Infinity

Okay so I've been considering the number infinity for the last couple of days and it's absolutely destroying my brain. Here's some questions that I hope the mathematicians of gamespot can help me with.

First of all is infinity a constant amount? For example if I subtracted 1 from infinity would I have a finite number or a number that is not the same as infinity. My guess is that infinity is not a constant amount and that any finite addition does not alter it's value. For example if infinity minus 1 did equal a finite number then we would also expect that finite number plus 1 to equal infinity which simply doesn't make sense when we are talking about a number that doesn't end.

It might follow that we cannot call infinity a rational number. What would happen then if we divided infinity by infinity or subtracted infinity by infinity? If we treated infinity like a rational number then we would conclude that the answer would be 1, but infinity clearly isn't a rational number. However does introducing two irrational numbers beget rational numbers? Would infinity minus infinity = zero and infinity divided by infinity = 1?

Here's another conundrum. When dividing the number one with rational numbers we know that the answer tends towards zero but never quite reaches it. What happens when we divide one by infinity, does the answer finally equal zero?

Or maybe we just flat out cannot treat infinity as a number. Maybe it's just more of an abstract concept that should be kept away from mathematics.

What does everyone think? If your confused after reading my post then it's most likely because I am very confused myself.

I think you are spending to much time thinking about that.

Infinity isn't a number.

infinite is the result of dividing by zero, there now everyone knows

I love the concept of infinity. I'll see if I can answer your question adequately. I apologize if I get a bit too detailed with answers; I tend to find it best to start with the most technical answer and then pare back if the person has troubles with that answer.

First of all is infinity a constant amount? For example if I subtracted 1 from infinity would I have a finite number or a number that is not the same as infinity. My guess is that infinity is not a constant amount and that any finite addition does not alter it's value. For example if infinity minus 1 did equal a finite number then we would also expect that finite number plus 1 to equal infinity which simply doesn't make sense when we are talking about a number that doesn't end.

domatron23

No, infinity is not a constant amount. In fact, it's not even really an "amount" at all. It's more simply a concept. There is no one single "infinity" as if it were the largest number; infinity is anything that is not finite. Generally speaking, infinity can be defined as that which is not explicitly enumerable. For example, the set {1, 2, 3, ..., 1,000,000,000,000}, while a very large set, is nonetheless finite because it is theoretically possible to explicitly enumerate every item in the set. Conversely, the set of natural numbers {1, 2, 3, ...} is infinite, because no matter how many items you list, there will always be an infinite number of items that you have not listed.

It might follow that we cannot call infinity a rational number. What would happen then if we divided infinity by infinity or subtracted infinity by infinity? If we treated infinity like a rational number then we would conclude that the answer would be 1, but infinity clearly isn't a rational number. However does introducing two irrational numbers beget rational numbers? Would infinity minus infinity = zero and infinity divided by infinity = 1?

domatron23

Ah, now we're running into the realms of calculus and limits. I'm not sure whether or not I can satisfactorily answer this question for you, but I'll do my best. In general, dividing infinity by infinity or subtracting infinity from infinity is undefined. In specific situations, however, it can be defined.

For example, suppose you have the function f(x) = x / x. As x approaches infinity, this would clearly approach infinity divided by infinity, but just from the usual form we can easily see that this function is always equal to 1 (except where x = 0). So when x approaches infinity, it clearly is still 1, and infinity divided by infinity here is 1. But then suppose you have another function g(x) = 2x / x. Same deal here: as x approaches infinity, this would also approach infinity. However, we know that this is always equal to 2, so in this case, infinity divided by infinity is 2.

And on and on these examples could go. This is why in the general case infinity divided by infinity (and subtraction too, by a similar reasoning) is considered undefined.

Here's another conundrum. When dividing the number one with rational numbers we know that the answer tends towards zero but never quite reaches it. What happens when we divide one by infinity, does the answer finally equal zero?

domatron23

If you have the function f(x) = 1 / x, yes, the limit of f as x approaches infinity is indeed considered to equal 0.

Or maybe we just flat out cannot treat infinity as a number. Maybe it's just more of an abstract concept that should be kept away from mathematics.

domatron23

That would be largely correct.

Wow, can't say I can help you but you've certainly aroused my curiosity. Maybe it'll be something I ask my math teacher.

I'd like to recommend a movie for you, it's called Pi and judging by your first post I presume you will see yourself in the main character, it's an indie film and my thoughts are that it was very interesting more than it was anything else, I'm not saying it's good or that it's bad but it certainly does get you thinking, and having an interest in numbers or math I'm sure you will appreciate this film more than most.

infinite is the result of dividing by zero, there now everyone knows

linkin_guy109

[QUOTE="linkin_guy109"]

infinite is the result of dividing by zero, there now everyone knows

xaos

i know it was worht a try :P

By the way, if you want your brain to truly hurt, I could also say that mathematicians recognize different degrees of infinity, as well. For example, the natural numbers {1, 2, 3, ...} is considered "countably infinite" because the elements in the set are enumerable, which is to say that you can list the elements in an order that then imply the remaining unenumerated elements. However, the real numbers are uncountably infinite: since you can have an infinite number of decimal places, it is impossible to enumerate the elements in any coherent order. The reason for this is that there are an infinite number of real numbers between any two real numbers.

