Thread: Infinity equations

say there is a theoretical highest number. call it n. but n+2=2 higher than n. correct? but say n=infinity. n+2=n. this is because infinity is always higher. since it is always higher, 2 is just another addition to infinity. infinity is always getting higher. infinity+infinity=infinity. but what about infinity divided by infinity? is it not 0? and also, take any number divided by infinity. what do you get? 0.0000000.......0000000001. infinite zeros, than one. if the zeros are infinite, how can you get the one at the end?does the 1 keep getting pushed back forever? and what is 2 - infinity? it is negative infinity. infinitly far away from zero, and infinitly away from any other number, even if it is negative. because infinite is not bigger. just farther away from the number line, whatever the direction.

So, when it is ∞ + 1, then we say ∞+1 is theoretically greater than ∞, or rather, ∞+1 approaches infinity faster than ∞. Now if we have (∞)/(∞²) obviously this approaches 0, just by logic, but if you don't want to use logic, or if it is more complicated than this, we use L'hospital's rule. L'hospital's rule says that if you have some function of x that has a limit of x->∞, then you take the derivative of the numerator and the derivative of the denominator and then take the lim of x->∞. If that wasn't clear here is the general rule:

If lim x->∞ f(x)= 0 and lim x->∞ of g(x)=0, then

lim x->∞ [f(x)/g(x)] = lim x->∞ [f'(x)/g'(x)]

If lim x ->∞ f(x)= ∞ and lim x->∞ g(x)=∞, then

lim x-> ∞ [f(x)/g(x)] = lim x->∞[f'(x)/g'(x)]

Example:

lim x->∞ [x/(x+1)²]

following the rule, we get lim x->∞ [1/2(x+1)] = 0

Again, the only difference between ∞² and ∞ is that ∞² approaches infinity much faster.

another question: what is infinity divided by 2? is it not still infinity? lets just say infinity is replaced by 6. 6/2=3. but if the six is now back to infinity, the 3 also increases with the 6. they both keep increasing.

when we say infinity we must understand that infinity is not a finite number. In my own experience I have found the word "infinity" to mean the overall limit of the set ofl real numbers.

another question: what is infinity divided by 2? is it not still infinity? lets just say infinity is replaced by 6. 6/2=3. but if the six is now back to infinity, the 3 also increases with the 6. they both keep increasing.

I think you have the correct idea. When we say infinity we are talking was again about the limit of the set of real numbers. Since this limit is never reached, infinity divided by two is approaching infinity slower than infinity. Since infinity divided by two and infinity both have the same limit, therefore the two are equal.

I'd like to propose a new question too. What is the derivative of infinity? Is infinity a constant? If not then also what is the integral?

I personally think we can't say infinity is a constant, but input would be nice.

infinity is not a constant. otherwise we would have to find another name for infinity+2, witch is infinity again. if infinity was a constant, infinity+2 would equal something other than infinity, witch we all agree it is infinity. infinity covers all numbers. infinity represents length from any number on the number line. it is infinitly far away from both 1 million and 0. than it cant be a constant.

As far as I know, infinty is undefined.[/code]

Originally Posted by Jimis

As far as I know, infinty is undefined.

and i am trying to define it.

[quote=" but what about infinity divided by infinity? is it not 0? and also, take any number divided by infinity. what do you get? 0.0000000.......0000000001. in.[/quote]
wouldnt infinity / infinity be 1 ?
?

isn't it rather pointless to perform opperations on infinity, a limit.

Wallaby aren't you taking calculus?

Well if you remember to take the overall integral, you have to calculate the integral from 0 to inf. where the eqn approaches 0 as it reaches inf.

not really no.

Infinity is a subject of debate.

The proposition of L 'hopital's rule and other facts are there in their place. but the discussions I have observed is a deterministic view of infinity. This is what i think is incorrect, why?

Because when we say infinity we are actually saying we dont know. Infinity is not a quantity to be approached at, it is a realm which is undefined. For if we could define infinity then it would not be infinity it would become finite

Think about it?

Regards

Rahim Ali

Originally Posted by Rahim

Infinity is a subject of debate.

The proposition of L 'hopital's rule and other facts are there in their place. but the discussions I have observed is a deterministic view of infinity. This is what i think is incorrect, why?

Because when we say infinity we are actually saying we dont know. Infinity is not a quantity to be approached at, it is a realm which is undefined. For if we could define infinity then it would not be infinity it would become finite

Think about it?

