A dumb question I know, but I was thinking about an interesting idea lately, It is based on the following assumptions:
Infinity times any positive, finite number is infinity
Any fraction of infinity is infinity
infinity is greater then any finite number
Ok so I was thinking about 1.9 repeating. you can increase 1.9 by adding another 9 behind the decimal an infinite number of times, so depending on how you look at it you are adding a fraction of infinity or multiplying infinity by a finite number(either way equating to infinity). But the catch is that 1.9 repeating could never exceed 2, a finite number.
I don't know, perhaps Im insane or just stupid but Im interested to hear others opinions on this.

just because a number is irrational does not mean it is equal to infinity. for example, pi is not equal to infinity. e is also not equal to infinity. get it?

your post is mostly rubbish and not worthy of further comment.

I understand and think your concept is correct yet, there is something not quite riht about it.

so depending on how you look at it you are adding a fraction of infinity or multiplying infinity by a finite number(either way equating to infinity).

This bit just doesn't make any sense.

its basically like 0.9 + 0.09 + 0.009 + 0.0009 ect for an infinite number times

Infinity is not a number, don't treat it like one.

my mathematics professor said himself that infinity is NOT a number.

it is a concept of something that is really large.

when talking about limits, you dont say the limit of a function as x goes to infinity is equal infinity, but rather "as x-> infinity, f(x) -> infinity" or "f(x) is unbounded"

it is a concept of something that is really large.



i think that's a bit of an understatement :D

i remember reading about there being different "sized" infinities

example an infinity made up of whole numbers versus one made up of numbers with decimals...

obviously the one with decimals could go 1, 1.1, 1.2... while the one made up of whole numbers would go 1, 2, 3... both are infinite but one is more "dense" if you will

i dont know, it thought it was an interesting bit to think about

and i think a consensus has been reached that infinity is a "concept" by most mathematicians

not a number or an illusion...

The different sizes of infinity is true, as was first shown by Georg Cantor who was a brilliant mathematician. To understand it though, you need to think of infinity as providing a cardinality, or a sense of the size of a set. So, the cardinality just tells you the number of elements in a set. So if your set is {1,5,6,13,3532} then the cardinality is 5 since there a 5 distinct elements in that set.

Now to compare sizes of sets, its very simple if the sets are finite, even if they have very different members. So, for example, one set could be {a,g,h,d,r} and another set could be {5, 2, 4, 1, 9}. Now by simply counting each set you can easily see they have the same number of elements. But the real way you compare them is by pairing off elements. So, for example, you can associate a with 5, g with 2, h with 4, d with 1, and r with 9. Then, since every element of the first set is paired off with exactly one element of the second set, and there are no repeats or anything left over without being paired off, then the sets have the same number of elements.

To be precise using common mathematical terminology, this pairing off is called a function, then since there are no repeats the function is called injective, and since there is nothing left over without being paired off the function is called surjective. When a function is injective and surjective, it is called bijective, and having a bijective function from one set to another set precisely says that both sets have the same cardinality, or the same number of elements.

This is all very simple with finite sets, but things are a little more complicated with infinite sets. For example, take two sets, A={1,2,3,4,5,....} and B={1,4,9,16,25,....}. So A is just all the positive integers, which is obviously infinite in number, and B is all the squares of the integers, which is also clearly infinite. On first look it might seem that there are less numbers in B than there are in A, since they are so much more spread out. However, you can associate to every number in A, precisely one number in B, namely its square. Clearly there are no repeats, and there are no numbers left over, hence there is a bijection from A to B, and these sets have the same cardinality. Or in other words, the infinity that describes the size of A is the same infinity that describes the size of B. (This infinity is always referred to as aleph-naught, so I'll do that from now on.)

It can also be shown without much difficulty, that the set of all possible fractions, (called the rational numbers) has the cardinality aleph-naught, so there are as many fractions as positive integers. However, consider the set of all real numbers. This is just all possible decimals, whether they be repeating or non-repeating. Recall a repeating decimal is something like .33333.... or .500000....., and so on. These are just the rational numbers and have cardinality aleph-naught.

However, decimals like pi and e, (and many, many others) do not ever repeat, and basically proceed at random with their digits. It can be shown (by a very simple yet ingenious argument done by Cantor, called Cantor's diagonal slash) that this set of real numbers cannot have a bijective function(or pairing off) with the positive integers. So, in a clear sense, there are more real numbers than integers. Thus, the infinity describing the size of the set of real numbers is "larger" than the infinity describing the size of the set of positive integers.

