In some systems of “numbers”. In other systems of “numbers” of interest, there’s zero, one, or two infinities. It depends on what you’re talking about. The same system of cardinal numbers that always comes up in this conversation happens to also have no fractional or negative numbers; it’s not the only system appropriate for discussing the idea of infinity in all contexts, just the one appropriate for discussing the idea of infinite sets modulo one-to-one correspondence.
No. 1/any-number-greater-than-1 yields a fraction between 0 and 1. 1/0.5 certainly isn’t between 0 and 1.
Well everyone knows that infinity plus one beats infinity, unless you call no infinity first. But you can’t really enhance zero. If someone says ‘I owe you nothing!’, try responding, ‘You owe me nothing plus one!’. It will have no effect and sounds stupid. Say ‘You owe me zero plus infinity!’ is also fruitless. So while zero represents the ultimate nothing, infinity speaks of a endless vastness that cannot be sufficiently analogized with material things, or their lack.
So infinity and zero can be considered opposites in their scope, but it forms a narrow inverse analogue.
As for which infinity to use? I prefer the infinite number of sets which are not members of themselves.
Let me just note that there’s no mathematical definition at all for “two sides of the same coin.” This isn’t a matter of there being several possible definitions in several different situations. There’s no such definition at all.
Yeah, that’s true; but I’m sure the OP was just informally asking something like “Is there merit to analogizing these?”. Still, so far as a rigorous definition of “two sides of the same coin” goes, let’s begin by defining a simplicial complex as… 
Two sides of the same coin up to homeomorphism:p
Mathemeticians recognize three orders, or “strengths” of infinity [by convention, denoted by the Hebrew letter “Aleph”]:
0[FONT=Times New Roman][SIZE=6]א [/SIZE][Aleph Sub-Zero]—The infinity of all counting numbers (1, 2, 3,…), all integers (-2, -1, 0, 1, 2,…), all rational numbers (1⁄1, 1⁄2, 2⁄1, 1⁄3, 2⁄2, 3⁄1,…)
1[FONT=Times New Roman][SIZE=6]א [/SIZE][Aleph Sub-One]—The infinity of all points on a line, all points in a plane, all points in a 3-Dimensional Space…
[/FONT]
2[FONT=Times New Roman][SIZE=6]א [/SIZE][Aleph Sub-Two]—The infinity of all possible curves.
[note: the subscript is usually on the right, but the formatting controls on this board will not allow it. :(]
One way of expressing this is that there are as many even counting numbers as all integers, as many integers as all numbers divisible by 10, or 1,000, or ten million!
But there are more points on a line one inch long than there are integers, and there are as many points [in the mathematical, zero-dimension sense] in the known universe as there are points on a one-inch-long line.
And there are more possible curves than there are points in the known universe.
The arithmetic of infinity can be confusing and counter-intuitive:
∞ + ∞ = ∞
∞ - ∞ = ∞
∞ × ∞ = ∞
∞ / ∞ = ∞
BUT
∞ ^ ∞ [infinity raised to an infinite power] is a higher-order infinity!
I don’t understand the assertion in the OP:
as any analog output may take on an infinite number of points within its span, and there are many other ways to demonstrate infinity in real-world, practical terms.
Other posters have pointed out the many ways in which you may have “zero” of any kind of countable item. Certain physical processes may not take on a value of “zero”, but that does not justify this assertion.
The assertion in the OP:
is just flat-out wrong. Taking the limit of 1/x as x approaches zero from the positive side of the number line approaches positive infinity. The limit as x approaches zero from the negative side approaches negative infinity. AT x = 0, however, the value of the function 1/x is equal to every number from +∞ to -∞! The standard way of expressing this is the function 1/x is undefined for x=0, because allowing division by zero permits a proof that any given number is equal to any other number.
[/FONT][/FONT]
The aleph-numbers are cardinalities; I reiterate that often cardinalities are not the number system of most relevance. However, even among the cardinalities, there are far more than three aleph numbers (there are infinitely many [more precisely, there is a “large” class of “small” alephs]). Furthermore, the (classical) number of points on a line is beth_1 and number of curves is beth_2; it is only under the generalized continuum hypothesis that the beth numbers happen to coincide with the aleph numbers, an assumption which is in many contexts unreasonable.
I don’t know why you say ∞ - ∞ = ∞ and ∞ / ∞ = ∞; in many contexts, one would want to take these expressions as indeterminate rather than necessarily equal to ∞, including in the context of cardinal arithmetic with subtraction and division being given as cancellation of addition and multiplication (in turn given by coproduct and product of sets).
Mostly because, in the case of most infinities (for example, the set of positive and negative integers), subtracting an infinite set (e.g., the even numbers) from the whole leaves an infinite set (the odd numbers) which is an infinity of the same order. Or, taking the infinite set of real numbers and dividing by any infinite number (any subset of the set of real numbers), the result will be an infinity of the same order as the reals.
There certainly are some “contexts” in which this would not be the most appropriate way of expressing it, but the level of understanding implied in the OP suggests it is an adequate visualization of the property of infinities. Many learned treatises have been written on the subject. It would not be practical to reprint major excerpts here.
Suppose you subtract the odd primes from the primes; both are countably infinite sets, but you won’t be left with an infinite set; you’ll be left with a one-element set. Suppose you subtract the integers from the integers; both are countably infinite sets, but you won’t be left with an infinite set; you’ll be left with a zero-element set. Suppose you subtract the integers greater than 7 from the integers greater than 4; three-element set. Etc. This is precisely the indeterminacy I was talking about.
