Is Zero x Infinity Really a Real Number?

[Grimble]

OK. I get it. Infinity is not just a very large number as in everyday usage, but strictly a quantity without boundary.
Then would it be correct to say that one cannot add something to an infinite quantity? As in Hilbert's Hotel?
For if every guest was moved on one room and another guest were added in room 1, then would that not mean that the number before that addition was less than infinite?
I know, I know, mathematically that might be nonsense; I am just thinking aloud.

But if ∞ means a number without limit, then what exactly does a mathematician understand zero to mean?
Is it a definite, precise number? Or is it more of an idea?

 

[Grimble]

And with respect to this:

For Example; one might say the number of molecules of water in the oceans is infinite ∞, yet one also knows that the number of hydrogen and oxygen atoms in those molecules is exactly 3 times as many, 3 ⋅ ∞ , yet that term is meaningless. Much better perhaps to say that ∞ is not a term that can be manipulated mathematically?

If we I had said "take an infinite number of water molecules... ?

 

[Mark44]

OK. I get it. Infinity is not just a very large number as in everyday usage, but strictly a quantity without boundary.
Then would it be correct to say that one cannot add something to an infinite quantity? As in Hilbert's Hotel?
For if every guest was moved on one room and another guest were added in room 1, then would that not mean that the number before that addition was less than infinite?

No, it wouldn't. If you add any finite number to infinity, you still get infinity. If you subtract any finite number from infinity, you get infinity.

I know, I know, mathematically that might be nonsense; I am just thinking aloud.

OK

But if ∞ means a number without limit, then what exactly does a mathematician understand zero to mean?
Is it a definite, precise number? Or is it more of an idea?

Zero is a defined, precise number. It is exactly equal to the number of elephants I own.

If you have $76 in your checking account, and write a check for $76, your account balance will be $0 (before the bank tacks on an NSF charge).

 

[Samy_A]

OK. I get it. Infinity is not just a very large number as in everyday usage, but strictly a quantity without boundary.
Then would it be correct to say that one cannot add something to an infinite quantity? As in Hilbert's Hotel?
For if every guest was moved on one room and another guest were added in room 1, then would that not mean that the number before that addition was less than infinite?
I know, I know, mathematically that might be nonsense; I am just thinking aloud.

Well, there you again are confronted with the fact that doing usual arithmetic with ∞ doesn't work.
For example, intuitively there are more integers than natural numbers, as a natural number is an integer, but the converse is not true.
However, mathematically, the set of integers (ℤ) is "as big" as the intuitively "smaller" set of natural numbers(ℕ). Why is that? Because we can map one set to the other with a one to one and onto function, for example:
##f: \mathbb N \to \mathbb Z##
##f(n)=n/2## if ##n## is even
##f(n)=-(n+1)/2## if ##n## is odd

(note, just to be clear: I include 0 in ##\mathbb N## here)

So adding elements to an infinite set doesn't necessarily make it bigger.

 

[Mark44]

And with respect to this:

If we I had said "take an infinite number of water molecules... ?

Putting aside the impossibility of finding an infinite number of water molecules, you would have an infinite number of hydrogen atoms and and infinite number of oxygen atoms.

Basically, if k is any positive, finite number:
##\infty + k = \infty##
##\infty - k = \infty##
##k \cdot \infty = \infty##
##\infty/k = \infty##

 

[jbriggs444]

Putting aside the impossibility of finding an infinite number of water molecules, you would have an infinite number of hydrogen atoms and and infinite number of oxygen atoms.

Basically, if k is any positive, finite number:
##\infty + k = \infty##
##\infty - k = \infty##
##k \cdot \infty = \infty##
##\infty/k = \infty##

To clarify what Mark44 knows perfectly well and has elected not to say out loud, these are rules for an arithmetic that has been extended to include ##\infty##.

For instance, these rules fit with cardinal arithmetic. In cardinal arithmetic if one has two disjoint sets A and B and if the cardinality of A is a and the cardinality of B is b then a + b is given by the cardinality of ##A \bigcup B##.

You get into difficulties if you get carried away try to pretend that cardinal arithmetic is just like ordinary arithmetic and that ##\infty## is just like any other number. For instance one might define things so that ##\infty + \infty = \infty## but then ##\infty - \infty## cannot be consistently defined. It is because of these sorts of difficulties that we often avoid defining arithmetic operations on infinite quantities.

