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#3815 11/30/05 03:33 AM
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Rob, let's review. You say:

Rob:" Let's go back to numbers YET AGAIN. Divide infinity by two infinity times but you'll never get to zero. That's the point I've been trying to make since I GOT HERE."

And I gave you a traditional example. The fraction
(x-c)/[(x-a)(x-b)] IS zero in the limit x->infinity. Alvernatively, as The FF said, you can use l'Hospital theorem, and you get the same thing, zero, although in this case l'Hospital is unnecessary.

Then you are not happy with this example, because it involves the concept of infinity, and you say:

Rob: "Forget infinity; divide the number 1 by 2 forever and you still wont reach zero, EVER! Do you need an education to realise that?"

Well Rob, even in this example you cannot forget the infinity, since mathematically, your latest example can be rewritten as:

lim[n->infinity]{1/(2^n)}=0

You only think that you avoided the concept of infinity, but in fact you haven't. You still need to define what infinity is, and without any further consideration, mathematically it can be defined as 1/(2^infinity)=0. And this definition is also consistent with the previous fraction,if you can see that.

In both math and physics, you need to understand the concept of infinity. Infinity is not like any other number (integer, rational or real). It has special properties, but nevertheless, it can be consistently defines, say as above.

So, unless you still have some problems with these limits, we can put these issues to rest.

.
#3816 11/30/05 12:37 PM
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"Rob, let me ask you something. Consider the polynomial fraction (x-1)/[(x-2)(x-3)]. What is the value of this fraction as x becomes infinitely large (in the dedicated lingo as x tends/goes to infinity)?"

The answer is 1, if x = infinity.
I don't know what it is when x is infinitely large.
I'll get back to you on that, I have to go.

#3817 11/30/05 02:52 PM
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""Rob, let me ask you something. Consider the polynomial fraction (x-1)/[(x-2)(x-3)]. What is the value of this fraction as x becomes infinitely large (in the dedicated lingo as x tends/goes to infinity)?"

"The answer is 1,"
"The answer is 0, because the question is implicitly asking for a limit."

"if x = infinity."

X can never equal infinity. That's why we use the clumsy limit notation instead of just writing infinity in for those variables.


"I don't know what it is when x is infinitely large."

I don't understand the distinction between x being equal to infinity and x being infinitely large. This stuff is not actually calculus at this point. So far this is stuff that's covered pretty thoroughly near the end of algebra II.

#3818 11/30/05 03:20 PM
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To Rob: Rob, the limit of the fraction as x->infinity (which is exactly the same as saying that x is infinitely large) is zero, not 1 as you say.

To The FF: You can actually include "infinity" in the real axis, and define R^=[-infinity, infinity] instead of R=(-infinity, infinity). Formally, you just include into the real line the accumulation point of all divergent sequences, or series. It doesn't matter that much whether you define the real line as compact or non-compact, at least not at this level. So formally, you can make such that the number infinity is included in the real line.

#3819 11/30/05 08:38 PM
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"So formally, you can make such that the number infinity is included in the real line."

I can't say that you're wrong, but I can say truthfully that I don't follow the argument. (In particular, I'm not able to decipher "include into the real line ..."

I've read many scientific, mathematical, and (mostly) technical papers over the past 25 or so years that made use of infinity in one way or another. Not one has ever used infinity is as anything other than a transfinite number. In particular, none has ever used infinity as a real number.

I'm aware that some kinds of software (mathematica, for example) treat it as if it were a number. I always considered this to be a notational convenience. (It pretty much says exactly that on one of Wolfram's mathworld pages.)

The most dependable sites I can find on the web say that infinity is not a number. In fact, the only ones that I've found so far that maintain it IS a number are a newage site and an astrology site.

Do you have a reference that clearly expresses this idea? Doesn't necessarily have to be on the web.

#3820 11/30/05 09:10 PM
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TFF, the real line is another expression for the set of real numbers (there is a natural isomorphism between the set of real numbers and the points along an infinite straight line).

Now, if you think in terms of a line, an infinite line is a non-compact set as defined traditionally (does not include the end points at infinity). You can simply compactify it by including these points, and make the real line a closed interval.

