Math question

[redwizard007]

Infinity... I get it. Sort of.

Infinity +/- any number = infinity

Right?

So what about infinity - infinity.

Is it unsolvable, zero, infinity, undefined? My math just isn't up to this.

[wumpus]

Infinity... I get it. Sort of.

Infinity +/- any number = infinity

Right?

So what about infinity - infinity.

Is it unsolvable, zero, infinity, undefined? My math just isn't up to this.

Presumably unsolvable, but with exceptions.

Infinity(alph one:number of reals) - Infinity(alph null:number of integers) = Infinity(presumably alph: one, but I'm not sure). [Edit: and conversely, infinity(number of integers) - infinity(number of reals) = -infinity] See Cantor's diagonal argument for that one.

For equal "orders" of infinity, you get things like Infinity+finite number-infinity-other finite number=?
I'd just call it "Not A Number".
[edit: you have to be specific about what you mean by "infinity", otherwise you are going to get "not a number" for most of the answers]

[Chronos]

In any context where it makes sense to ask the question, infinity - infinity will be indeterminate. So will 0*infinity, infinity/infinity, or 0/0. That is to say, it's not unanswerable; it's just that you can't tell which of the infinite number of possible answers is the intended one.

Since infinity + 1 = infinity, it is fair to say that infinity - infinity = 1. But since infinity + 2 = infinity, it's also fair to say that infinity - infinity = 2, and so on. And similarly for all of the other indeterminate forms.

It should also be emphasized, of course, that there are plenty of contexts where it doesn't even make sense to ask the question. There are many different sets of objects which can be labeled "numbers", and something might be a number in some contexts but not others. You can speak, for instance, of the real numbers, and infinity is not a real number, and so in that context, you can't do math on infinity at all. But there are also contexts like the projective real numbers, which do include infinity as a number, and where you can do math with infinity (but that math ends up missing some properties that real-number math has, as a consequence). And there are even contexts where you can interpret things like "turn the left face of a Rubik's cube 90º clockwise" as a number.

[eggynack]

Infinity... I get it. Sort of.

Infinity +/- any number = infinity

Right?

So what about infinity - infinity.

Is it unsolvable, zero, infinity, undefined? My math just isn't up to this.
It's an indeterminate, which means that it can take on any value, inclusive of both infinity and negative infinity. I generally like to think of it in terms of producing mappings between infinite sets and seeing what remains. So, you map the set of natural numbers to itself, and whatever remains is the result of the subtraction. Obviously this mapping can yield no elements in either set by just matching each element in set A to itself in set B. However, you can also map each element of set A to itself times 2 in set B, which would leave nothing in A and infinite elements of B, or you can do the exact inverse. We can pretty fairly call this process "A-B, which would cause the subtraction to yield either infinity or negative infinity. A mapping from each element of A to itself +1 would yield one element left over in B, which here corresponds to -1 as a result. The means of achieving any other integer is pretty trivial.

[Khedrac]

Infinity... I get it. Sort of.

Infinity +/- any number = infinity

Right?

So what about infinity - infinity.

Is it unsolvable, zero, infinity, undefined? My math just isn't up to this.

To simplify a bit, you appear to be making the same mistake a lot of people make, especially science fiction authors - you are mistaking infinity for a number.

+ and - are really only defined for numbers, and infinity isn't a number.
So, for "infinity - infinity" to equal anything you need to define what you mean by "-", and that definition should give you an answer.

Note: there is quite a lot of maths where there are operations defined for things that aren't numbers (like different 'sizes' of infinity), some of them are even commonly referred to a "+" and "-" but they are not the same operators that are used with numbers.

For example, consider matricies (matrixes).
If A is the two by 2 matrix with a "1" in every position then A + A is clearly defined, and so is A - A. However A + 1 is meaningless.
{1 1} + {1 1} = {2 2}
{1 1} + {1 1} = {2 2}
But
{1 1} + 1 = ?
{1 1}

infinity - 1 isn't really defined, but it is close enough that people think it is infinity, looking at infinity - infinity just shows the lack of definition.

[warty goblin]

One of the tricky things about infinity is that it's a grossly overloaded symbol. There's the number theoretic infinity, which is basically about size of sets, the set {1} has cardinality 1, the set{1, 2} has cardinality 2, the set {1, 2, ...} is countably infinite. Even here it's overloaded, because the cardinality of {1, 2,...} is a lesser infinity than the number of real numbers in [0,1].

There's also infinity that comes from limits of functions, i.e. limit as x -> infinity of x. You can talk about differences of functions that have infinite limits, and although they don't have to exist, the question itself is not ill formed due to the machinery of limits.

There's probably some deep mathematical linkage between those two ideas, but I don't know it. Thankfully, I'm a statistician, a profession which firmly believes that 30 is 'close enough' to infinity to be getting on with, most of the time, so I don't have to care.