Regards

Rahim Ali

well, i guess in all the equations above we aproached infinity as a always fluctuating number, except for infinity/infinity=1. then it is a predetermined number. if we treated it as a always rising number, and in some other equations treated infinity as different numbers, then infinity/infinity acually equals infinity! against all we have been taught. there may be no way to use signs and infinity together, but we might as well try.

Regards x 2


ME

I dont think Infinity/Infinity = 1. It is undefined, Just like 0/0.

i don't think the opperations of adding subtracting multiplying or dividing by infinity have an impact on the outcome.

if something is infinite then no matter how many pieces you cut it into it must by definition be infinite.

What if infinity was just another set of numbers, kinda like the imaginary numbers.... (-1)^1/2 is just considered to be 'i' and the rest of the number in question is normal, and because of the 'i' the number is in a new set of imaginary numbers... Now what if considering 1/0 could count as a NEW letter like 'w', then the rest of the number is normal with this 'w' attached to it meaning that it is in the infinite number system.

infinity is a concept not a number and thus cannot be treated with 'normal' mathematical treatment. it must be understood that infinity cannot be attained by addition or multiplication but can only be addressed in the limit of things as a concept.

thus infinity has some interesting conceptual properties:

infinity+anything=infinity
infinity+(-infinity) is a meaningless statement
any subset of an infinite set is an infinite set

hopes this helps or at least facilitates more thought on the subject.

Originally Posted by lance

thus infinity has some interesting conceptual properties:

infinity+anything=infinity
infinity+(-infinity) is a meaningless statement
any subset of an infinite set is an infinite set

Can you explain this last one? For example, if the set of all numbers is considered an infinite set, then there is subset that contains the element 1, which is itself non-infinite. Perhaps this is not in line with what you mean.

There is a question as to whether or not infinity exists within the real world, outside of the context of mathematics, wherein it only exists in theory and by definition.

The actual statement i believe is that every infinite set contains an idempotent proper subset, i.e. you can find a proper subset of the infinte set that you can place in a 1-1 correspondence with the original infinite set. This is actually the definition of an infinite set btw.

If you believe that mathematics is discovering things instead of inventing them then infinities actually exist (what ever that may mean). Infinity is about as real as the concept of a natural number in a sense, the abstract idea of the number 2 is just as mysterious as that of infinity.

Originally Posted by river_rat

The actual statement i believe is that every infinite set contains an idempotent proper subset, i.e. you can find a proper subset of the infinte set that you can place in a 1-1 correspondence with the original infinite set. This is actually the definition of an infinite set btw.

This sounds more familiar.

If you believe that mathematics is discovering things instead of inventing them then infinities actually exist (what ever that may mean).

I am afraid that I don't know what it means. If the universe is infinite, whatever that might mean, then I accept the there could be infinity in the real world. Otherwise, no. The notion of an infinite set of numbers cannot be considered to have a real correlate, I believe.

Infinity is about as real as the concept of a natural number in a sense, the abstract idea of the number 2 is just as mysterious as that of infinity.

I agree that the number 2 is not so much meaningful in itself, but in its ability to be used as an identifier of relationship and structure, such as a counter.

Questioning if infinity exists rapidly puts you in the area of metaphysics, so bare with me if this gets technical.

You first have to ask yourself what existence means, as it is quite a strange idea as any ontological argument (an argument for the existence of God from first principles) shows. The idea of two somethings makes sense, no one will look at you funny if you talk about two penguins for example but counting and the nautral numbers are very different concepts and it is a big step to go from talking about two chairs, two fingers, two rocks to just talking about two. It took mankind a long time to take that leap, and the abstraction of zero took even longer. Now if these abstractions actually existed independant of mankind thinking about them and making them is the heart of the question if mathematics is discovered or invented. Did man discover this abstraction, was it sitting there waiting for him or did he invent it and it has no existence other than that in our heads. However, how does a chair have an existence other than in our heads in a sense? We dont have direct interaction between the world and our minds, so a chair exists as the sum of a group of sensual stimuli and we should ask ourselves how is this more real then the abstraction of a number. But we are heading in the direction of platonic forms etc and thus metamathematics and metaphysics.

Originally Posted by river_rat

Questioning if infinity exists rapidly puts you in the area of metaphysics, so bare with me if this gets technical.

