Is Zero x Infinity Really a Real Number?

[Grimble]

This seems a very simple case to me, yet I have heard it said that the answer is some undefined real number.

Yet zero times anything means no iterations of whatever the object is; whether that be a real number , an imaginary number or an undefined number.

Whatever it is I don't see how one can get away from the fact that what is specified is zero iterations of that number! So how can zero times anything not be zero?

Looked at from the other side, it means an infinite (that is an undetermined number) times nothing and however many times one iterates this process the answer must still be nothing (or zero)!

 

[Samy_A]

Is this about limits?

##0.\infty## is undefined in ##\mathbb R##.
But, if you have two sequences, ##(a_n)## and ##(b_n)##, with ##\displaystyle\ \lim_{n\rightarrow +\infty}a_n=0##, and ##\displaystyle\ \lim_{n\rightarrow +\infty}{b_n}=\infty##, then ##\displaystyle\ \lim_{n\rightarrow +\infty}a_nb_n## could be ##0##, another real number, ##\infty##, or undefined.

For example, take ##a_n=n, \ b_n=1/n##, then ##\displaystyle\ \lim_{n\rightarrow +\infty}a_n=\infty##, ##\displaystyle\ \lim_{n\rightarrow +\infty}b_n=0##, and ##\displaystyle\ \lim_{n\rightarrow +\infty}a_nb_n=1##.

Or: ##a_n=n\sin²(n), \ b_n=1/n##, then ##\displaystyle\ \lim_{n\rightarrow +\infty}a_n=\infty##, ##\displaystyle\ \lim_{n\rightarrow +\infty}b_n=0##, and ##\displaystyle\ \lim_{n\rightarrow +\infty}a_nb_n=\lim_{n\rightarrow +\infty}\sin²(n) ## is undefined (limit doesn't exist).

 

[HallsofIvy]

What you say about "0", "Yet zero times anything means no iterations of whatever the object is; whether that be a real number , an imaginary number or an undefined number" simply makes no sense. Multiplication itself is defined as an operation on numbers- we first define multiplication on "natural numbers", then extend that definition to "integers", then "rational numbers", "real numbers", and "imaginary numbers" but it makes no sense to even talk about defining multiplication on "undefined numbers"! We simply can't say anything about what happens when we multiply any number, including 0, by an "undefined number". Now, there are ways of extending the real or complex numbers to include "infinite" (as well as "infinitesimal") numbers so that we have defined "infinity" but how we define multiplication of such numbers, and the result of such a multiplication, depends on exactly how we have defined those numbers- and there several different ways to do that.

 

[Grimble]

No.It is not about limits nor any particular realm of mathematics; it is purely about the logical meaning of zero x anything. That whatever it is that one multiplies by zero. one is saying that one wants none of whatever it is, therefore the result has to be zero.

 

[Samy_A]

No.It is not about limits nor any particular realm of mathematics; it is purely about the logical meaning of zero x anything. That whatever it is that one multiplies by zero. one is saying that one wants none of whatever it is, therefore the result has to be zero.

You have to define what multiplication by zero means. For a real number ##x##, clearly ##x.0=0##.
But if you want to look at ##0.\infty##, you have to define what you mean (I gave the example of limits because that is a context where I sometimes see expressions as ##0.\infty## used informally).

You are of course entitled to define ##0.\infty=0## if you so wish, but the real question is if this is useful.
For example, with this definition the product rule for limits isn't true anymore (see my previous post for examples).

The point of course is that you can't do usual arithmetic with ∞.

EDIT: weird, @HallsofIvy 's post only became visible 25 minutes after he posted it.

 

[Grimble]

Excuse me if this seems a bit basic, ∞ is a symbol representing something, a quantity that is undefinable. but a quantity nonetheless.
Zero x ∞ therefore means zero times that quantity, zero iterations of that quantity - however big it is -or else there has to be a limit beyond which the quantity ∞ means something tangible beyond being a mere quantity?
How can zero times any quantity be other than zero?