Real numbers are, thus, infinite in two degrees: the numbers themselves can be infinitely long in both directions, and you can have an infinite number of them. This makes them "more infinite" than the set of natural numbers; though both are infinite, it is provable that there are more real numbers than there are natural numbers.

i think...... i love pot haha i think about crazy **** like that all the time too, dudesovereign_22

That just made me laugh out loud...I wasn't expecting that.

I've thought about the concept of infinity before too...and I simply cannot comprehend it. We humans understand finite things. Maybe because our lives and the limited part of the universe we can presently grasp are finite.

Wow, can't say I can help you but you've certainly aroused my curiosity. Maybe it'll be something I ask my math teacher.

I'd like to recommend a movie for you, it's called Pi and judging by your first post I presume you will see yourself in the main character, it's an indie film and my thoughts are that it was very interesting more than it was anything else, I'm not saying it's good or that it's bad but it certainly does get you thinking, and having an interest in numbers or math I'm sure you will appreciate this film more than most.

-theCHUD-

Yeah I've seen that before, it's really weird but intriguing. Best of all though the dude puts a power drill through his head so in my books it's a good movie.

Very interesting, cheers Gabu. The first two answers you gave make a lot of sense but the third raises a few more questions. If 1/infinity=0 then shouldn't 0*infinity=1?domatron23

Ah, that's a good catch. What you have to recognize is that you can't actually say "1 / infinity" in any meaningful fashion because infinity is not a number that you can divide another by. It's more just a concept, as I said. You can say that 1 / x approaches 0 as x approaches infinity (i.e., gets bigger and bigger), but you can't really talk about "1 / infinity" in any meaningful fashion.

0 * infinity is considered to be undefined in the same way that infinity divided by infinity is also undefined. Suppose you have two functions f(x) and g(x) such that the limits of f and g as x approaches infinity are 0 and infinity, respectively. It can then be shown that the limit of 1 / f(x) would then be infinity. Since we also know that f(x) * g(x) is the same as g(x) / (1 / f(x)), we see that 0 * infinity is actually an identical case as infinity / infinity.

I should note that 0 * infinity can be equal to 1 depending on what the 0 and infinity are. For example, if you have (1 / x) * x, that clearly is 0 * infinity as x approaches infinity, but just dividing x by x gets you that it's equal to 1. However, as above, this is not necessarily the case, which is why 0 * infinity is considered undefined.

finite number/infinity=0 infinity/finite number=infinity 0*infinity!=1 as it is in the indeterminate form infinity/infinity !=1 as it is in the indeterminate form infinity-1=infinitytratyu92

What about infinity times i (sqrt(-1))? Try it! lol I wana know!

[QUOTE="domatron23"]Very interesting, cheers Gabu. The first two answers you gave make a lot of sense but the third raises a few more questions. If 1/infinity=0 then shouldn't 0*infinity=1?GabuEx

Ah, that's a good catch. What you have to recognize is that you can't actually say "1 / infinity" in any meaningful fashion because infinity is not a number that you can divide another by. It's more just a concept, as I said. You can say that 1 / x approaches 0 as x approaches infinity (i.e., gets bigger and bigger), but you can't really talk about "1 / infinity" in any meaningful fashion.

0 * infinity is considered to be undefined in the same way that infinity divided by infinity is also undefined. Suppose you have two functions f(x) and g(x) such that the limits of f and g as x approaches infinity are 0 and infinity, respectively. It can then be shown that the limit of 1 / f(x) would then be infinity. Since we also know that f(x) * g(x) is the same as g(x) / (1 / f(x)), we see that 0 * infinity is actually an identical case as infinity / infinity.

*head explodes*

What you're saying does make sense but it's just putting me into a head-spin. Here is my dilemma and the actual point behind this thread.

I've been trying to assess pascal's wager and the problem of an infinite afterlife weighed against a finite present life and the probabilities that comes with each. Basically I've been trying to work out if an infinite value multiplied by an infinitely small probability (1/infinity) equals a finite number. However I fear that is an exercise in futility.

*head explodes*

What you're saying does make sense but it's just putting me into a head-spin.

domatron23

Sorry. :oops: I unfortunately doubt if I could explain it any better. The whole idea of infinity seems weird because... well, it is weird. We're trying to understand something infinite through what we can understand with our finite minds, which obviously causes problems.