However, back on topic, a lot of the confusion concerning infinity comes from trying to interpret it as a number, which as has already been said, it is not. So, in your original post, it makes no sense to say infinity multiplied by a number, or a fraction of infinity. The proper way to interpret it, in most cases, is as a cardinality, or size of a set, as above.

Why do unthinking people always try to philosophize when they are drunk.

wow... my brain hurts, but that was useful. So the entire paradox is in fact based in an incorrect assumption, that infinity is a number.

The different sizes of infinity is true, as was first shown by Georg Cantor who was a brilliant mathematician. To understand it though, you need to think of infinity as providing a cardinality, or a sense of the size of a set. So, the cardinality just tells you the number of elements in a set. So if your set is {1,5,6,13,3532} then the cardinality is 5 since there a 5 distinct elements in that set.

Now to compare sizes of sets, its very simple if the sets are finite, even if they have very different members. So, for example, one set could be {a,g,h,d,r} and another set could be {5, 2, 4, 1, 9}. Now by simply counting each set you can easily see they have the same number of elements. But the real way you compare them is by pairing off elements. So, for example, you can associate a with 5, g with 2, h with 4, d with 1, and r with 9. Then, since every element of the first set is paired off with exactly one element of the second set, and there are no repeats or anything left over without being paired off, then the sets have the same number of elements.

To be precise using common mathematical terminology, this pairing off is called a function, then since there are no repeats the function is called injective, and since there is nothing left over without being paired off the function is called surjective. When a function is injective and surjective, it is called bijective, and having a bijective function from one set to another set precisely says that both sets have the same cardinality, or the same number of elements.

This is all very simple with finite sets, but things are a little more complicated with infinite sets. For example, take two sets, A={1,2,3,4,5,....} and B={1,4,9,16,25,....}. So A is just all the positive integers, which is obviously infinite in number, and B is all the squares of the integers, which is also clearly infinite. On first look it might seem that there are less numbers in B than there are in A, since they are so much more spread out. However, you can associate to every number in A, precisely one number in B, namely its square. Clearly there are no repeats, and there are no numbers left over, hence there is a bijection from A to B, and these sets have the same cardinality. Or in other words, the infinity that describes the size of A is the same infinity that describes the size of B. (This infinity is always referred to as aleph-naught, so I'll do that from now on.)

It can also be shown without much difficulty, that the set of all possible fractions, (called the rational numbers) has the cardinality aleph-naught, so there are as many fractions as positive integers. However, consider the set of all real numbers. This is just all possible decimals, whether they be repeating or non-repeating. Recall a repeating decimal is something like .33333.... or .500000....., and so on. These are just the rational numbers and have cardinality aleph-naught.

However, decimals like pi and e, (and many, many others) do not ever repeat, and basically proceed at random with their digits. It can be shown (by a very simple yet ingenious argument done by Cantor, called Cantor's diagonal slash) that this set of real numbers cannot have a bijective function(or pairing off) with the positive integers. So, in a clear sense, there are more real numbers than integers. Thus, the infinity describing the size of the set of real numbers is "larger" than the infinity describing the size of the set of positive integers.

However, back on topic, a lot of the confusion concerning infinity comes from trying to interpret it as a number, which as has already been said, it is not. So, in your original post, it makes no sense to say infinity multiplied by a number, or a fraction of infinity. The proper way to interpret it, in most cases, is as a cardinality, or size of a set, as above.

do you actually expect anyone to read all of tHat?

If they're interested in infinity, set theory, or math in general then sure. Otherwise, it would probably be pretty dull, so no.

Also, I probably should have mentioned that the part that has the most direct relevance to this thread is the last paragraph. All the preceding parts were mainly there to clarify the post before my long one from house, although all the rest of it does at least clarify the statement in my last paragraph that infinity should be thought of as a cardinality, and that infinity does have an objective existence and is not merely an illusion.

Ask Chuck Norris. He has counted to infinity twice and once backwards.

do you actually expect anyone to read all of tHat?

I read it all, and found it very interesting.

I read it all, and found it very interesting.

Such things are great. Everybody finds philosophy interesting, we all want to understand the mysteries of the universe, but barely anyone takes the time for it, unless they are drunk.