Hm, references. Try something on modules, rings, fields,algebraic topology something like that. I will have to look for a more explicit reference.
But the ideea is simple to follow, algebraically speaking. Sure, it is a transfinite number. But you have to define how it behaves withe respect to the real numbers and the operations defined on the real numbers (infty +infty=infty, 1+infty=infty, infty*infty=infty, and so on and so forth). And the only problems you have are infty-infty, infty/infty, 0*infty and the similar. So you might as well treat it as a number, with these potential problems in mind.

#3821 11/30/05 10:21 PM
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"But you have to define how it behaves withe respect to the real numbers and the operations defined on the real numbers (infty +infty=infty, 1+infty=infty, infty*infty=infty, and so on and so forth). And the only problems you have are infty-infty, infty/infty, 0*infty and the similar. So you might as well treat it as a number, with these potential problems in mind."

That's part of my problem. If you include Inf with the reals (or integers), they violate the criteria of a ring. 0*Inf must equal 0 for it to be a ring. Nor is it a field, since there is no additive inverse for Inf. It is, however, a semiring. Alas, my background is in engineering math and not pure math, so I'm out of my depth beyond this point. It could be that forming a semiring is sufficient.

Nevertheless, we see at http://mathworld.wolfram.com/SurrealNumber.html the term "surreal number," which includes the real numbers and the transfinites, and also the term "Omnific Integer." The implication, I think, is that the transfinites are distinct from the reals (or the integers).

I know this part of my comment amounts to argument from authority. Wolfram could be wrong. Or he could be outdated. But if one is going to argue from authority, I reckon we ought to argue from the good one - and this is the best one I could find on the web.

#3822 12/01/05 10:43 AM
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"I don't understand the distinction between x being equal to infinity and x being infinitely large."

3.3333333333... is infinitely large, but it is not infinity. Actually, maybe it is...

#3823 12/01/05 12:44 PM
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Rob:"3.3333333333... is infinitely large, but it is not infinity. Actually, maybe it is... "

No Rob, this is not an infinitely large number. A practical definition for an infinitely large number, let's call it, say N , is something like this N-a~=N for whatever a a finite number (~= meaning "approximately the same", "almost the same").

"Largeness" of a number is not related to the number of its decimals, but to the ordering relation on the set of the real numbers (the ordering relation saying that any number is larger than another number and smaller than yet another number like 3.2<3.333333<3.4 ) So your example is incorrect.

#3824 12/01/05 12:48 PM
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TFF:"That's part of my problem. If you include Inf with the reals (or integers), they violate the criteria of a ring. 0*Inf must equal 0 for it to be a ring. Nor is it a field, since there is no additive inverse for Inf. It is, however, a semiring. Alas, my background is in engineering math and not pure math, so I'm out of my depth beyond this point. It could be that forming a semiring is sufficient.

You are right, including Infty in the real line will spoil your ring and field structure. So what? It is sufficient to have all structures valid (additive group and multiplicative semigroup ? this is mainly the ring structure) lest 1 point, the Infty. You have gotten to the same thing I was saying. You include Infty and you keep in mind that certain operations are not uniquely defined. This is in fact what you do in calculus, you just call it differently, and the differences are purely academic (for this purpose, that is). In fact, in calculus, you use Infty as a real number, when appropriate (when the operations are unambiguous like 1*Infty). When the operations are ambiguous, you deal with them differently, and you don?t particularly care too much for the fact that such operations are multivalued (in the sense that sometimes Infty/Infty in an expression can give you 1 and some other times, in a different expression it yields say 5).

TFF:?Nevertheless, we see at http://mathworld.wolfram.com/SurrealNumber.html the term "surreal number," which includes the real numbers and the transfinites, and also the term "Omnific Integer." The implication, I think, is that the transfinites are distinct from the reals (or the integers).?

As I said, I agree with you that it spoils the algebraic structures. At one point, and not all operations. Algebraically speaking. Not that it matters much, unless indeed you pursue a different goal, where such structures are fundamental.