 

[Samy_A]

Excuse me if this seems a bit basic, ∞ is a symbol representing something, a quantity that is undefinable. but a quantity nonetheless.
Zero x ∞ therefore means zero times that quantity, zero iterations of that quantity - however big it is -or else there has to be a limit beyond which the quantity ∞ means something tangible beyond being a mere quantity?
How can zero times any quantity be other than zero?

I'll play along. Let's define ##0.\infty=0##.
Using similar logic,we should define ##x/\infty=0## for any ##x \in \mathbb R##.
There is nothing forbidden with these definitions, but the problem is you can't do arithmetic with them.

For example: ##2/\infty=0##, but ##0.\infty=0##, therefore we get ##(2/\infty).\infty=0##. Simplifying by removing the infinity for numerator and denominator, we get ##2=0##. That's tongue in cheek, but it shows that defining ##0.\infty=0## doesn't lead very far.

##\infty## is not simply a "very big number", so applying usual number rules and logic to it doesn't work.

 

[Mark44]

Excuse me if this seems a bit basic, ∞ is a symbol representing something, a quantity that is undefinable. but a quantity nonetheless.

No, ∞ does not represent a quantity. A "quantity that is undefinable" is not a quantity.

Zero x ∞ therefore means zero times that quantity, zero iterations of that quantity - however big it is -or else there has to be a limit beyond which the quantity ∞ means something tangible beyond being a mere quantity?
How can zero times any quantity be other than zero?

Zero times any specific number is zero, but the expression ##0 \cdot \infty## is one of several indeterminate forms that show up in calculus, in the study of limits. This expression is indeterminate because different expressions having this form can come out to be anything.

 

[Grimble]

I find your thought processes interesting - and illuminating! That it is in applying rules that deal with more advanced realms of mathematics that we find the difficulties.
Surely what we are dealing with here, at its very basic level is the very foundation of maths and that is counting. When we are counting zero of anything has a very definite meaning, whether it is numbers or apples; for in counting we are not necessarily defining what it is we are counting. In this case it is how many instances of the term ∞ we have.
And as I said in the OP, if one takes it the other way round (again from the point of view of counting, it doesn't matter how many times one adds zero to zero the answer is and always will be zero! So adding zero an infinite number of times can still be nothing (excuse the pun) but zero.

When you, Samy_A, argue: " For example: 2/∞=0, but 0.∞=0, therefore we get (2/∞).∞=0. Simplifying by removing the infinity for numerator and denominator, we get 2=0. That's tongue in cheek, but it shows that defining 0.∞=0 doesn't lead very far". - (sorry I can't see how to do quotes...)
you are rounding your result - for 2 divided by any number will always be a +ve real number however small. Yes it is undetermined yet will always be >0.

As you can see I am not a mathematician (pretty obvious isn't it?) yet I would say (using my logic) that if one wrote ∞.2/∞ the result would not be 2, because ∞ is indeterminate and the two terms of ∞ are not the same number.
In maths does ∞ = ∞ ? It surely is undetermined yet has certain definite properties? It is very large, positive, and a real though undefined number.

(Is this more of a philosophical question? About the nature and meaning of terms? I admit my approach is that of a determined Ockhamite!)Reference https://www.physicsforums.com/threads/zero-x-infinity.849936/

Reference https://www.physicsforums.com/threads/zero-x-infinity.849936/[/SUB]

 

[Samy_A]

I find your thought processes interesting - and illuminating! That it is in applying rules that deal with more advanced realms of mathematics that we find the difficulties.
Surely what we are dealing with here, at its very basic level is the very foundation of maths and that is counting. When we are counting zero of anything has a very definite meaning, whether it is numbers or apples; for in counting we are not necessarily defining what it is we are counting. In this case it is how many instances of the term ∞ we have.
And as I said in the OP, if one takes it the other way round (again from the point of view of counting, it doesn't matter how many times one adds zero to zero the answer is and always will be zero! So adding zero an infinite number of times can still be nothing (excuse the pun) but zero.

