Infinity Questions

[The_King_of_Durf]

For whatever math expert wanders across this thread... can you answer these questions or add some more interesting ones?

∞+∞=
∞-∞=
∞*∞=
∞/∞=
∞^∞=
√∞=

∞+(-∞)=
∞-(-∞)=
∞*-∞=
∞/-∞=
∞^-∞=
√-∞=

-∞+(-∞)=
-∞-(-∞)=
-∞*-∞=
-∞/-∞=
-∞^-∞=

K+∞=
K-∞=
K*∞=
K/∞=
K^∞=
K√∞=

∞-K=
∞/K=
∞^K=

Thanks!

[13_CBS]

∞+∞=
∞-∞=


From what I know of infinity, I don't think those two operations are legal.

[Boo]

From what I know of infinity, I don't think those two operations are legal.

From what I know about infinity, a) = infinity b) = 0.

Of course, I'm BSing my way through it. :smalltongue:

[13_CBS]

AFAIK, infinity, first and foremost, cannot be treated like a normal number.

Examples: there are at least two kinds of infinity--the set of all even numbers, and all real numbers (there are, of course, lots more kinds of infinity). What's the answer when you add those two different infinities together? Subtract them? Multiply them? No, it's not going to be 1.5 infinity or something or other. I'm pretty sure it's mathematically illegal to add or subtract with infinity.

[Coplantor]

∞+∞= ∞
∞-∞= cant remember
∞*∞= ∞
∞/∞= 1
∞^∞= ∞
√∞= ∞

∞+(-∞)= cant rememer
∞-(-∞)= ∞
∞*-∞= -∞
∞/-∞= cant remember
∞^-∞= 0+?
√-∞= there's not

[Shishnarfne]

The trouble with infinity is that there are many different ways of generating it, which leads to different properties based on which one you take... Yes, even infinities of the same cardinality can get different values to these questions...


∞+∞=still infinite
∞-∞=ANYTHING
∞*∞=still infinite
∞/∞=ANYTHING
∞^∞=still infinite
√∞=still infinite

∞+(-∞)=ANYTHING (see #2)
∞-(-∞)=still infinite (see #1)
∞*-∞=still infinite
∞/-∞=ANYTHING
∞^-∞=Not well defined
√-∞=Imaginary and still infinite

-∞+(-∞)=still negative infinite
-∞-(-∞)=ANYTHING (see #2)
-∞*-∞=still infinite
-∞/-∞=ANYTHING
-∞^-∞=not well defined

K+∞=still infinite
K-∞=negative and infinite
K*∞=infinite (unless K=0, when ANYTHING)
K/∞=zero
K^∞=infinite for |K|>1, zero for |K|<1, 1 for K=1, not defined for K=-1
K√∞=Infinity to any non-zero power is still infinite

∞-K= still infinite
∞/K=still infinite
∞^K=still infinite (unless K<0)

Assuming K is a constant.

[Haruki-kun]

http://i264.photobucket.com/albums/ii173/Guanlong93/Posters/Motivator4.jpg

....sorry, I had to do it.:smallbiggrin:

[Player_Zero]

What's your basis behind this postulation? You can't really do infinity minus infinity as far as I am aware because infinity isn't a number. You can use limits and whatnot to say that, for example, 1/x as x=> 0 tends to infinity, but it doesn't mean that you can compute operators like that at all.

[Griever]

K^∞=infinite for |K|>1, zero for |K|<1, 1 for K=1, not defined for K=-1

This is the one I wanted to comment on. You would first of all have to rewrite this to be the limit as b approaches infinity of K^B, and only then would it be (kinda) solvable.

1^∞ is indeterminate as you can't have a number to the power of a concept.

[averagejoe]

AFAIK, infinity, first and foremost, cannot be treated like a normal number.

Examples: there are at least two kinds of infinity--the set of all even numbers, and all real numbers (there are, of course, lots more kinds of infinity). What's the answer when you add those two different infinities together? Subtract them? Multiply them? No, it's not going to be 1.5 infinity or something or other. I'm pretty sure it's mathematically illegal to add or subtract with infinity.

Well, the natural numbers are used more often than the even numbers as a type of infinity, but it turns out the two are bijective anyhow.