So, you are saying that subtracting the infinte set of even numbers from the infinite set of integers will not result in an infinite set of odd numbers?
I seem to remember stating,
and there are most definitely examples wherein this may not apply, but, as a general rule, the arithmetic expressions for infinity apply for most situations. I can, if you’d like, post vast sections of advanced mathematics texts on this subject, but I was trying to keep it simple and understandable. Unfortunately, simplicity sometimes brings with it insufficiency. To detail every caveat and qualification would get tedious, and I think beyond the scope of this discussion.
Obviously, sometimes, removing a infinite set from an infinite set of the same order results in an infinite set. I’m not denying that. But just as well, sometimes, indeed quite often, it results in a finite set. That’s the whole point of my saying it’s indeterminate!
Why would anyone claim that most of the time, removing an infinite set from an infinite set results in an infinite set when so many examples, including the most basic examples (remove the integers from the integers; remove the positive naturals from the naturals; remove (0, 1) from [0, 1]; etc.), do not do so? Yes, removing the even integers from the integers results in an infinite set of odd integers. Why should that particular subtraction uniquely define ∞ - ∞ as ∞, as opposed to the other mentioned just as natural subtractions producing 0, 1, and 2? (And just as easily, whatever else…)
If you would like to post vast sections of advanced mathematics texts which address the specific point of why I should consider removing the even integers from the integers somehow more canonical than removing the integers from the integers, go ahead. But I’ve been able to make my argument for indeterminacy quite simply.
More to the point, what’s yours?
The point is that Silmandil was trying to claim that since 1/x approaches ∞ as x approaches 0 from the positive side, therefore 1/0 must be defined as ∞. But by the same argument, since 1/x approaches -∞ as x approaches 0 from the negative side, therefor 1/0 must be defined as -∞. So just looking at the limit on one side doesn’t produce a wholly consistent definition of 1/0.
Thanks for that correction. Of course, you’re right. I had already realized my statement was simplistic because of the negative numbers moving to the other side of zero, and thus forcing my universe to go from -1 to 1.
Wouldn’t you just then say that infinity doesn’t have a sign, the same way zero doesn’t?
That is indeed what is done in the projective reals (you can think of these as representing ratios (such as slopes of lines), infinity representing the ratio of any non-zero quantity to zero (such as in the slope of a vertical line) just as zero represents the ratio of zero to any non-zero quantity (such as in the slope of a horizontal line); a ratio of two items with the same sign is positive and of two items with opposite sign is negative, but infinity, like zero, is neither of those, in this system).
Lumpy, sure, you could do that, but then you’d be changing the standard definition of infinity. As Indistinguable pointed out, you can do that in the projective real numbers, but that’s rather different from the standard real numbers (with plus and minus infinity added). The OP seemed to be saying that using the standard definition of infinity you could say that 1/0 equals infinity, and that’s not true… What I’m trying to do here is stick to the standard definitions of these terms (i.e., the definitions that most people learned in high school). (If I had not stuck to the standard definitions, you would instead be saying, "What’s wrong with you mathematicians? Why do you always have to make up your own definitions just to confuse me?)
Using the standard definitions, most people would say the following:
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + . . . = 2
(-1) + (-1/2) + (-1/4) + (-1/8) + (-1/16) + (-1/32) + . . . = -2
1 + 1 + 1 + 1 + 1 + 1 + . . . = ∞
(-1) + (-1) + (-1) + (-1)+ (- 1) + (-1) . . . = -∞
and they would say that this is at least a consistent definition for ∞ and -∞.
If you were to tell them that instead the last equation should be this:
(-1) + (-1) + (-1) + (-1)+ (- 1) + (-1) . . . = ∞
they would probably say, “Well, that’s not very consistent. I finally figured out what infinity means, and now you’re changing the definition.”
C’mon, Wendell, that’s no worse than having 1 + 1/2 = 3/2 and 1 + 1/2 = 6/4 simultaneously. Clearly, if one is going to take ∞ as unsigned the way 0 is unsigned, then ∞ will equal -∞ and there will be no inconsistency. I doubt most high school math classes bother to discuss the differences between the affinely extended and projectively extended reals, but the right thing to do as people begin to grasp and voice the reasons why one might want to consider an unsigned infinity in the context of meromorphic functions is to point out precisely that there are these different systems, and how each is relevant and useful in some contexts, despite their differences, and what the motivations for each are, and… To have this whole discussion, basically, but without the refrain of “Don’t do that!”.
The layperson who thinks “Hey, it looks like the graph of 1/X has gone off to negative infinity and then returned right back at positive infinity; seems like it’d be worthwhile to think about identifying the two as though in a loop” deserves a pat on the back, not the stern voice of authority saying “That’s not how it’s done (except when it is, but only by trained mathematicians; you don’t get to.).” That layperson has just demonstrated more mathematical insight than most students ever get to experience; how tragic it would be to stamp it out rather than use it as a launching pad.
What do you mean by “the standard definition of infinity,” though? I can’t recall seeing a definition of “∞” per se, at a high school or early college level—only an explanation of what the symbol is used to mean in various contexts.
When a beginning calculus textbook says something like “the limit of such-and-such = ∞,” there’s a temptation to believe, since the infinity symbol is on one side of an equals sign, that infinity is an actual thing that you can talk about being equal or not equal to other things or performing operations on or whatever. But really, saying that some limit “= ∞” is just a convenient notation for saying that the limit doesn’t exist because the function gets arbitrarily large.
You could work with a system where ∞ was definited as a point or “number” with a particular status and properties, but as far as I’m aware, this isn’t typically done in Calculus 1-level math.