I consider that inifnity in the universe is within the realm of physics. Infinity in mathematics exists by definition.

You first have to ask yourself what existence means, as it is quite a strange idea as any ontological argument (an argument for the existence of God from first principles) shows. The idea of two somethings makes sense, no one will look at you funny if you talk about two penguins for example but counting and the nautral numbers are very different concepts and it is a big step to go from talking about two chairs, two fingers, two rocks to just talking about two. It took mankind a long time to take that leap, and the abstraction of zero took even longer. Now if these abstractions actually existed independant of mankind thinking about them and making them is the heart of the question if mathematics is discovered or invented. Did man discover this abstraction, was it sitting there waiting for him or did he invent it and it has no existence other than that in our heads.

I agree that this is interesting. The primary value of numbers is to express relationships in nature that makind discovered. Numbers do not exist in nature, but were invented/developed as a method of expressing relationships and structure.

However, how does a chair have an existence other than in our heads in a sense? We dont have direct interaction between the world and our minds, so a chair exists as the sum of a group of sensual stimuli and we should ask ourselves how is this more real then the abstraction of a number.

I guess you could consider it metaphysical to question whether or not the source of our sensual stimuli exists or not. However, in general I do not. Furthermore, whether or not chairs really have some form of existence, I find it extremely useful to assume/pretend that they do when I want to sit down.

Im not doubting that chairs exist Hermes, just if their existence is any more privileged than that of numbers, infinity etc as in the end our only interactions with those entities are internal. If things exist is not a question of physics, it is an assumption of physics and thus cannot be answered by physics. An axiom of physics is that the objective world exists, it follows certain "laws" and we can determine what those laws are.

As a different example, do you believe that the continuum hypothesis has an objective truth value?

Originally Posted by river_rat

Im not doubting that chairs exist Hermes, just if their existence is any more privileged than that of numbers, infinity etc as in the end our only interactions with those entities are internal. If things exist is not a question of physics, it is an assumption of physics and thus cannot be answered by physics. An axiom of physics is that the objective world exists, it follows certain "laws" and we can determine what those laws are.

As a different example, do you believe that the continuum hypothesis has an objective truth value?

You have made something of a conceptual jump here. While you were talking about the objective existence of things I could follow, but what do you mean by objective truth value? I cannot think how any truth in mathematics could be called subjectively true. A subjective truth value might be something that I would assign to a statement like this:

"The world is a horrible place."

Originally Posted by river_rat

Im not doubting that chairs exist Hermes, just if their existence is any more privileged than that of numbers, infinity etc as in the end our only interactions with those entities are internal. If things exist is not a question of physics, it is an assumption of physics and thus cannot be answered by physics. An axiom of physics is that the objective world exists, it follows certain "laws" and we can determine what those laws are.

These are good points.

As a different example, do you believe that the continuum hypothesis has an objective truth value?

I do not, in general, believe in objective truth. However, such a concept is possible in mathematics, where truth is dependent upon definitions and so has the potential to reach objective truths. I am not sure if this is your question or not.

u can make infinite into a certain number, take any number exept 0 and power it up to a negative infinity, then u get 0

mmm, i thought i had replied here - go figure.

When i was talking about "subjective truth" i was talking about statements that should have a truth value : like the axiom of choice, continum hypothesis etc but are independant in a sense of the foundations of mathematics in that assuming them either true or false does not result in a contradiction. So in a sense we can arbitrarily assign a truth value to these statements and either assignment is correct according to mathematics. My question was if one of these statements are correct in the sense that it actually has an objective truth value and we cannot work out what it is, or do there exist propositions which are both true and false according to the whim of the person thinking about them?

I have a rigorous definition of infinity:

Infinity is big. You just won't believe how vastly, hugely, mind- bogglingly big it is. I mean, you may think it's a long way down the road to the pharmacy, but that's just peanuts to Infinity.

8)

Lol - a hitchhikers guide fan?

Originally Posted by river_rat

Lol - a hitchhikers guide fan?

For life! 8)

Advice:
If you are really interested in this, then I suggest a course in number theory, real analysis, and set theory.

Okay, so here is the scoop. Any set of numbers has what is called "cardinality." Cardinality is nothing more than the number of elements in a set. For example, take the set of even numbers between and including 0 and 10: {0, 2, 4, 6, 8, 10}. The cardinality of this set is 6 (it has 6 elements).