Spoilered for length.

For those not in the know: two sets A, B are bijective if there exists a mapping (or, if you like, function) that goes from A to B such that 1) every element in B gets mapped to, and 2) no element in B gets mapped to twice. To restate these more precisely, 1) for every element b in B, there exists an a in A such that f(a)=b and 2) for a and c in A, f(a)=f(c) implies that a=c. So, for example, the natural numbers (that is, the numbers 1, 2, 3, 4...) are bijective with the positive evens by the mapping f(n)=2n. Every number in the positive evens gets mapped to (for any e in the positive evens, the number e/2 is what maps to it) and no two numbers map to the same place (if 2a=2b then a=b). However, the natural numbers are not bijective with the real numbers (the numbers most people think of as "numbers," including the trancendentals such as the square root of two or pi.) There is no function from the natural numbers to the real numbers that will map to every real number; there will always be "holes" in the mapping, or numbers that just get left out. For example, it might be the case that f(n)!=2 for all n in the natural numbers.

Since talking about the "amount" of elements in infinite sets is meaningless, the concept of cardinality is more useful. For example, a set is called denumerable if it has a bijection with the natural numbers, countable if it is finite or denumerable (finite should be obvious) and uncountable if it is not denumerable. One can think of these as "varieties" of infinity.

If any of this is unclear then ask questions (unless you don't want to :smalltongue:).

Back to the OP's questions. Most of them make no sense or are undefined, and most of the others depend on what sort of space you're in and what sort of math you're using.

[Mando Knight]

For whatever math expert wanders across this thread... can you answer these questions or add some more interesting ones?

∞+∞=
∞-∞=
∞*∞=
∞/∞=
∞^∞=
√∞=

∞+(-∞)=
∞-(-∞)=
∞*-∞=
∞/-∞=
∞^-∞=
√-∞=

-∞+(-∞)=
-∞-(-∞)=
-∞*-∞=
-∞/-∞=
-∞^-∞=

K+∞=
K-∞=
K*∞=
K/∞=
K^∞=
K√∞=

∞-K=
∞/K=
∞^K=

Thanks!

∞
indeterminate
∞
indeterminate
∞
∞

indeterminate
∞
-∞
indeterminate
0
∞i

-∞
indeterminate
∞
indeterminate
0

∞
-∞
∞
0
∞ unless |K|< 1, in which case K^∞ = 0, if K = 1, then K^∞ = 1, if K = -1 then K^∞ is indeterminate through oscillation.
∞

∞
∞
∞

(assuming you can raise a number to the power of ∞, i.e. you can multiply it by itself indefinitely.)

[someonenonotyou]

here something i always woundered if even numers go 2 4 6 ... infinity and odd numbers go 1 3 5 ... infinity went you put them together woulden't that be like double infinity?

[averagejoe]

here something i always woundered if even numers go 2 4 6 ... infinity and odd numbers go 1 3 5 ... infinity went you put them together woulden't that be like double infinity?

No. Actually, both sets are bijetive (see the spoiler above, or just thinking it as "having the same number of elements only not really cos we're talking about infinity") with the natural numbers (the numbers 1, 2, 3, 4...), even though the natural numbers is what you get when you combine those sets. So, if you call the natural numbers N, the even numbers E, and the odd numbers O, then going from N to E uses the mapping f(n)=2n, and from N to O uses the mapping g(n)=2n-1. Similarly, each has an inverse going the other way. You get from E to N with h(n)=n/2 and from O to N with i(n)=(n+1)/2. However, even so, OUE=N. (The U means that you take the elements of both sets. So, OUE is a set with all the elements of O and all the elements of E.)

Yep, you get weird stuff when you talk about infinity. And this one isn't even that weird.

[Tirian]

here something i always woundered if even numers go 2 4 6 ... infinity and odd numbers go 1 3 5 ... infinity went you put them together woulden't that be like double infinity?