[maruahm]

I'm an (applied) mathematician. "∞ - ∞" is a meaningless expression without context. What precisely do you mean by "∞?" How do you imagine the subtraction symbol '-' to apply to infinities?

I'm not being nitpicky, these are things that a mathematician would ask themselves before they begin to try to evaluate "∞ - ∞." Consider if I asked, "What is apple - orange?" Suppose I defined apple and orange to be a collection of properties, e.g. apple = {round, thin skin, size of hand, yummy, has core} and orange = {round, thick skin, size of hand, yummy, citrus-flavored}, and I defined the operation '-' to remove all the things in the left-hand collection which are also in the right-hand collection, e.g. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {1, 3, 5, 7, 9, 11} = {2, 4, 6, 8, 10}. Then apple - orange = {thin skin, has core}.

If you're saying, "Well, that's a really arbitrary way to define subtraction on fruits," then you'll know how mathematicians feel about subtraction on infinities.

As it happens, we do have ways to deal with subtractions on infinities, but they're all very context-specific. In general, if we see "∞ - ∞" in, say, a research article without context, we won't know what it means.

[Kato]

Note: in applied cases (i.e. the only ones that should matter, you high and mighty pure mathematicians :smalltongue:) there is most often one answer to infinity - infinity, depending on what led to the equation. I know the terms 'bigger' and 'smaller' infinity are ugly, but if you discuss things like the limit of polynomial terms it's often possible to give an answer, e.g. because in things like 'x^2-x' the limit might suggest 'infinity - infinity' but actually easily resolves to infinity.
(of course I know many of you know this but I'm not sure what OP's level of knowledge or intent was when stating the question)

[jayem]

Note: in applied cases (i.e. the only ones that should matter, you high and mighty pure mathematicians :smalltongue:) there is most often one answer to infinity - infinity, depending on what led to the equation. I know the terms 'bigger' and 'smaller' infinity are ugly, but if you discuss things like the limit of polynomial terms it's often possible to give an answer, e.g. because in things like 'x^2-x' the limit might suggest 'infinity - infinity' but actually easily resolves to infinity.
(of course I know many of you know this but I'm not sure what OP's level of knowledge or intent was when stating the question)

I wanted to say if you know you have (x+5-x) then the obvious answer ought to have precedence. However the more I thought about it the more dubious became.

[Jay R]

Most mathematical uses of "∞" are really just an abbreviated version of "the limit of this function as the value of x grows without bound." So to answer the question, you need to know what actual function you're talking about

As x grows without bound, 2x − x (which can be described as "∞ − ∞") = x, which grows without bound, and is therefore "∞".

As x grows without bound, (x + 2) − x (which can be described as "∞ − ∞"), remains zero.

As x grows without bound, (x + 1/x) − x (which can be described as "∞ − ∞") equals 1/x, which approaches zero.

But infinity by itself has no direct numerical meaning, so "∞ − ∞" has no direct numerical meaning.

[JNAProductions]

Most mathematical uses of "∞" are really just an abbreviated version of "the limit of this function as the value of x grows without bound." So to answer the question, you need to know what actual function you're talking about

As x grows without bound, 2x − x (which can be described as "∞ − ∞") = x, which grows without bound, and is therefore "∞".

As x grows without bound, (x + 2) − x (which can be described as "∞ − ∞"), remains zero.

As x grows without bound, (x + 1/x) − x (which can be described as "∞ − ∞") equals 1/x, which approaches zero.

But infinity by itself has no direct numerical meaning, so "∞ − ∞" has no direct numerical meaning.

Wouldn't the bolded formula equal 2?

[maruahm]

Wouldn't the bolded formula equal 2?

Yep. I think the poster meant it's a finite value. Insofar as infinities are concerned, zero might as well be two. :smalltongue:

[warty goblin]

Yep. I think the poster meant it's a finite value. Insofar as infinities are concerned, zero might as well be two. :smalltongue:

"Lissen up. I've seen a lot of young numbers, all bright eyed and bushy tailed, think they can be the next big thing in town. Well, round these 'ere parts, yer infinite, or yer nothing. So either yer got what it takes, or get yerself back off to Finitesburg."

[Kato]

"Lissen up. I've seen a lot of young numbers, all bright eyed and bushy tailed, think they can be the next big thing in town. Well, round these 'ere parts, yer infinite, or yer nothing. So either yer got what it takes, or get yerself back off to Finitesburg."

:smallbiggrin:
Finite numbers just aren't trying hard enough.

[Brother Oni]

"Lissen up. I've seen a lot of young numbers, all bright eyed and bushy tailed, think they can be the next big thing in town. Well, round these 'ere parts, yer infinite, or yer nothing. So either yer got what it takes, or get yerself back off to Finitesburg."