There is a something philosophical buried here. A common picture is that mathematics is the process of discovering the properties and relationships of some kind of pre-existing set of numbers. This is the Platonic view. From this standpoint, axioms of arithmetic are taken to be true and obvious statements. A more modern perspective is that it is the process of discovering the properties that a set of numbers would have if it existed and adhered to a particular set of axioms. The axioms need not be "true", so long as the result is an interesting and consistent theory [the notion of truth is interesting in its own right]. Some sets of axioms entail the existence of ##\infty##. Some sets of axioms explicitly deny it. You can do useful mathematics either way. There is no need (and no way) to determine which set of axioms is correct.

From the modern perspective, simply writing down ##\infty## is somewhat ambiguous unless context allows us to figure out what set of axioms or system of arithmetic you are talking about.

 

[Grimble]

So, ∞ refers to a set without a boundary. One may add or subtract, multiply or divide it by any quantity and it will still be a set without a boundary or without a limit (however one says that mathematically - I am not a mathematician so please don't get all pedantic; I am trying to express ideas here - correct my terminology or language by all means, but please don't say I am meaning something other than what I an doing my best to express... )

So when I say an infinite number of water molecules, that comprises an infinite number of oxygen atoms and an infinite number of hydrogen atoms...
two infinite sets of items.
And I get it that the total number of items from two infinite sets is still infinity, that is taking the two sets of items and adding them together;
just as if we take one set of the two we will still have an infinite total;
but how many do we have if we take neither infinite set? Is the answer to that not 0?
What I am envisaging here is 2 x ∞ = ∞; 1 x ∞ = ∞; but 0 x ∞ = 0 in my simplistic way

 

[Mark44]

So, ∞ refers to a set without a boundary.

Not a set. From the wikipedia article (https://en.wikipedia.org/wiki/Infinity):

Infinity (symbol: ∞) is an abstract concept describing something without any bound and is relevant in a number of fields, predominantly mathematics and physics. In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as natural or real numbers.

Note that this article does not describe infinity as a set.

One may add or subtract, multiply or divide it by any quantity

You can add any finite number to infinity, or subtract any finite number from infinity, with the result still being infinity. You can multiply infinity by or divide infinity by any positive finite number, the result is still infinity. If you multiply or divide by a negative finite number, you get negative infinity.

Multiplication of infinity by zero is not defined.

If n is any real number:
##\infty + n = \infty##
##\infty - n = \infty##

If n is any positive real number:
##\infty * n = \infty##
##\infty / n = \infty##

If n is any negative real number:
##\infty * n = -\infty##
##\infty / n = -\infty##
Intentionally omitted here is the case where n = 0.

Also
##\infty + \infty = \infty##
##\infty * \infty = \infty##

Intentionally omitted are ##\infty - \infty## and ##\infty / \infty##, which are two of several types of indeterminate forms.

and it will still be a set without a boundary or without a limit (however one says that mathematically - I am not a mathematician so please don't get all pedantic; I am trying to express ideas here - correct my terminology or language by all means, but please don't say I am meaning something other than what I an doing my best to express... )

So when I say an infinite number of water molecules, that comprises an infinite number of oxygen atoms and an infinite number of hydrogen atoms...
two infinite sets of items.
And I get it that the total number of items from two infinite sets is still infinity, that is taking the two sets of items and adding them together;
just as if we take one set of the two we will still have an infinite total;
but how many do we have if we take neither infinite set? Is the answer to that not 0?
What I am envisaging here is 2 x ∞ = ∞; 1 x ∞ = ∞; but 0 x ∞ = 0 in my simplistic way

No. You asked this in your first post of this thread, and the answer is still no. Please go back and reread this thread.

 

[micromass]

A lot of the confusion here arises because there simply isn't one thing called infinity. There many multiple related and unrelated concepts in mathematics which get the name infinity. There is not something like the infinity.

In many systems of infinity (like those used in calculus and elementary real analysis), something like ##0\cdot \infty## is left undefined. However, in many others (transfinite numbers, hyperreal numbers, surreal numbers, affine real line in measure theory, etc), we do have ##0\cdot \infty = 0##.

So do you want to argue that ##0\cdot \infty = 0##. Sure, go ahead. A lot of mathematicians use this succesfully. But you need to be aware that it only holds in certain systems and not in others. So you need to be clear with which notion of infinity you're working.