That is a fantastic thing to wonder about, and was one of the questions that vexed mathematicians of the late nineteenth century. Let's stop talking about "infinity" for a moment because, as someone pointed out above, it really isn't very well defined. But if we were to call ω the smallest number that was larger than every natural number 0, 1, 2, 3, ..., is ω the largest number or are there numbers that are even larger? After years of banging his head against the wall, Georg Cantor proved that ω = ω + 1, ω = ω + ω, ω = ω * ω, and a couple of other equivalences, but 2^ω is larger. A very intuitive exploration of this "paradox" is called Hilbert's Hotel (http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel).

[someonenonotyou]

*head aspoleds*
...what???

[averagejoe]

*head aspoleds*
...what???

Basically, no, you don't get "double infinity" or whatever. You actually get precisely the same variety of infinity.

[Shishnarfne]

This is the one I wanted to comment on. You would first of all have to rewrite this to be the limit as b approaches infinity of K^B, and only then would it be (kinda) solvable.

1^∞ is indeterminate as you can't have a number to the power of a concept.

I'm going to say that this is practically the only way that you can EVER do much arithmetic with infinity. In general, we can talk about the cardinality of infinity (such as being able to define a 1-1 mapping between the positive integers and the rational numbers while unable to do with the irrational ones), and we can talk about limits of expressions as a term heads to infinity, but in general we cannot do math directly with infinity. However, the limits of expressions are sufficiently useful that we frequently use them, refering to them in shorthand as "going to infinity" or "behavior at infinity".

And I say that no matter how many times you multiply one by itself, it stays one!:smallyuk:

[Ponce]

For some of the operations, you may have the option to use a limit. You can't stricly solve ∞/∞ but if you have two functions, f(x) and g(x), and you notice that as x approaches, for example, infinity, f and g approach infinity, then if you determine which of f and g approach infinity more quickly as x increases then you can determine what the limit of f/g is as x approaches infinity.

If f approaches infinity faster, then f/g = ∞,
If g approaches it faster, then f/g = 0,
If they do it at the same rate, f/g=1.

[valadil]

*head aspoleds*
...what???

Just remember that infinity isn't a number. It's a concept. Infinity means that numbers don't end. Don't let people pretend infinity is a number and your head won't hurt so much. Seriously, it's not hard. I figured this out in the second grade. Trying to add 1 + "infinity" is like adding 1 to "Dodge Caravan."

[Tirian]

Just remember that infinity isn't a number. It's a concept. Infinity means that numbers don't end. Don't let people pretend infinity is a number and your head won't hurt so much. Seriously, it's not hard. I figured this out in the second grade. Trying to add 1 + "infinity" is like adding 1 to "Dodge Caravan."

Uhhh ... no. It is slightly confusing because there are three different definitions for infinity, all of which have different symbols. But I don't think that any of them are any less "conceptual" than pi or -4 or i or any other number that can't be the number of sheep in a field.


∞ is either an extended real number that represents the (only) upper bound of the set of real numbers or an extended complex number that represents the "missing" point in the Riemann sphere that is the other side of the diameter from 0.

Aleph-zero (and I am being thwarted trying to express that symbol in this post without resorting graphics, sorry), is the size of the set of natural numbers {0, 1, 2, 3, ...}.

ω (lowercase omega) is an ordinal number indicating the smallest number greater than all natural numbers.


All of these concepts are well-defined, and in each one there is a limited set of arithmetic operations that are well-defined in some generally understood sense. And, yes, you can add one to each of them, so it's not much like adding one to a Dodge Caravan unless there is an extension of mathematics to SUVs that I am not aware of.

And, just in case all of these concepts are too clear, in surreal mathematics ω is a fully-fledged number and ω+1, ω-1, and the cube root of ω are all different fully-fledged surreal numbers, all of which are larger than any natural number.

[Oregano]

To quote myself from another thread.


I'm going to say 3.


In other words, I have no idea and you seem so much smarter than me.:smallfrown:

[Deathslayer7]

you can treat infinity like a real number. There are ways of subtracting and adding infinity, but that is not until you get way into calculus. I myself still don't know how to do that, and i am in calc 3.

But again the thing to remember is that infinity is a concept, not a number.

[averagejoe]

To quote myself from another thread.


In other words, I have no idea and you seem so much smarter than me.:smallfrown:

It's not necessarily smarts, it's just a specialized knowledge.