Anyhow, its good to find this kind of threads here. Means there are still people here seeking wisdom.

The different sizes of infinity is true, as was first shown by Georg Cantor who was a brilliant mathematician. To understand it though, you need to think of infinity as providing a cardinality, or a sense of the size of a set. So, the cardinality just tells you the number of elements in a set. So if your set is {1,5,6,13,3532} then the cardinality is 5 since there a 5 distinct elements in that set.

Now to compare sizes of sets, its very simple if the sets are finite, even if they have very different members. So, for example, one set could be {a,g,h,d,r} and another set could be {5, 2, 4, 1, 9}. Now by simply counting each set you can easily see they have the same number of elements. But the real way you compare them is by pairing off elements. So, for example, you can associate a with 5, g with 2, h with 4, d with 1, and r with 9. Then, since every element of the first set is paired off with exactly one element of the second set, and there are no repeats or anything left over without being paired off, then the sets have the same number of elements.

To be precise using common mathematical terminology, this pairing off is called a function, then since there are no repeats the function is called injective, and since there is nothing left over without being paired off the function is called surjective. When a function is injective and surjective, it is called bijective, and having a bijective function from one set to another set precisely says that both sets have the same cardinality, or the same number of elements.

This is all very simple with finite sets, but things are a little more complicated with infinite sets. For example, take two sets, A={1,2,3,4,5,....} and B={1,4,9,16,25,....}. So A is just all the positive integers, which is obviously infinite in number, and B is all the squares of the integers, which is also clearly infinite. On first look it might seem that there are less numbers in B than there are in A, since they are so much more spread out. However, you can associate to every number in A, precisely one number in B, namely its square. Clearly there are no repeats, and there are no numbers left over, hence there is a bijection from A to B, and these sets have the same cardinality. Or in other words, the infinity that describes the size of A is the same infinity that describes the size of B. (This infinity is always referred to as aleph-naught, so I'll do that from now on.)

It can also be shown without much difficulty, that the set of all possible fractions, (called the rational numbers) has the cardinality aleph-naught, so there are as many fractions as positive integers. However, consider the set of all real numbers. This is just all possible decimals, whether they be repeating or non-repeating. Recall a repeating decimal is something like .33333.... or .500000....., and so on. These are just the rational numbers and have cardinality aleph-naught.

However, decimals like pi and e, (and many, many others) do not ever repeat, and basically proceed at random with their digits. It can be shown (by a very simple yet ingenious argument done by Cantor, called Cantor's diagonal slash) that this set of real numbers cannot have a bijective function(or pairing off) with the positive integers. So, in a clear sense, there are more real numbers than integers. Thus, the infinity describing the size of the set of real numbers is "larger" than the infinity describing the size of the set of positive integers.

However, back on topic, a lot of the confusion concerning infinity comes from trying to interpret it as a number, which as has already been said, it is not. So, in your original post, it makes no sense to say infinity multiplied by a number, or a fraction of infinity. The proper way to interpret it, in most cases, is as a cardinality, or size of a set, as above.


Good post
+1 epic
Sorry I love math.

The problem with your idea is that you have an infinite number of infinitesimal values. And, believe it or not, that is a finite number. It's the basis of calculus.

A dumb question I know, but I was thinking about an interesting idea lately, It is based on the following assumptions:
Infinity times any positive, finite number is infinity
Any fraction of infinity is infinity
infinity is greater then any finite number
Ok so I was thinking about 1.9 repeating. you can increase 1.9 by adding another 9 behind the decimal an infinite number of times, so depending on how you look at it you are adding a fraction of infinity or multiplying infinity by a finite number(either way equating to infinity). But the catch is that 1.9 repeating could never exceed 2, a finite number.
I don't know, perhaps Im insane or just stupid but Im interested to hear others opinions on this.

What you are doing has not much to do with infinity. Rather, you are talking about the concept of a limit. If you turned your 1+0.9+0.09..etc into a function, as x approaches infinity, you would find that the limit is 2. No, it the graph itself will hypothetically never reach 2, but as far as limits are concerned, at x-> infinity, y = 2.

In another words, what you're doing is putting infinity into the input. The 2 is the OUTPUT. Can't compare those two.

Why do unthinking people always try to philosophize when they are drunk.