When you apply your counting logic to set ##0.\infty=0##, what you really do is apply a limit (even if you don't use the term).
Yes 0 times 7 is 0. 0 times 700000000000000 is still 0, and 0 times 700000000000000000000000000 is also just 0.
But then you jump from large numbers to infinity, and claim that ##0.\infty## must be 0.

Mathematically, what you do is claim:
If ##0.a_n=0##, and ##\displaystyle\ \lim_{n\rightarrow +\infty}{a_n}=\infty##, then ##0.\displaystyle\ \lim_{n\rightarrow +\infty}{a_n}=0##.
But as I showed in my first post, you can use other sequences, and "show" likewise that ##0.\infty=1## or any other value (or undefined).

When you, Samy_A, argue: " For example: 2/∞=0, but 0.∞=0, therefore we get (2/∞).∞=0. Simplifying by removing the infinity for numerator and denominator, we get 2=0. That's tongue in cheek, but it shows that defining 0.∞=0 doesn't lead very far". - (sorry I can't see how to do quotes...)
you are rounding your result - for 2 divided by any number will always be a +ve real number however small. Yes it is undetermined yet will always be >0.]

So what should 2/∞ be as a non zero real number?
If you set 2/∞=a>0, then you get ∞=2/a. Even if a is very small, 2/a will be big but finite.

 

[WWGD]

In some cases, like in measure theory, you have loosely that the product is 0 (countably infinity union of sets of measure zero has measure zero). You can see this by having the set ##S_i## have measure ##\epsilon/2^i ## , then letting ##\epsilon \rightarrow 0 ##. But this is cheating.

 

[Mark44]

I find your thought processes interesting - and illuminating! That it is in applying rules that deal with more advanced realms of mathematics that we find the difficulties.
Surely what we are dealing with here, at its very basic level is the very foundation of maths and that is counting. When we are counting zero of anything
has a very definite meaning, whether it is numbers or apples; for in counting we are not necessarily defining what it is we are counting.

If you are counting specific things, then zero of them means that you have no (= 0) of those things. However, and as I said before, the expression ##0 \cdot \infty## is indeterminate, and does not fall into the category of "counting things."

In this case it is how many instances of the term ∞ we have.
And as I said in the OP, if one takes it the other way round (again from the point of view of counting, it doesn't matter how many times one adds zero to zero the answer is and always will be zero! So adding zero an infinite number of times can still be nothing (excuse the pun) but zero.

When you, Samy_A, argue: " For example: 2/∞=0, but 0.∞=0, therefore we get (2/∞).∞=0. Simplifying by removing the infinity for numerator and denominator, we get 2=0. That's tongue in cheek, but it shows that defining 0.∞=0 doesn't lead very far". - (sorry I can't see how to do quotes...)
you are rounding your result - for 2 divided by any number will always be a +ve real number however small. Yes it is undetermined yet will always be >0.

There is no rounding going on here. All that Samy_A did was multiply both sides of the equation by ##\infty##.

As you can see I am not a mathematician (pretty obvious isn't it?) yet I would say (using my logic) that if one wrote ∞.2/∞ the result would not be 2, because ∞ is indeterminate

No, ##\infty## is not indeterminate. The point is that it makes no sense at all to do arithmetic with ##\infty##.

and the two terms of ∞ are not the same number.
In maths does ∞ = ∞ ?

Not necessarily. If we're talking about the sizes of sets (or cardinality), the set {1, 2, 3, ... } would seem to be larger than the set {2, 4, 6, ...}, but both sets have the same cardinality. This means that there is a one-to-one mapping between the two sets so that each number in the first set gets paired with a number in the second set, and vice versa.

It surely is undetermined yet has certain definite properties? It is very large, positive, and a real though undefined number.