 

[Grimble]

Thank you, Micromass, that certainly makes sense!
Thank you too, Mark44, but if set is the wrong term for what I am trying to refer to, what is the correct term? Is there one?
(I was reading the Wiki entry https://en.wikipedia.org/wiki/Infinite_set...)

 

[UncertaintyAjay]

I only wanted to try to proof we can compare different undefined values and set that they can be equal.

This made me think of Cantor's infinite cardinal numbers for sets. I suggest you google transfinite numbers. You'll find it interesting. It's one of my favorite things in math. Admittedly i don't know too much math. I hope to study it for undergrad but still... My favourite thing in math. Also check out Hilbert's Hotel Infinity. Saying that it's awesome is a massive, massive understatement.

 

[Mark44]

Thank you too, Mark44, but if set is the wrong term for what I am trying to refer to, what is the correct term? Is there one?
(I was reading the Wiki entry https://en.wikipedia.org/wiki/Infinite_set...)

Look at this Wiki article - https://en.wikipedia.org/wiki/Infinity

 

[Grimble]

Thank you Mark44, but in the introduction of that article it reads: "Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[1] For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable.[2]"
Which I feel is very much the use of 'set' that I was taking...

 

[Mark44]

Thank you Mark44, but in the introduction of that article it reads: "Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[1] For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable.[2]"
Which I feel is very much the use of 'set' that I was taking...

My objection is to what you wrote in post #42, quoted below. ∞ does not "refer to" a set, but rather to something more akin to a number. You don't add something to infinity and get a set.

So, ∞ refers to a set without a boundary. One may add or subtract, multiply or divide it by any quantity and it will still be a set without a boundary or without a limit

You are confusing the concepts of infinity and infinite set. To help you understand the difference between these two concepts, consider the set {1, 3, 5}, a set with three elements. The set ({1, 3, 5}) and its size (3) are two entirely different things.

The set of natural numbers, {1, 2, 3, ... } is an infinite set. Its cardinality, a measure of the number of elements in this set is infinity (∞). Again, you need to distinguish between a set of numbers, and the size of the set. As an aside, the cardinality of the natural numbers, {1, 2, 3, ...} is ##\aleph_0##, (Aleph-null), a particular size of infinity.

 

[bahamagreen]

If n is any real number:...

Just wondering...
What about infinity plus 5+3i ?
Is that infinity, but a different kind of infinity than adding a real number?
I mean, is one case an infinite number of reals and the other case forced to be an infinite number of complex numbers (some without an imaginary part)?
Or is infinity always the count despite the variation in contents?
If so, is infinity always characterized as the count in terms of integers?

 

[micromass]

Just wondering...
What about infinity plus 5+3i ?
Is that infinity, but a different kind of infinity than adding a real number?
I mean, is one case an infinite number of reals and the other case forced to be an infinite number of complex numbers (some without an imaginary part)?
Or is infinity always the count despite the variation in contents?
If so, is infinity always characterized as the count in terms of integers?

There is no one thing called infinity. If you want to know that ##\infty + 5 + 3i## is then you need to specify which formalism you're working with.

 

[Georgers]

It seems to me that with regard to Hilbert's Hotel that there is something "illegal" about taking (pushing) someone out of his room temporarily to create a vacancy. One element will always be temporarily outside the set (that is, not in a room). Also doesn't this method rely on an assumption that there is actually an empty room somewhere along the line (at the end?) or that one will eventually be created? Must be one or the other I think.

It seems like another method of creating a room for the new guest in the Hilbert Hotel would be for the manager to politely ask Room 1 to move up to Room 2 and then Room 2 would politely ask Room 3 to move up and so on. They would then wait until the room above them was vacated before moving. Using this method the new guest never would get a room. Seems like it's just a matter of good manners whether this works or not!

 

[Mark44]

It seems to me that with regard to Hilbert's Hotel that there is something "illegal" about taking (pushing) someone out of his room temporarily to create a vacancy. One element will always be temporarily outside the set (that is, not in a room). Also doesn't this method rely on an assumption that there is actually an empty room somewhere along the line (at the end?) or that one will eventually be created? Must be one or the other I think.