Seriously, why bitch about this? Better than the alternative.

do you actually expect anyone to read all of tHat?

I read it... acolyte

Infinity is not an illusion. Numbers do began to become irrelevant at a point though. It boils down to why would you ever need to use the number 1,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000.

Actually large numbers like that do have relevance to the world, in many ways. For instance, cryptography uses huge prime numbers, larger than the number in the previous post even in order to encode messages. Also, there's a thread in this forum now about Graham's number, which is unimaginably larger than any number you could possibly write down in standard notation,(there's not even enough material in the entire universe to write this number down) and this number was actually used in a mathematical proof.

The correct answer is, yes its an illusion. Technically speaking all numbers are illusions. They are a construct of nothing and something, that is it.

since time is an illusion, a man made concept then anything affiliated with time is also an illusion.

but if you want to get technical and assume time is just the passing of space, then yes there is an infinity. space is ever expanding, into what? nobody knows.

The correct answer is, yes its an illusion. Technically speaking all numbers are illusions. They are a construct of nothing and something, that is it.

Numbers are terms used to describe quantities. Sure they don't physically exist; the number five does not have a physical location, shape, or size. But it is certainly not an illusions.

I have five apples in my basket. This is not an illusion.

Infinity is also not an illusion. It is a concept.

1.9 recurring is not infinity, it is an irrational number. Though, if you're counting the 9's then I guess you could say it is infinitley long.

space is ever expanding, into what? nobody knows.

Why does it even have to be expanding into anything? Alot of people struggle with the idea of infinity, never ending and lack of purpose.

1.9 recurring is not infinity, it is an irrational number.

1.999... is equal to two, which is clearly rational. ;)

More generally, any real number with a decimal representation which is eventually repeating is rational. Try proving this if you are interested.

More generally, any real number with a decimal representation which is eventually repeating is rational. Try proving this if you are interested.

Wow, that just brought me back to real analysis, which is not a good thing, lol. I actually remember proving exactly that, or it might've been the converse of that statement, I forget which. Either way I remember it being a bitch to do rigorously, without using any common intuitive notions about decimals and fractions. We started from scratch and constructed real numbers as equivalent classes of Cauchy sequences of rational numbers. Needless to say, it made proving anything about real or rational numbers, like the above, quite a bitch.

1.999... is equal to two, which is clearly rational. ;)

Haha, I always argue the opposite. Would you care to explain. Thanks.

Haha, I always argue the opposite. Would you care to explain. Thanks.

Theres many proofs of the fact that 1.9 repeating equals 2, but here's a pretty good one, especially if your familiar with geometric series. (This actually proves .9 repeating equals 1, which is basically the same thing).

So, as you might know, a geometric series is a series of the form, a+a*r+a*r^2+a*r^3+a*r^4+.... and so on to infinity, where a and r are real numbers. Clearly, this series will converge only if -1<r<1. This series sums to a/(1-r) which can be easily shown by the following. Let s=1+r+r^2+r^3+... Then a*s is the sum of the series. Also, r*s=r+r^2+r^3+... Now subtract r*s from s. This gives s-r*s=(1+r+r^2+...)-(r+r^2+r^3+...)=1. Thus, since s-r*s=1, then s(1-r)=1 so s=1/(1-r). And finally the total sum is a*s=a/(1-r).

By multiplying the above series by r, you get a*r+a*r^2+a*r^3+...=a*r/(1-r). Now set a=9 and r=.1 . Then the series a*r+a*r^2+a*r^3+... becomes .9+.09+.009+.0009+... which is clearly .9999..., so its .9 repeating. This equals a*r/(1-r), so putting in a=9 and r=.1 in this gives .9/.9 which equals 1, hence the whole series is equal to 1, and since the whole series is also equal to .9 repeating, then .9 repeating equals 1. Since 1 is rational, then .9 repeating is rational. (The case 1.9 repeating equaling 2 is the same thing, just add 1 to everything)

I have always considered it to be somewhat pointless to "prove" that .999...=1, or any of its equivalents, because it is a direct result of the construction of the real numbers.

You see, the entire point behind constructing the real numbers (as opposed to just using rationals) is to ensure that limits behave nicely. More specifically, we want to be able to work in a space where any sequence that eventually restrict its values to any arbitrarily small interval will be guaranteed to have a unique limit (such sequences are referred to as Cauchy sequences). A space in which Cauchy sequences are guaranteed to have a limit is called complete. The completeness of the real numbers is essential to the proof's of many well known theorems from calculus, including the intermediate value theorem.