But then, think of topology. You include Infty on the real line as a topological space without any problem, as the accumulation point of certain sequences. Just a different viewpoint, with no algebraic structure.

TFF:"I know this part of my comment amounts to argument from authority. Wolfram could be wrong. Or he could be outdated. But if one is going to argue from authority, I reckon we ought to argue from the good one - and this is the best one I could find on the web."

I am fine with refs, as long as they can be checked. But in the end, think about it. It is a matter of viewpoint and application. For algebraic structures, it doesn't make sense to include Infty in the real line because it spoils the structures, in calculus, you can do it as a matter of practicallity, if you remember that there are certain finer issues regarding multiplication and additive inverse, and in topology, you don't exactly care for any of these issues.

#3825 12/01/05 03:20 PM
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I agree that it's largely a matter of efficacy. The only reason I'm tied up on the ring/field thing is that I know that there are certain mathematical benefits to such things.

I suppose I had thought that the issue was decided long ago. I was warned back in algebra II that using Inf as a number was an expediency that was "not technically correct." I've seen people do this in lectures and presentations before, but never in a journal article. OTOH, we often *think* this way.

Topology is something I've studied cursorily on my own. It's not something that's covered in engineering curricula. I don't recall using algebra with it, but I was reading mostly on a baby level. Regardless, I can see the advantage in regular calculus of treating Inf as a number, if no other reason than to simplify notation of more complicated problems.

I'll take your word for it till I have a chance to look into this in more detail. I'm a pretty slow guy - and I gotta work through it at my own pace.

#3826 12/01/05 03:26 PM
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"3.3333333333... is infinitely large, but it is not infinity. Actually, maybe it is..."

You're having the same problem my youngest daughter had when I first taught her about pi. Neither pi nor 3.33... is "infinitely large." You "could" say that - and I understand your reasoning. But that's a sure fire way to confuse yourself.

Maybe a clearer way to think of it is:
"It has an infinite number of digits, BUT it's value is not infinitely large." Don't get discouraged. It took us (humanity) till the 1800s (Georg Cantor) to figure this stuff out - and the concepts were so strange and counter-intuitive that many brilliant mathematicians of the day rejected them outright.

#3827 12/01/05 06:06 PM
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To: TFF

Thank you for misunderstanding Pasti and stating the obvious.

Coming from you; your comments are condescending. And that is the nicest thing I can say about them.


DA Morgan
#3828 12/01/05 06:16 PM
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DA said: "Thank you for misunderstanding Pasti and stating the obvious."

No idea what you're talking about, which is to say that I don't know what you're talking about and I suspect that you, as usual, don't either.

If you check back, I was responding to Rob, not Pasti on that last post ... apparently it wasn't all that obvious to him. But unlike you who looks for every opportunity to say something nasty and condescending, I prefer to clarify the issue for him.

Regardless of which post you've just responded to, next time, before you post, DA, try taking a crow bar and prying your head from its current location.

#3829 12/01/05 07:34 PM
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TFF:"Topology is something I've studied cursorily on my own. It's not something that's covered in engineering curricula. I don't recall using algebra with it, but I was reading mostly on a baby level."

Think about it, it's really simple. You don't need too much topology.Topologically,the numbers are simply points on a line (the real line), and as such you don't need to define addition and multiplication of points (you don't need algebraic structures like groups, semigroups or fields/korps). You only need some ordering relation between points. So nothing stpos you to include -Infty and Infty into the real line as the inf and supp of the set of points. In this way you have just compactified the real line.

TFF:"Regardless, I can see the advantage in regular calculus of treating Inf as a number, if no other reason than to simplify notation of more complicated problems."

That was (part) of my point. The other part of the point was that as you said (better that I did), you can define a consistent algebraic structure over the real numbers that includes Infty - the semiring - in spite of the traditional wisdom that claims that in fact Infty is not a number. It is for a semiring, it isn't for a unitary ring or a field, depends on the aplication, context, etc.

TFF:"I'll take your word for it till I have a chance to look into this in more detail. I'm a pretty slow guy - and I gotta work through it at my own pace."

Fine by me, but hopefully the above cleared a bit what I wanted to say from the topological viewpoint.