(Is this more of a philosophical question? About the nature and meaning of terms? I admit my approach is that of a determined Ockhamite!)Reference https://www.physicsforums.com/threads/zero-x-infinity.849936/

Reference https://www.physicsforums.com/threads/zero-x-infinity.849936/[/SUB]

 

[WWGD]

Maybe, OP, you mean to say: ## lim_{x \rightarrow \infty} x .0 =0 ## . Then this is correct.

 

[Mark44]

Maybe, OP, you mean to say: ## lim_{x \rightarrow \infty} x .0 =0 ## . Then this is correct.

Based on what the OP wrote earlier --

As you can see I am not a mathematician (pretty obvious isn't it?)

-- I don't think this is the case.

 

[WWGD]

Based on what the OP wrote earlier --
<snip>.

I meant to say that the sequence : ## 1.0=0, 2.0=0,..., n.0=0 ...## converges to ##0##.

 

[Mark44]

I meant to say that the sequence : ## 1.0=0, 2.0=0,..., n.0=0 ...## converges to ##0##.

You confused me at first, but I think this is what you meant: ##1 \cdot 0=0, 2 \cdot 0=0,..., n \cdot 0=0 ...##

 

[WWGD]

You confused me at first, but I think this is what you meant: ##1 \cdot 0=0, 2 \cdot 0=0,..., n \cdot 0=0 ...##

Yes, sorry, did not know how to tex the ##a \cdot b ##.

 

[Samy_A]

More generally, Grimble is trying to make sense of ∞. Nothing wrong with trying, but it so happens that infinity (even countable infinity) behaves very differently than finite numbers, even very large finite numbers.

@Grimble , have you heard about Hilbert's hotel? It's a rather amusing illustration of how infinity is different from finite numbers. Take a look at it.

 

[WWGD]

More generally, Grimble is trying to make sense of ∞. Nothing wrong with trying, but it so happens that infinity (even countable infinity) behaves very differently than finite numbers, even very large finite numbers.

@Grimble , have you heard about Hilbert's hotel? It's a rather amusing illustration of how infinity is different from finite numbers. Take a look at it.

Sorry, I may have been inadvertently throwing the discussion off track.

 

[Samy_A]

Sorry, I may have been inadvertently throwing the discussion off track.

Oh, that's not what I meant.

I was just rebounding on this post:

Maybe, OP, you mean to say: ## lim_{x \rightarrow \infty} x .0 =0 ## . Then this is correct.

 

[Gjmdp]

An interest proportion about ∞*0=∞*0 is that 0/0=∞/∞ Which of course doesn't equal 1, but it proofs that this fractions equal the same. It's curious how infinite has many characteristics in common with 0, being opposite concepts .

 

[WWGD]

An interest proportion about ∞*0=∞*0 is that 0/0=∞/∞ Which of course doesn't equal 1, but it proofs that this fractions equal the same. It's curious how infinite has many characteristics in common with 0, being opposite concepts .

Actually, neither of those ratios is even defined, so they cannot be the same.

 

[Mark44]

An interest proportion about ∞*0=∞*0 is that 0/0=∞/∞ Which of course doesn't equal 1, but it proofs that this fractions equal the same.

Actually, neither of those ratios is even defined, so they cannot be the same.

To expand on what WWGD said, since the two expressions aren't defined, then they aren't equal, so you haven't proven anything.

It's curious how infinite has many characteristics in common with 0, being opposite concepts .

Exactly which characteristics of ##\infty## do you think you have found?
BTW The word is "infinity", a noun, not "infinite", an adjective.

 

[WWGD]

Ultimately, @Gjmdp , it is up to you to provide a layout in which your results show themselves to be productive and meaningful. I don't see how to do that. Can you explain how you would do that?

 

[micromass]

Ultimately, @Gjmdp , it is up to you to provide a layout in which your results show themselves to be productive and meaningful. I don't see how to do that. Can you explain how you would do that?

Very right. I don't mind somebody coming up with new and novel concepts, or to make a theory that deviates from the normal one. But one needs to make sure it is actually somewhat useful.