It is easy to see that the rational numbers are not complete. For example, consider the sequence whose n'th term is given by the first n digits of the square root of 2. If this sequence had a rational limit, it would have to be sqrt(2) (why?), but there is no rational number whose square is 2 (again, try to prove this). This equivalent to the statement that the continuous function given by f(x)=x^2-2 has no rational roots, which proves that the intermediate value theorem does not on the rational numbers.

One standard way to define the real numbers is to define each real number x by the set of Cauchy sequences which "should" have x as a limit. The definition in terms of decimal representation is equivalent to this, and basically amounts to picking one particularly nice sequence from each of the afore mentioned sets.

Now, concerning .999...=1. Let us denote .999... by x. It is of course the case that x<=1. It is also obvious that every term of the sequence whose n'th term is given by by n nines after the decimal point is less than x. One is clearly a limit of this sequence, and the above inequalities guarantee that x also is. By one of the defining properties of the reals, that limits are unique, it follows that x=1.


Wow, that just brought me back to real analysis, which is not a good thing, lol. I actually remember proving exactly that, or it might've been the converse of that statement, I forget which. Either way I remember it being a bitch to do rigorously, without using any common intuitive notions about decimals and fractions. We started from scratch and constructed real numbers as equivalent classes of Cauchy sequences of rational numbers. Needless to say, it made proving anything about real or rational numbers, like the above, quite a bitch.

Well, once you prove that the decimal definition is equivalent to that definition (a good starting point for this is mentioned in the above wall of text), you can use decimals freely.

Infinity is not a number, don't treat it like one.

Nice.


Shut that nigger up huh?

Theres many proofs of the fact that 1.9 repeating equals 2, but here's a pretty good one, especially if your familiar with geometric series. (This actually proves .9 repeating equals 1, which is basically the same thing).

So, as you might know, a geometric series is a series of the form, a+a*r+a*r^2+a*r^3+a*r^4+.... and so on to infinity, where a and r are real numbers. Clearly, this series will converge only if -1<r<1. This series sums to a/(1-r) which can be easily shown by the following. Let s=1+r+r^2+r^3+... Then a*s is the sum of the series. Also, r*s=r+r^2+r^3+... Now subtract r*s from s. This gives s-r*s=(1+r+r^2+...)-(r+r^2+r^3+...)=1. Thus, since s-r*s=1, then s(1-r)=1 so s=1/(1-r). And finally the total sum is a*s=a/(1-r).

By multiplying the above series by r, you get a*r+a*r^2+a*r^3+...=a*r/(1-r). Now set a=9 and r=.1 . Then the series a*r+a*r^2+a*r^3+... becomes .9+.09+.009+.0009+... which is clearly .9999..., so its .9 repeating. This equals a*r/(1-r), so putting in a=9 and r=.1 in this gives .9/.9 which equals 1, hence the whole series is equal to 1, and since the whole series is also equal to .9 repeating, then .9 repeating equals 1. Since 1 is rational, then .9 repeating is rational. (The case 1.9 repeating equaling 2 is the same thing, just add 1 to everything)

Seriously....What the fuck....
You lost me from here a+a*r+a*r^2...

I'm not doubting you...just...What the fuck....from what I got of that formula, 0.9 hasn't been proven to equal 1 at all. But as I say, it fucked my brain.

Considering infinity, every number is a fraction of infinity for example 1 is 1/infinity, the fact that infinity is not a number proves to be ultimate predicament, how can a number be a fraction of a never ending line of numbers? Inifinity is literally a figure with no bounds. From what I learned in Maths though the mathmatical idea of infinity is one that places the idea of infinity as an actual number and uses it to problem solve, much like pi is used but to a larger degree. Usually infinity as a symbol is used to depict a real number which has not been given a static position, and isn't needed to be known to solve the problem.

Just to sum up, you cannot use mathmatical formulae to label infinity because in maths infinity is acompletely different thing to its literal meaning.

An illusion eh? Ok OP. What's the biggest number? :D


That thing about cardinality and some infinities being bigger has always tripped me out lol. I mean, I get it, but... talk about mindfuck. No wonder Cantor went nutty lol.