What does dividing by zero really mean?

Just as clarification for democoder, I never said that 1/0=inf. I said that zero and infinity can be defined as limits in terms of each other in a consistent and simple way.

You're right in pointing out that algebraically 1/0 is undefined. The trouble is that in some systems zero itself is undefined (such as has been pointed out in string theory, or even considering heisenberg uncertainty's not allowing any measurable quantity to go definitely to zero). My point was that zero and infinity are intimately related concepts and I tried to discuss that without going into needless detail on what the algebra has to say about it.

 

Not really. For example lim (x->0) 1/x is undefined because when x approaches from the positive side of 0 the limit is infinity while on the negative side it is -infinity.

No, infinity and zero are not related at all. You said there are systems without zero, well there are plenty of systems without infinity either such as a finite group or ring. In fact, infinity is not a number in the normal real number system, while 0 is.

 

There are systems without a zero and systems without infinity, and they're never simultaneously defined (0*inf is undefined) - expressing them in terms of each other makes this point. It's more philosophy than hard math, but the concepts are different sides of the same coin. That's the point I was trying to make and that's the point my math professor was trying to make when he discussed it in lecture. Call it meta-math if you will, but it's not a rigorous idea.

At any rate, take it or leave it, but I thought it was worth bringing up.

 

Ahh but there are systems where they are both defined. Take the projective plane as an example. (Projective plane is basically your standard Cartesian plane union "infinity". It's topologically equivalent to a sphere.)

 

I honestly don't know anything about that. How is that they're symultaneously defined on the projective plane though? What is the definition of 0*inf in that system?

 

That's still a nonsense statement. The projective plane is the plane union it's closure (i.e the set of all limit points of the plane). "infinity" is the only limit point missing.

 

So does that tear down what I was saying or not? I'm not really following you.

 

Infinity isn't a number, it's a limit. That's why it just breaks everything to try to say 1/0 is infty, because lim(x -> 0+) = infty and lim(x -> 0-) = -infty. Clearly, positive and negative infinity are quite different concepts! It works for 1/(x^2), but again as a limit. 1/(0^2) is still not infinity. Saying 1/infty is 0 is fine because you can only take a left-hand limit of 1/x as x grows arbitrarily large.

Sqrt(-1) also isn't meaningless at all, because numbers are in themselves are pure abstractions. The problem with the real numbers is that they're not algebraically complete. i simply gives us the algebraic closure of the reals. The complex numbers are nice because they're sort of R^2 as a field instead of just a ring.

Just remember, infinity is not a number. It's a limit.

 

infty is a symbol, and you can define operations on it. My first post was badly worded or vague ( sometimes means in a context where you deal only with limes of positve values - I had the Radius of Convergence in my mind ), read my second post where I basically say the same as you.

 

If we are talking the reals, zero and infinity are not related or "flip sides" of the same coin, or any of the other constructions mentioned. I realize what you are trying to allude to (something like, both absolute zero and absolute infinity are both something one can only approach, but never realize in the real world. Or, more figuratively, something like "zero represents the absolute absense of something and infinity represents the absolutely absense of nothing") But unfortunately, this vague intuitive notion doesn't translate to the real number system as constructed.

1 / 0 is undefined, it is not infinity. It is undefined because there isn't one limit. Likewise, 0/0 is indeterminate.

As for projective planes, it depends on how they are constructed. Not all of them need to include a line "at infinity" When I worked on projective planes we defined them in terms of balanced incomplete block designs/steiner triples/latin squares.

For example, the projective plane of order 2 (the fano plane) consists of the following "points":
{1,2,4}, {2,3,5}, {3,4,6}, {4,5,7}, {5,6,1}, {6,7,2}, {7,1,3}

Those 3 numbers are not "coordinates" in X,Y,Z space, so don't be fooled. This is not 3-dimension space, it's pure combinatorics.

Given that set of sets, which we call "points", and there are seven of them, we say that any two points wherein the set has a member in common form a line (1,2,4 and 2,3,5 have '2' in common). Notice how every point in the above list has exactly one other point which a set member in common.

Those numbers weren't chosen by chance. They insure that every 2 points form a line, and every two lines define a point. So {1,2,4},{2,3,5} make a line, and {4,5,7},{7,1,3} unique determine a point. In this case, both {1,2,4},{2,3,5} and {4,5,7},{7,1,3} are lines which share the common point {6,7,2}

That is, {1,2,4},{2,3,5},{6,7,2} all share one line (the line with common member 2) and {4,5,7},{7,1,3},{6,7,2} all share one line (the line with common member 7). Together, they share point {6,7,2} in common.

This is a projective plane, yet there is no special "infinity" element added.

 

A plane of 7 points? A line with 2 points? Wtf?

I never cared much for combinatorics. I was much more interested in topology and analysis.

 

Actually the combinatorics definition is more general, just like abstract algebra or ZFC set theory is more general than real analysis.

See http://en.wikipedia.org/wiki/Projective_plane

I've always liked abstract algebra, number theory, combinatorics, and the other "finite" and symbolic maths because of the level of abstraction as well as the connections with notions of computability, codes, cryptography, etc.

The only analysis class I really enjoyed was Complex Analysis.

With finite projective planes there are actually significant unsolved problems as to the "existence" of projective planes of order N. For example, we think N must be a prime power, but it isn't proven.

If there is one branch of math that is most "spooky" to me it's the theory of Finite Simple Groups. Simple Groups are like the periodic table of elements for groups or like the primes compared to composites. In that, ultimately, every group can be decomposed and shown to be isomorphic to a bunch of simple group factors.

But why should there be only a finite number of classes of fundamental "building blocks" for all groups, and why should this number be 18 types of infinite familities of groups plus 26 "sporadic groups" Why are there 26 and only 26 sporadic groups? Why is the largest sporadic group, the Monster Group, intimately related to string theory? There are 26 sporadic groups and 26 dimensions in some string theories, coincidence? How come there is no sporadic group bigger than the monster, and what is the significant of the prime numbers used to construct it?

Why should the fundamental structure of all possible algebraic groups you can define and a separately developed string theory of the physical universe have any "deep" connection whatsoever? One is pure deductive logic from *first principles*, the other relies on physical observations to produce a model consistent with the standard model. The monster group is "weird" and unique, and it's amazing that other branches of math were found to be related to this "ultimate" group.

These are the kinds of deep mysteries that keep me very interested in mathematics.

 

That is what I was digging at democoder. Probably not a point worth considering since it doesn't have any direct connection to the real number system as it is. In fact, it was in a math history class that my math professor briefly discussed the thought and he mentioned it after having discussed historical number systems that didn't include zero.

Now that I think about it, that was the last math class I ever took - exluding mathematical methods for physics type courses.

 

If we "Indians" hadn't invented Zero ("Shunya" in Sanskrit) and Decimel system....then..

 

Then we'd be using the Babylonian sexagesimal system which was invented 2000 years before the Indian zero. Or we'd be using the Ptolemic zero, or the Roman Numeral nulla. Or, the Mexican/Mayan zero.