#3830 12/01/05 07:55 PM
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The word "topologically" kinda threw me off. I thought you meant something more complicated than that. I see where this is heading.

The implication, then, is that Inf and -Inf are now taken as actual points on the number line.

I'm not familiar with 'compactification.' Is this a way of circumventing the common notion of orderliness, usu we think something along the lines of given a point x and a positive c, there exist x-c and x+c, such that x - c < x < x + c, obviously not true when x=Inf. The "compactification" is a way of getting around this common idea?

#3831 12/02/05 06:55 AM
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TFF:"The implication, then, is that Inf and -Inf are now taken as actual points on the number line."

Yep.

TFF:"I'm not familiar with 'compactification.' Is this a way of circumventing the common notion of orderliness, usu we think something along the lines of given a point x and a positive c, there exist x-c and x+c, such that x - c < x < x + c, obviously not true when x=Inf.The "compactification" is a way of getting around this common idea?"

No, this is not the ideea. BTW, you have already introduced an additive group structure on the real line when you wrote the ordering relation the way you did above. Think points - not coordinates - and sets of points. And even in your case it works if you define Infty=supp{<R>}, -Infty=inf{<R>} with respect to this ordering relation, where <R> is a s previously the real line with the infties included.

The ideea of compactification is something along these lines: (0,1) is a non-compact interval while [0,1] is a compact interval (you have included the ends of the interval in the latter set. Same thing with the real line, you include the infties and you end up with the compactified real line.

#3832 12/02/05 11:12 PM
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Ric: Edge of the universe:

Site: antwrp.gsfc.nasa.gov/apod/apo50925.html

Explanation: Analyses of a new high-resolution map of microwave light emitted only 380,000 years after the Big Bang appear to define our universe more precisely than ever before. The eagerly awaited results announced last year from the orbiting Wilkinson Microwave Anisotropy Probe resolve several long-standing disagreements in cosmology rooted in less precise data. Specifically, present analyses of above WMAP all-sky image indicate that the universe is 13.7 billion years old (accurate to 1 percent), composed of 73 percent dark energy, 23 percent cold dark matter, and only 4 percent atoms, is currently expanding at the rate of 71 km/sec/Mpc (accurate to 5 percent), underwent episodes of rapid expansion called inflation, and will expand forever. Astronomers will likely research the foundations and implications of these results for years to come.
There is an egg shaped picture offered in the above, which would, I think, cause many to imply that the universe has a positive shape and that our chicken got there first. We have narrowed down the origins to within 1% (no small thing) and the rate of expansion, about 71 km/sec., is concluded accurate to 5% of what may be happening in real time. With out the accepted credentials normally offered I must offer one layman?s view. Any average person reading and looking at the depiction should think that there is some edge to the universe. If the universe is constantly expanding we might wonder into what are we expanding? Possibly our universe is compressing some other universe that is compressing some other universe, etc.?

I suppose we can chalk it up to my ignorance of the relativity of space-time and I will concede that up front. Possibly we could best convey the idea by saying that the CONTENT of the universe is constantly expanding and that would offer some of us the prospect that the universe itself is everywhere so there is no need for an outside or an edge. If space, our idea of the universe is everywhere, then we are only concerned with the contents and edges play no part in our understanding.
So, while there is no apparent proof that our universe has an edge, or limit, there is evidence that the contents of our universe become larger as we improve vision.
jjw

#3833 12/05/05 03:27 PM
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If the universe is infinite, then it has no edge.
If the universe isn't infinite, and there are others, then the edge of this universe is about as interesting as the edge of a bubble in liquid, or an explosion in air.

#3834 12/05/05 09:14 PM
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Rob wrote:
"If the universe is infinite, then it has no edge.
If the universe isn't infinite, and there are others, then the edge of this universe is about as interesting as the edge of a bubble in liquid, or an explosion in air."

Your assumption seems to be that a finite universe has an edge, a barrier, a point beyond which one can not travel. Your assumption is not necessarily valid.

Consider a black hole and assume you were in the center. As you travel in any direction you will find that there is no edge, there is no barrier, there is, however, a very finite volume.


DA Morgan
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