 

[Gjmdp]

Very right. I don't mind somebody coming up with new and novel concepts, or to make a theory that deviates from the normal one. But one needs to make sure it is actually somewhat useful.

Well, I think it's interesting, because I may found the equality of 2 undefined different values. How much times 0 contain itself is equal to how much times infinity contains itself.

 

[micromass]

Well, I think it's interesting, because I may found the equality of 2 undefined different values.
How much times 0 contain itself is equal to how much times infinity contains itself.

Glad you find it interesting, I guess. Just don't expect anybody else to adopt your system, since it is not really useful, even to pure mathematicians. But yeah, as something to entertain yourself with, it's cool. But it doesn't check out mathematically.

 

[Gjmdp]

Glad you find it interesting, I guess. Just don't expect anybody else to adopt your system, since it is not really useful, even to pure mathematicians. But yeah, as something to entertain yourself with, it's cool. But it doesn't check out mathematically.

Ok thanks for your attention, is that I'm A 15-year old doing self-studying in Mathematics and Physics and I'm not a PhD who knows what's interesting and what isn't about Mathematics. I only wanted to try to proof we can compare different undefined values and set that they can be equal. Because I thought this would we interesting I posted it, but really, sorry for the inconveniences.

 

[Samy_A]

Ok thanks for your attention, is that I'm A 15-year old doing self-studying in Mathematics and Physics and I'm not a PhD who knows what's interesting and what isn't about Mathematics. I only wanted to try to proof we can compare different undefined values and set that they can be equal. Because I thought this would we interesting I posted it, but really, sorry for the inconveniences.

There is no inconvenience.

But you haven't proved that 0/0=∞/∞, you merely stated it. Proving would mean deducing that equation from other known facts.
Still, you can, as others have said, just set 0/0=∞/∞. But then? What can you do with that definition? People here think not much, that's based on experience. We all can be wrong. I would gladly be wrong, as playing with ∞ as a "real number" could be mathematical fun, if you can find some context where it has meaning and is consistent.

This thread introduced ∞ in the context of multiplication of ever larger numbers by 0.
As @WWGD remarked, this can be mathematically written as ##\displaystyle \lim_{x \rightarrow \infty} x .0 =0##.
The catch is that, although we see the ∞ in that equation, the equation doesn't tell us anything about ∞, because what that equation actually means is:
∀ ε>0 ∃ y∈ℝ: ∀x∈ℝ, x>y: |x.0-0|<ε
That's a silly way to write ##\displaystyle \lim_{x \rightarrow \infty} x .0 =0##, but you see that there is no ∞ in there. It is a statement about sufficiently large real numbers, not about ∞.

 

[votingmachine]

First:
0x=0 for all x. I don't see any reason to prove it. I think I could wrestle thru an inductive proof for all integers, but it also seems obvious.

It seems that the loose math that was used relies on combining limits with other associative (or commutative?, algebraic?) properties. I don't know if that is valid.

EG:

(x=>infinity)limit(2x)=infinity
(x=>infinity)limit(3x)=infinity
And infinity=infinity so 2x=3x and 2=3

or

(x=>0)limit(2x)=0
(x=>0)limit(3x)=0
And 0=0 so 2x=3x and 2=3

in both cases the use of loose math may well be too loose. Infinity is one of those symbols that may be too easy to throw around.

 

[Mark44]

First:
0x=0 for all x. I don't see any reason to prove it. I think I could wrestle thru an inductive proof for all integers, but it also seems obvious.

It seems that the loose math that was used relies on combining limits with other associative (or commutative?, algebraic?) properties. I don't know if that is valid.

EG:

(x=>infinity)limit(2x)=infinity
(x=>infinity)limit(3x)=infinity
And infinity=infinity so 2x=3x and 2=3

The conclusion above is not valid (which is probably the point you're trying to make).
Just because ##\lim_{x \to \infty} f(x) = L## and ##\lim_{x \to \infty} g(x) = L##, it doesn't follow that f(x) = g(x).

or

(x=>0)limit(2x)=0
(x=>0)limit(3x)=0
And 0=0 so 2x=3x and 2=3

Similar comment as above.

in both cases the use of loose math may well be too loose.

