographic image of the point. The number 0 on the number line maps vertically through the center of the circle up to the top point. So the point 0 is its own stereographic image. It is the only such point. All other stereographic images are on the unit circle and not on the original number line. Play with this a bit and you will see that every point on the number line has a single image on the unit circle and every point (except one!) on the unit circle is the image of a point on the number line (just draw a line from the top point of the circle through the selected point on the unit circle and extend it to cross the number line). The one point on the unit circle that is not the image of a point on the number line is the top point itself. This will be infinity. On the unit circle, you can travel from 0 (bottom of unit circle), through 1 and on to larger numbers. At the top of the unit circle you cross infinity and go through negative values back to 0. Notice what points on the original number line correspond to these points.

You will not be able to handle anything like X/0 in any number system that tries to preserve basic rules of algebra. That is because 0*X must be 0. The only exception to this is a number field containing only the number 0 (where 1 = 0 and all X = 0). That is insufficient for analysis. So you should give up on X/0. If you try to push on with X/0 = infinity, then you have to put up with algebra busters like infinity -1 = infinity = infinity +1 and infinity - infinity = nothing-in-particular. BAH! HUMBUG!

If you are going to have infinite numbers then have many different infinite numbers, so x-1 < x < x+1 and x - y always has a definite value, even when x or y have infinite value. Solitary infinity is like doing math with "finity". What is finity + finity? Just finity? What is finity - finity? Just finity? That is hardly good math. Likewise, infinity + infinity, infinity - infinity and so on are silly.

A number system that includes distinct infinite numbers is the Robinson extension to the real numbers (commonly called "hyperreal" numbers). It takes some time to explain these thoroughly, but beyond all the ordinary real numbers are positive infinite numbers and also ahead of all the real numbers are negative infinite numbers. If X is a positive infinite number, then 1/X is a positive infinitesimal number. Likewise, if X is a negative infinite number, then 1/X is a negative infinitesimal number. Other numbers are sums and differences of finite, infinite and infinitesimal numbers. This still preserves the rules of algebra, so X/0 is still impossible, but X/Y is a valid number when Y is an infinitesimal number not 0. That is practically as good as having a value for X/0 without putting up with the algebra-busting.

 

[Zurtex]

No,You misunderstood infinity is an equal distance from every number and every number is contain on the afore mentioned circle. So the only way for that to work is for infinity to be a point at the center of the circle.

Assuming the circle is in 2D space, if the circle is in 3D space there is a line that runs through the centre of the circle where looking at each point individually they are the same distance to every point on the circle. Thus there an infinite number of points and by your definition an infinite of of infinities.

But this all seems rather abstract and meaningless to me.

 

[matt grime]

You've not explained in what ambient space all these objects lie, what metric you are using, why straight lines are circles, why infinity is equidistant from all other points, why infinity must then be a point in this ambient space (and what are all the other points?). Amongst other issues.

 

[Hagie190]

Just pick some point on the real number "circle" to represent number 0. It really doesn't matter which one you pick, because all the points (numbers) on that "circle" have the same fixed distance from your "circle" center (infinity). A fundamental problem with this idea is that you must be able to go from "zero" to "one" and on to very large numbers (without bound) by traveling around that "circle". Somehow you must end up back at 0 again after going around the "circle". You even have to magically pass through negative numbers before returning to 0. Where do you change from positive numbers to negative numbers? It isn't at infinity, because infinity is not on the "circle" of numbers but in the center. Give this idea up!

Wow you really crushed that idea, but I am not out of "fight" yet. I never thought about it but what if infinity was the line Zurtex was referring to? In that Infinity Would be a vairable (spelling?), and could change so that Infinity1+infinity2= infinity3. This would be like x+y=z an open problem that can't be solved but doesn't "kill" the basic rules of algebra (Thanks zurtex for the idea)

As for your problem with the sign change, have you everlooked at a X^-1 graph? You'll see that 0^-1 equals both positve and negative infinity. so in that case there has to be some sign change. So my thought is that this is no different. However you were right in saying that the center is not the defines infinity. Instead, r+x= infinity where x is any number on the circle and r is the distance from x to the point on the line.

 

[Hagie190]

You've not explained in what ambient space all these objects lie, what metric you are using, why straight lines are circles, why infinity is equidistant from all other points, why infinity must then be a point in this ambient space (and what are all the other points?). Amongst other issues.

