Is it possible that positive infinty and negatice infinity are the same thing?

Question

The idea of numbers being in a loop has always interested me. I first began thinking about this when trying to understand tan90 as when it appraches from the left it tends towards positive infinity and as it approoaches from the right it tends towards negative infinity. I know my hypothesis is totally abstract but then again is infinity itself and mathematics in general abstract. This notion first began when reading about a closed universe and thought that if it is theoretical for a universe to be closed why not the number system that defines it. I imagine the numbers zero and positive/negative infinity as being different poles on a sphere. where positive infinity and negative infinity are a whole circumference apart from each other but are also at the same point.

Answer 1

The handy thing about mathematics is that it is what you want it to be. As was mentioned, the complex number system came essentially from someone like you asking, "Yeah, but what if we COULD take the square root of something negative?" Now the complex numbers had a long and arduous journey to being accepted, and an eventual geometric use for them helped out a lot.

I should probably stress that in our current "popular" number system, the real numbers, infinity (whether positive or negative) is NOT included. Infinity is merely a concept of when things grow without bound.

That being said, there are a few number systems that admit a "number" infinity. It seems useful to define infinity=1/0, since that's how limiting behavior tends to work. Then we've allowed some division by 0...it should then come as no surprise that we can define 5/0 as 5*1/0=5*infinity. So we would likely want to call this infinity as well, so that any real positive number times infinity is infinity. 0*infinity might need some consideration. Then perhaps, naively, we would say that a negative divided by 0 is negative infinity. But then
-infinity = (-1)/0 = -(1/0) = 1/(-0) = 1/0 = infinity.

So if we do allow most of our usual operations on fractions, we MUST define positive and negative infinity to be the same. However, we could construct a system that does not allow the moving of the negative I did above, and we could allow a distinction between positive and negative infinities.

If we allow two infinities, this is the "extended real number line": http://en.wikipedia.org/wiki/Extended_real_number_line
Notice that 1/0 is not defined as either + or - infinity; we just allow the elements of infinity to exist.

If we allow just one infinity, then the system is called the "real projective line": http://en.wikipedia.org/wiki/Real_projective_line
http://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html
The mathworld site has a good picture that confirms that your idea of 0 and infinity being antipodal points on a circle (you don't really need a sphere--although with a sphere you could create a complex number system with infinity). It also points out some of the differences between these two systems.

It is interesting to note that we still don't have every operation defined. In either of the systems I've named, adding/subtracting infinities is undefined, as is 0/0. There is a system proposed by James Anderson which allows 0/0 to be considered a new number, which he calls "nullity." His system also allows for - and + infinities, and he calls it the "transreal" number system. As far as I can tell, his system is consistent (not self-contradictory), but not well-regarded by the general math community.
http://en.wikipedia.org/wiki/James_Anderson_%28computer_scientist%29#Transreal_arithmetic

In your tan(x) example, yes, the projective reals would allow tan(x) to be a continuous function. But notice we need to have tan(x) defined where cos(x) is zero, so we need to have a system that includes infinity as a number, not just an abstraction. That is, just using the real numbers and saying that "positive and negative infinity are the same" won't do it.

One last thing: adding infinities to our "popular" number system is not a strictly good thing. Some of our favorite things about the real numbers start to go haywire. For example, I mentioned that infinity-infinity is not defined, while we're used to having x-x=0 for all x. In the projective reals, we lose our ability to compare elements as less than or greater than, since the system in essence forms a circle.

So yes, it is possible, and has been done. I can't imagine these additions will ever become mainstream, but they're useful for certain applications (suppose that the universe "loops" onto itself, like the projective reals).

I hope I've been clear enough to be of some help.

Answer 2

In high school math, infinity is not a number. It's just a direction when taking limits. (By the way, there was a typo in an otherwise excellent answer above -- functions don't converge "to" infinity, they converge "at" or "as x goes to" infinity.)

More precisely, infinity and -infinity are two different directions when taking limits.

Answer 3

No proof
suppose infinity =negative infinity
devide both sides by infinity
0=-1
therefore infinity .ne. negative infinity

BTW, are you smoken?

Answer 4

If you could find a use for such numbers, all you would need to do is define it as such. For example, we know that no two real numbers squared can give a negative number, so we constructed imaginary numbers that solved the euaqtion, and a whole new branch of mathematics, complex analysis emerged.

Answer 5

I think it is possible that pos. and neg. infinity are the same but I don't know for sure. This is an interesting question.

Answer 6

Well, infinity is not a number.

That said, when we add inifinity to the number line, sometimes we add two points (+infinity and -infinity) and sometimes we add a single point at infinity.

The single point solution yields what we call the 'real projective line.' 'Projective' in this instance means that it is related to 'Projective Geometry,' where there are no parallel lines (all lines that look parallel essentially meet 'at infinity.')

However, when dealing with limits in the real numbers, it does not make sense to make -infinity and +infinity equal because the limit converges to one or the other.

You might want to look into a field called 'Point Set Topology.' Adding a single point at infinity to the real line is an example of what is called 'compactification' of a space.

The big 'geometric' thing you lose when you add single points at infinity to every line is that you no longer have the following rule:

Given three points on a line, there is only one that is
between the other two.

If you add a single point at infinity to the line, and choose points x=0, y=1, and z=infinity, then y=1 is between x=0 and z=infinity, and x=0 is between z=infinity and y=1.

There is no 'right answer' here. The real line is not something we can test to see if it comes around and hits itself at infinity or not.

When we define things in mathematics, we often define them so they'll be most useful. Outside of Projective Geometry and point set topology, the Real Projective Line is not so useful, so we choose to distinguish +infinity with -infinity.

In the case of limits, however, we have more complicated problems is +infinity=-infinity.

For example, in general, if the sequences:

(x1,x2,....,xn,...)

and

(y1,y2,...,yn,...)

Both converge to "+infinity", then their sums converge to "+infinity":

(x1+y1,...,xn+yn,...)

But we can't say that if (y1,y2,...yn,...) converges to -infinity.
So we distinguish in this case because there are important rules that we want to state about convergence that require us to distinguish between +infinity and -infinity.

Answer 7

I don't think that they are the same, but couldn't they be complete mirrors?
I like your hypothesis, and now that I think about it, you could be right. But I don't think that this is that abstract, if you think about it mathematically.

Answer 8

DOES NOT COMPUTE!

DOES NOT COMPUTE!

Answer 9

It is an interesting theory. No one can know the answer for certain, since it is a theory. However, I would say anything is possible. And so is this. Good Luck ! :)

Answer 10

Interesting. Like if it a big chain, as such, and are at opposite ends but also meet. That would also lead to other theories, like, since they do meet at a point, then would tan90 be continuous? Is it not an asymptote? Just alot of new things possible. As of now, I don't think that is true, but anything is possible ^.^