Infinity is one of those symbols that may be too easy to throw around.

 

[Grimble]

Hmmm.
As a non mathematician, I see a difference in philosophy (if that is the right term? (As every term seems to have particular meanings (yes, meanings, plural!) in mathematics))
[It is so much easier to write about mathematics because one may use nested parentheses - or is that not de rigueur?] hehehe!

Yes, back to the point, a difference in philosophy or treatment of the term ∞; for it may be taken, logically, as meaning a very large, yet indeterminate term - as can be seen in Hilbert's hotel; yet, for mathematicians there seems to also be, a compulsion to treat ∞ as a definite term representing a definite, very large yet specific value! I see this as a real problem with mathematical terminology; in using the same term to define many different types of discrete entities. For ∞ can represent many different sets of numbers, depending on what they represent.

So it depends on how one is thinking - mathematically - as to how one treats that symbol; as to what properties one endows it with.

One can see this in the comments above where so many attempts are made to define the undefinable. One really has to consider what it represents before deciding how to treat it.

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?

To me 0 however it is used is still, at its root a counting term representing a total absence of what ever is being counted, so 0 ⋅ ∞ means no instances of whatever ∞ represents.

To use ∞ as a mathematical term that can be subject to mathematical operations is like trying to stack water! Mathematical terms have precise meanings but ∞ doesn't, for ∞ represents an idea...

Reference https://www.physicsforums.com/threads/zero-x-infinity.849936/

 

[Samy_A]

Hmmm.
As a non mathematician, I see a difference in philosophy (if that is the right term? (As every term seems to have particular meanings (yes, meanings, plural!) in mathematics))
[It is so much easier to write about mathematics because one may use nested parentheses - or is that not de rigueur?] hehehe!

Yes, back to the point, a difference in philosophy or treatment of the term ∞; for it may be taken, logically, as meaning a very large, yet indeterminate term - as can be seen in Hilbert's hotel; yet, for mathematicians there seems to also be, a compulsion to treat ∞ as a definite term representing a definite, very large yet specific value! I see this as a real problem with mathematical terminology; in using the same term to define many different types of discrete entities. For ∞ can represent many different sets of numbers, depending on what they represent.

No. The Hilbert hotel example shows that ∞ does not behave as a "very large" term.
I don't know which mathematician has "a compulsion to treat ∞ as a definite term representing a definite, very large yet specific value!".

So it depends on how one is thinking - mathematically - as to how one treats that symbol; as to what properties one endows it with.

One can see this in the comments above where so many attempts are made to define the undefinable. One really has to consider what it represents before deciding how to treat it.

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?

Indeed, that's why just setting 0.∞=0 is meaningless in the context of real analysis, or counting, as you did in the OP.

To me 0 however it is used is still, at its root a counting term representing a total absence of what ever is being counted, so 0 ⋅ ∞ means no instances of whatever ∞ represents.

We can continue to go in circles, but what purpose does it serve?
Unless you can do something interesting with the definition 0.∞=0, it is simply not useful.
That 0.∞=0 can be deduced from some counting logic is wrong, or, more precisely, has no meaning.

 

[DocZaius]

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 one were to say the number of molecules of water in the oceans is infinite, then one would be plainly wrong. You seem to think that ∞ is an actual number, but simply one that we cannot determine because it is so very big. As if infinity is a member of the set of counting numbers, but just a really really big one. I am not 100% sure if that's your conception of infinity. But if it is, it's not the same one that others are using in this thread. If you realize ∞ is not a number as 5, or 823, then you might begin to see the problem with understanding just what "0 ⋅ ∞" even means.

 

[Mark44]

Already dealt with by DocZaius, but bears repeating.

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.

The number of molecules of water in all of the oceans is a very large number, but is finite. Let's call it N. Then those N water molecules comprise 2N hydrogen atoms and N oxygen atoms.