For the record I really don't like explaining myself.

Ok the answers to your questions are in order

1. What difference does it make?
2. I don't know what metric is.
3. every school of geometry (besides Euclidean) accepts that lines can't exist exept in a finite space and that lines in there truest since are in fact circles read a book it can explain it better than me.
4. If you understand the concept of infinity it is a equal distance from from all numbers. Once again read a book it can explain it better than me.
5. Void, due to my previous post.

 

[quartodeciman]

You'll see that 0^-1 equals both positve and negative infinity

No, I don't see that. I see negative values of 1/X going more and more negative and I see positive values of 1/X going more and more positive. That tells me that 1/X is not going to any particular value at all at X=0. Furthurmore, if +infinity = -infinity, then I expect +infinity+infinity = +infinity-infinity=0. I don't think that is very good. Or is algebra dead?

I also don't see why +1 and +2 have the same distance from a positive infinite number x. I expect x-1 > x-2 because -1 > -2. But that's just me. :)
Even my projective example has the distance of the infinite point (top of the projective circle) to point +1 larger than the distance of the infinite point to +2. This is because the distances are arcs on the projection circle and they are unequal.

You can do what you are doing if you want to. I just don't see where it is going.

 

[matt grime]

infinity is not part of the real numbers and doesn't have an arithmetic that follows that of the reals so you cannot add infinity to itself like that.

in the analytic situation of having a one point compactification there is one point at infinity, there are two point versions.

 

[quartodeciman]

Hurkyl reply #2: "In the real numbers ... infinity is not a number"
Hagie190 reply #3: "infinity is an equal distance from every number"
Hagie190 reply #7: "r+x= infinity where x is any number on the circle and r is the distance from x to the point on the line"
Hagie190 reply #8: "infinity...equal distance from all numbers"

OK. It looks like no one thinks "infinity" is supposed to be a number. I will agree. Furthurmore, no one seems to think there are any infinite numbers outside of and beyond the real numbers. In that case, I will stop talking about such. It is curious that people want to talk about "going to infinity" as a paraphrase of "increasing without bound" or "decreasing without bound", which boil down to purely numerical relationships.

 

[matt grime]

it is not curious to speak of going to infinity - it follows from the proper definition of the word infinite in the English language meaning not finite, the limit is not finite. unfortunately many people get the mistaken impression that there is actaully such a point physically lying above the others and don't even notice they are making a presumption that the real numbers are physical points.

 

[MiGUi]

matt grime said: word infinite in the English language meaning not finite

In spanish, the meaning is the same (if you had any doubt xD)

...

Infinity is a hole where mathematicians and physicists throw the equations that don't works.

Ok, let's be serious!

Cantor's theory is pretty cool for this subject...

Maybe infinity can be defined as the cardinal of numbers that are between "x" and "x + dx"

 

[quartodeciman]

So, it is "going to a not finite"- what? Not a number? It seems more like "going beyond finite things- numbers". All of them (finite numbers) eventually get left behind.

"The limit is not finite" = "no actual limit exists"?

Physical points are like smudges of graphite and ink, serving only as rough cartoons for a conceptual situation.

 

[matt grime]

dx is not a real number, so there are no real numbers between x and dx. Just stick to infinity as the noun back formed from the adjective infinite, meaning not finite. a sum is an infinite sum if there are not a finite number of terms, the meaning of the word is contextual, stop thinking it is actaully a number of some description and you're fine. the problem comes from the, typically, lax way that mathematicians abuse notation. this causes no issue to people who have a mathematical mind.

 

[MiGUi]

Ok, if you don't want "dx" use "Dx" (D = Delta) or maybe, x and x + 1

 

[matt grime]

the cardinality of any open interval of real numbers is c, the continuum.

 

[matt grime]

"The limit is not finite" = "no actual limit exists"?

no, the sequence 1,-1,1,-1... has no limit, ie it diverges (oscillates), but it is not the same behaviour as the sequence 1,2,3,4... which also diverges (tends to infinity)

 

[quartodeciman]

CORRECTION: "The limit is not finite" -> "no actual limit exists".

Yes, there are many other ways a limit can fail to exist.

For f(x) = sin(π/(2x)), what happens as x -> 0?