"And ye be wanting to stay away from de edges of the set, little number, 'cause out there, there be monsters. They look harmless, that tiny dot for de eye, but once one of thems got its claws into ye, ye forever be damned to a twilight existence, forever one of them. An imaginary number."

[Chronos]

Until another one also gets its claws into you. Then you're real again.

[Kato]

Until another one also gets its claws into you. Then you're real again.

Nay. It's not as simple as that. Sure, if ye mate with it, yer offspring will be of the grotesque kind, but their children might turn out normal again. But if clings to ye, you're an abomination, something that is both yet neither, but just adding another will not cure ye.

[Khedrac]

Until another one also gets its claws into you. Then you're real again.


Nay. It's not as simple as that. Sure, if ye mate with it, yer offspring will be of the grotesque kind, but their children might turn out normal again. But if clings to ye, you're an abomination, something that is both yet neither, but just adding another will not cure ye.

Can we not just say that "its complex"?

[wumpus]

Can we not just say that "its complex"?

NAY! Any arithmetic between reals, integers, and infinity won't be complex until you take a power less than 1 to a negative number. Only then will the complex plane open up to you.

Sure, the infinity may itself be complex (thanks, overloading), but unless noted you shouldn't assume it.

[Jay R]

Wouldn't the bolded formula equal 2?

Oops. You're right.

I changed the formula from x - x, and then didn't change the rest of the sentence.

Good catch. Thanks.

[Mister Tom]

The question here is actually... "which infinity"? (And possibly which Mathematician).

I would say that if you define subtraction as the unique inverse of addition, then for any finite number x, infinity - x is the infinity you started with (because infinity plus x is also the infinity you started with, and nothing else added to x gives you the same answer).

Infinity - infinity is a bit messier because there are _lots_ of infinities. (For example, there are the same number of even numbers as whole numbers... but there are more numbers on a number line than either.)

but if infinity A is larger than infinity B, then A- B = A (because A + B = A etc.) A - A is undefined, because A + anything smaller than A = A, so you don't have a strict inverse.

Confused? Congratulations, you're one of today's lucky ten thousand. Google "hilbert's hotel wikipedia" for more details. Hope this helps :-)

[Radar]

While everything meaningful was said about the initial question, I just wanted to say it is a funny feeling to see all this and remember that for quite many years doing infinity-infinity calculations was my bread and butter.

[shawnhcorey]

Infinity... I get it. Sort of.

Infinity +/- any number = infinity

Right?

So what about infinity - infinity.

Is it unsolvable, zero, infinity, undefined? My math just isn't up to this.

Ah no. Infinity is not a number. Trying to apply any math to it is undefined.

[Caerulea]

Ah no. Infinity is not a number. Trying to apply any math to it is undefined.
I mean, it could be a number. https://wikimedia.org/api/rest_v1/media/math/render/svg/306c55e6bc96d94db729ff5821c8f45a34c72bce0 is a cardinal describing the size the set of integers. It is a cardinal number, that happens to be large than anything in the set of integers, and is thus infinite. Not sure what https://wikimedia.org/api/rest_v1/media/math/render/svg/306c55e6bc96d94db729ff5821c8f45a34c72bce0-https://wikimedia.org/api/rest_v1/media/math/render/svg/306c55e6bc96d94db729ff5821c8f45a34c72bce0 would be though. Probably not defined.

—Caerulea

[shawnhcorey]

I mean, it could be a number. https://wikimedia.org/api/rest_v1/media/math/render/svg/306c55e6bc96d94db729ff5821c8f45a34c72bce0 is a cardinal describing the size the set of integers. It is a cardinal number, that happens to be large than anything in the set of integers, and is thus infinite. Not sure what https://wikimedia.org/api/rest_v1/media/math/render/svg/306c55e6bc96d94db729ff5821c8f45a34c72bce0-https://wikimedia.org/api/rest_v1/media/math/render/svg/306c55e6bc96d94db729ff5821c8f45a34c72bce0 would be though. Probably not defined.

—Caerulea

ℵ₀ may be transfinite but it is not a number. It cannot be in the set of numbers, rational or real. If it was, it would be too small.

[Caerulea]

ℵ₀ may be transfinite but it is not a number. It cannot be in the set of numbers, rational or real. If it was, it would be too small.
I was under the impression that transfinite numbers, while certainly being not part of the reals, are still numbers.

—Caerulea

[Radar]

Ah no. Infinity is not a number. Trying to apply any math to it is undefined.
Some operations are undefined, some are perfectly fine but one has to be careful, what kind of infinity one deals with. Aside from transfinite numbers the most common infinity is defined by limits and as such it is perfectly fine to make any calculations with. Simply some of the operations will not yield convergent solutions.

And yes, transfinite numbers are still numbers.