Now, with the definition of cardinality in hand, let's talk about infinity. For our set, let's take the positive integers (1, 2, 3,... infinity). This is denoted by a "fancy" capital Z+. (I say "fancy" because on the chalkboard it is written with two diagonal lines instead of one.) The cardinality of Z is... you guessed it... infinity. It is denoted (the cardinality that is) by the Hebrew letter "aleph." More precisely, aleph-null (the letter aleph with a subscript zero).

Now, let's take for our set, all the integers (the positive, negative, and zero). You may think that we just doubled our set and added another element (zero) so it must be larger. Can anyone guess what the cardinality of this set is?

It is still infinity (aleph-null)!!!

This kind of set, such as the positive integers and all integers is further defined as being "countably infinite." For example, we can start with the number zero, then increment one positive and one negative, and keep repeating this forever as we count (0, 1, -1, 2, -2, ...). We can count them. And the cardinality of the two sets I gave as examples can be proven to be the same if we can define a "one-to-one and onto mapping" between the two sets (which we can).

Now, you may have guessed that there are sets that are infinite and NOT countable. For example, there are an infinite number of numbers between 0 and 1. Now try to count them. Here, I'll get you started: 0. But where do we go from there? See, we can't count them!

Okay, almost done, I promise!
Now let's take as our set the set of all real numbers (R). (Z and Z+ are subsets of this set by the way, and R is a subset of C - the set of all complex numbers. Just an aside....) With the set of all real numbers, R, in our mind's grasp, can anyone guess what the cardinality of it is?

Yup. Infinity. But...
Do you think it is the same as the infinity of Z?

The answer is NO!!!!!!!

The cardinality of R is not equal to the cardinality of Z!!!
(I believe the cardinality of R is denoted by an aleph-one (debatable - read about the "continuum hypothesis"))

So how would we prove this you might ask? Well, we can't define a one-to-one and onto mapping between the two sets. Mathematically, you start by assuming you CAN and do a proof by contradiction. (Of course there's more to it than just that but forgive me, I didn't plan on doing this today when I woke up this morning.)

The moral of all this is that sometimes infinity does not even equal infinity. There are different "degrees" of infinity!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!



To specifically answer the original post:
"say there is a theoretical highest number. call it n. but n+2=2 higher than n. correct?"
- not if n = infinity. The two sets have the same cardinality.

"infinity+infinity=infinity."
---- = aleph-null. I demonstrated this with the example of Z and Z+.

"but what about infinity divided by infinity? is it not 0?"
---- It is undefined. Not 1, and not 0. But, n/infinity = 0, (n finite).

"and also, take any number divided by infinity. what do you get?"
---- See previous.

"0.0000000.......0000000001. infinite zeros, than one. if the zeros are infinite, how can you get the one at the end?does the 1 keep getting pushed back forever?"
---- Ask what is 0.99999999999.... It is = 1. It's basically the same question as the one you asked.

"and what is 2 - infinity? it is negative infinity."
---- This is the same as -infinity + 2, which is still -infinity.

"infinitly far away from zero, and infinitly away from any other number, even if it is negative. because infinite is not bigger. just farther away from the number line, whatever the direction."
---- This is true. Of course, keep in mind (in light of my previous discussion) that on the real axis (number line), if we are working in some system of units if you want to call it a distance (you said "farther away") so we don't have to worry about different degrees of infinity (the alephs) since we are only working in one set (miles, centimeters, etc.). Can you guess which set (which aleph) it would be?


Further reading:

For the layman: (the books I really liked are
From Here to Infinity, Ian Stewart
White Light, Rudy Rucker (a sci-fi)
Infinity and the Mind, Rudy Rucker

Advanced:
Any textbook you can buy used (and cheap!) on number theory, set theory, and real analysis. A warning though - unless you are VERY serious about learning (how to do) this stuff, you would probably be wasting your money. (And you would have to AT LEAST know calculus in order to peruse such a book....)

Keep thinking,
w

A thing you have to remimber is that infinty is a ideal reresentation that no one can truely understand and the only reason we have the rules such as x/infinty is zero is becouse the largest number we can understand. would create such a small number that would not change the over all out as we are concerned or can comprehind[/list][/code]

suggesting someting really stupid maybe
maybe infinity could be a zero for another stage
we have 2
posotive and negative
i dont really know the proper name for it lol
but could there be like a third stage?
which if accpeted(not likely)after would make all teh new stuff come out
ahh then i wud faint