There is no assumption about there being an empty room at the "end." By asking each person to move to the room with the next higher room number, a space for a new guest is created at room 1. Also, as the Hilbert Hotel is strictly a thought experiment, the room switches can occur simultaneously, and there is no concern about asking all those people to move.

It seems like another method of creating a room for the new guest in the Hilbert Hotel would be for the manager to politely ask Room 1 to move up to Room 2 and then Room 2 would politely ask Room 3 to move up and so on. They would then wait until the room above them was vacated before moving. Using this method the new guest never would get a room. Seems like it's just a matter of good manners whether this works or not!

 

[Ssnow]

If you admit that ##0^0## is undefined then ##0^0=0^{1-1}=0^1\cdot 0^{-1}=0\cdot \lim_{x\rightarrow 0^{+}} x^{-1}=0\cdot \lim_{x\rightarrow 0^{+}}\frac{1}{x}=0\cdot (+\infty)## (the same happen for ##x\rightarrow 0^{-}## with ##-\infty##). Thus also ##0\cdot \infty## is undefined ...

Ssnow

 

[jbriggs444]

If you admit that ##0^0## is undefined then ##0^0=0^{1-1}=0^1\cdot 0^{-1}=0\cdot \lim_{x\rightarrow 0^{+}} x^{-1}=0\cdot \lim_{x\rightarrow 0^{+}}\frac{1}{x}=0\cdot (+\infty)## (the same happen for ##x\rightarrow 0^{-}## with ##-\infty##). Thus also ##0\cdot \infty## is undefined ...

Unfortunately, that argument is a string of errors.

If ##0^0## is undefined then you cannot use it in an equality.
Since ##0^{-1}## is also undefined, you also cannot use it in an expression.
The law of exponents (##a^{b+c}=a^ba^c##) is only viable for strictly positive bases.
When a limit fails to exist, you cannot use it in an expression.
The limit of a product is not necessarily equal to the product of the limits. [For finite limits that exist, the result holds]

You cannot determine what a definition will say from first principles. Definitions are created by humans and are arbitrary. If I want to define ##0 \cdot \infty## as ##\frac{\pi}{2}##, no argument you make can prove that definition incorrect. Argument can only show that the definition is inconsistent with other things that you want to remain true.

 

[Ssnow]

Unfortunately, that argument is a string of errors.

yes sure, as many heuristics proofs ...

I want to observe that there are mathematicians of the opinion to consider ##0^0## defined equal to ##1## ( this is not true in general ...) at least in particular cases, this happens when you think a value for the self-exponential ##f(x)=x^x## at ##x=0## ...

 

[dkotschessaa]

To clarify what Mark44 knows perfectly well and has elected not to say out loud, these are rules for an arithmetic that has been extended to include ##\infty##.

Here's how Rudin puts it:

I'm not sure this is actually ever used later.

 

[dkotschessaa]

I'm still a sucker for threads about infinity.

 

[S.G. Janssens]

I'm not sure this is actually ever used later.

For example, in the extended reals every increasing sequence has a limit. This is convenient in measure theory, for instance (but not only) when formulating the monotone convergence theorem.

 

[Logical Dog]

yes I also was of op views of multiciplication being glorified addition. (primitive view)

but then I saw the sequences and their limits combined and it is surprising;

 

[dkotschessaa]

Isnt multiplication just glorified addition? so according to my primitive views,=) if so how you can keep adding zero infinitely to get zero IMO.

Sometimes primitive ideas are the most informative.

I could see defining ## x * \infty = x + x+ x + \cdots ## i.e an infinite sequence. Perhaps this has already been said in this thread in a different way.

-Dave K

 

[Logical Dog]

Sometimes primitive ideas are the most informative.

I could see defining ## x * \infty = x + x+ x + \cdots ## i.e an infinite sequence. Perhaps this has already been said in this thread in a different way.

-Dave K

yes but that series it would (?) seem to diverge to infinity or minus infinity, and converge to zero if x=0. IIRC

 

[Logical Dog]

this man HATES much of set theory and calculus, and real numbers...very interesting debate:

 

[dkotschessaa]

yes but that series it would (?) seem to diverge to infinity or minus infinity, and converge to zero if x=0. IIRC

Well, that's would I would expect it to do!

 

[dkotschessaa]

this man HATES much of set theory and calculus, and real numbers...very interesting debate:

Yeah, there are some.

I've considered being a finitist if only to be contrary.