Here is my dilemma and the actual point behind this thread.

I've been trying to assess pascal's wager and the problem of an infinite afterlife weighed against a finite present life and the probabilities that comes with each. Basically I've been trying to work out if an infinite value multiplied by an infinitely small probability (1/infinity) equals a finite number. However I fear that is an exercise in futility.

domatron23

That's a very interesting question. Pascal's Wager is a very neat topic, at least in terms of the mathematical ideas that it raises. Unfortunately, however, I would say that the question you ask probably is ultimately unanswerable. As before, I would probably say that an infinite value multiplied by an infinitely small probability is... well, undefined.

Just for curiosity's sake, I presume the idea you have is that you agree with the payoff as infinite, but you consider the probability infinitely small that you'll actually get that payoff when you die?

[QUOTE="domatron23"]

*head explodes*

What you're saying does make sense but it's just putting me into a head-spin.

GabuEx

Sorry. :oops: I unfortunately doubt if I could explain it any better. The whole idea of infinity seems weird because... well, it is weird. We're trying to understand something infinite through what we can understand with our finite minds, which obviously causes problems.

It's the concept itself that's causing me such discomfort, your explanations have been very well presented.

Here is my dilemma and the actual point behind this thread.

I've been trying to assess pascal's wager and the problem of an infinite afterlife weighed against a finite present life and the probabilities that comes with each. Basically I've been trying to work out if an infinite value multiplied by an infinitely small probability (1/infinity) equals a finite number. However I fear that is an exercise in futility.

domatron23

That's a very interesting question. Pascal's Wager is a very neat topic, at least in terms of the mathematical ideas that it raises. Unfortunately, however, I would say that the question you ask probably is ultimately unanswerable. As before, I would probably say that an infinite value multiplied by an infinitely small probability is... well, undefined.

Just for curiosity's sake, I presume the idea you have is that you agree with the payoff as infinite, but you consider the probability infinitely small that you'll actually get that payoff when you die?

That's right, the assumptions are that if you believe in God and are correct then you will recieve an infinite length of happiness (heaven for arguments sake) and that the probability that you have chosen the correct God to worship is infinitely small (based on the theory that there are an infinite number of possible Gods and that only one of these Gods is the true one).

[QUOTE="tratyu92"]finite number/infinity=0 infinity/finite number=infinity 0*infinity!=1 as it is in the indeterminate form infinity/infinity !=1 as it is in the indeterminate form infinity-1=infinityRikusaki

What about infinity times i (sqrt(-1))? Try it! lol I wana know!

[QUOTE="Rikusaki"]

[QUOTE="tratyu92"]finite number/infinity=0 infinity/finite number=infinity 0*infinity!=1 as it is in the indeterminate form infinity/infinity !=1 as it is in the indeterminate form infinity-1=infinitytratyu92

What about infinity times i (sqrt(-1))? Try it! lol I wana know!

lol thx anyways *hugs*

It's the concept itself that's causing me such discomfort, your explanations have been very well presented.

domatron23

Ah, well, yes, the concept is definitely strange, as I said.

That's right, the assumptions are that if you believe in God and are correct then you will recieve an infinite length of happiness (heaven for arguments sake) and that the probability that you have chosen the correct God to worship is infinitely small (based on the theory that there are an infinite number of possible Gods and that only one of these Gods is the true one).

domatron23

Ah yes, now this is bringing back memories of Pascal's Wager... it definitely is a very interesting question.

It actually reminds me of the situation where you shoot an arrow into a target. Suppose it takes one second to get to the target. It then takes half a second to get halfway, and a quarter of a second to get three-quarters of the way there, and an eighth of a second to get seven-eights of the way there, et cetera, et cetera. But the arrow very clearly does eventually get all the way to the target, so evidently the infinites must get so infinitely small that they effectively become zero.

And one might imagine the same sort of thing to happen here, where if a probability gets smaller and smaller but the payoff gets bigger and bigger, then once it reaches infinity, one might intuitively expect the expected payoff to still be a finite value, same as how the number of partitions gets bigger and bigger but the length of time needed to advance through each of them gets smaller and smaller, resulting in a finite amount of time to cover the whole length.

But that's just hand-waving, unfortunately. I can't really say one way or another on the subject; if it were a solvable problem, you can bet your boots that mathematicians would have solved it by now.

Ah yes, now this is bringing back memories of Pascal's Wager... it definitely is a very interesting question.

It actually reminds me of the situation where you shoot an arrow into a target. Suppose it takes one second to get to the target. It then takes half a second to get halfway, and a quarter of a second to get three-quarters of the way there, and an eighth of a second to get seven-eights of the way there, et cetera, et cetera. But the arrow very clearly does eventually get all the way to the target, so evidently the infinites must get so infinitely small that they effectively become zero.

And one might imagine the same sort of thing to happen here, where if a probability gets smaller and smaller but the payoff gets bigger and bigger, then once it reaches infinity, one might intuitively expect the expected payoff to still be a finite value, same as how the number of partitions gets bigger and bigger but the length of time needed to advance through each of them gets smaller and smaller, resulting in a finite amount of time to cover the whole length.

But that's just hand-waving, unfortunately. I can't really say one way or another on the subject; if it were a solvable problem, you can bet your boots that mathematicians would have solved it by now.

GabuEx

Your arrow problem seems kind of similar to Achilles vs the tortoise. It would seem like common sense and mathematical logic aren't always in harmony.

I would probably hesitantly admit that Pascal's wager is a valid argument from a mathematical point of view but it's rather problematic as an equation when we introduce one or two infinite values.

Your arrow problem seems kind of similar to Achilles vs the tortoise. It would seem like common sense and mathematical logic aren't always in harmony.

domatron23

Oh yes, that's basically the same thing. And yes, that is definitely true, although whenever that's the case, it's usually the case that the attempted mathematical logic is faulty somewhere.

I would probably hesitantly admit that Pascal's wager is a valid argument from a mathematical point of view but it's rather problematic as an equation when we introduce one or two infinite values.

domatron23

Yes, I would agree with that assessment of it.

[QUOTE="tratyu92"]finite number/infinity=0 infinity/finite number=infinity 0*infinity!=1 as it is in the indeterminate form infinity/infinity !=1 as it is in the indeterminate form infinity-1=infinityRikusaki

What about infinity times i (sqrt(-1))? Try it! lol I wana know!

I'm pretty sure you'd still get infinite.

Why can't there be more OT topics like these?

[QUOTE="domatron23"]

Your arrow problem seems kind of similar to Achilles vs the tortoise. It would seem like common sense and mathematical logic aren't always in harmony.

GabuEx

Oh yes, that's basically the same thing. And yes, that is definitely true, although whenever that's the case, it's usually the case that the attempted mathematical logic is faulty somewhere.

That logical quandary only exists when you consider Achilles in relation to the tortoise (in which case he would be in a race to reach where the tortoise IS, rather than where he WILL BE, which could be considered 'overtaking him'). If you measure the time it takes them to reach a certain point in space, then Achilles can most definitely take over the tortoise.

That logical quandary only exists when you consider Achilles in relation to the tortoise (in which case he would be in a race to reach where the tortoise IS, rather than where he WILL BE, which could be considered 'overtaking him'). If you measure the time it takes them to reach a certain point in space, then Achilles can most definitely take over the tortoise.

mikeg0788

Well, the question is basically "if Achilles keeps halving the distance between him and the tortoise on and on into infinity, how could he pass the tortoise?"

The answer to that is basically that although there are an infinite number of halvings that take place, the time that each takes also goes towards 0, and the two basically "cancel out".

[QUOTE="mikeg0788"]

That logical quandary only exists when you consider Achilles in relation to the tortoise (in which case he would be in a race to reach where the tortoise IS, rather than where he WILL BE, which could be considered 'overtaking him'). If you measure the time it takes them to reach a certain point in space, then Achilles can most definitely take over the tortoise.

GabuEx

Well, the question is basically "if Achilles keeps halving the distance between him and the tortoise on and on into infinity, how could he pass the tortoise?"

The answer to that is basically that although there are an infinite number of halvings that take place, the time that each takes also goes towards 0, and the two basically "cancel out".

Good point. Basic series, i guess.

God, I'm glad I don't have to study those any more. What a headache.

[QUOTE="Rikusaki"]

[QUOTE="tratyu92"]finite number/infinity=0 infinity/finite number=infinity 0*infinity!=1 as it is in the indeterminate form infinity/infinity !=1 as it is in the indeterminate form infinity-1=infinitymikeg0788

What about infinity times i (sqrt(-1))? Try it! lol I wana know!

I'm pretty sure you'd still get infinite.

Why can't there be more OT topics like these?

It would be infinity(i).

Now if it were i^infinity....

I would say there is no such thing as infinity.

the universe is not proven to be infinite nor many scientists dont belive so eiter as it keeps growing and gaining weight it will collapse upon itself, so it has an end, thus ending time, so time isnt infinite either, i cant think of anything that is infinite

I would say there is no such thing as infinity.

the universe is not proven to be infinite nor many scientists dont belive so eiter as it keeps growing and gaining weight it will collapse upon itself, so it has an end, thus ending time, so time isnt infinite either, i cant think of anything that is infinite

hormagaunt

I agree but I was referring to theoretical/supernatural things here (pascal's wager).

Would an infinitely improbable event (heaven exists) multiplied by the value of that event occuring (infinity) yield a finite expected gain?