A Mathematical Paradox?

Are there really people that believes ...
* that 007 and 7 are two different numbers?
* that 1.0 and 1.00 are two different numbers?
* that 0.999... and 0.9999... are two different numbers?

Originally posted by FabianFnas
Are there really people that believes ...
* that 007 and 7 are two different numbers?
* that 1.0 and 1.00 are two different numbers?
* that 0.999... and 0.9999... are two different numbers?

Yes. In statistical analysis there can be a very clear difference between 1.0 and 1.00.

There is no difference between 007 and 7.

http://www.purplemath.com/modules/rounding2.htm

Originally posted by twhitehead
No, it means it gets closer and closer the more terms you take. Its the same concept as used in calculus. In calculus infinities are also carefully avoided.

So earlier you wrote:

"instead, we define the sum to be the number that the sequence converges to."

So, in other words: "we define the sum to be the number that the sequence gets closer and closer to."

I agree with this.

I still don't see why we therefore should say "at infinity the number the sequence is is 1", when we could also say "at infinity the number the sequence gets infinity close to is 1".

Sorry, I just don't get it.

Originally posted by FabianFnas
So without proof it is just an opinion.

For those having a stringent mathematical proof, however, are infinitly closer to a truth.

Calm down Fabian, this is not a debate. I'll be the first to admit that I'm in the minority. I've already made this crystal clear. But I'm not going to say "yeah, I guess it's true" when every fiber in my body says it isn't so. I'm don't care that that makes me look stupid or whatever.

Originally posted by DeepThought
Do you think there is any difference between 1 and 01, or 001 or ···001 where the ellipsis is intended to convey an infinite number of leading zeros. If not then why do you think that trailing zeros after the decimal point make a difference to the value of a number?

Like I said previously, yes there can be a difference between 1 and 1.0. No, there is, as far as I am aware, no difference between 07 and 7. Other than when needing to sort the numbers 1 to 10 in excel.

The post that was quoted here has been removed

Yes, the "limit" is a term that I understand in this regard. But in my opinion it remains so that the convergent series gets infinitely close to this limit (at infinity), but never reaches it.

I will read your wiki link later.

I do wish I had made more of an effort to get good at maths when I was younger.

Originally posted by Great King Rat
I still don't see why we therefore should say "at infinity the number the sequence [b]is is 1", when we could also say "at infinity the number the sequence gets infinity close to is 1".

Sorry, I just don't get it.[/b]

Either you say "at infinity the number the sequence is is 1" or you say "as one approaches infinity the sums gets infinitely close to 1". By other words either you use a language that is static in both ends of the sentence or you use a language that is sentence at both ends of the sentence. Mixing static and dynamic language is just plain worng.

Maybe a simpler example will make my point come across clearly. Take the expression f(x)=1/x and let take the limit x tends to infinity. Hence you could say:
1 - At infinity f(x)=0
2 - As x approaches infinity f(x) approaches 0.

Saying something like at infinity f(x) gets infinitely close to 0 doesn't make because in the first half of the sentence one is using language that suggests that are at infinity and are not moving while on the second part of the sentence we are using language that suggests movement. It seems semantics but actually it is profound.

Did this make any sense to you?

PS: I know I haven't really answered your objection but first I want to make sure that the language that we usually use at mathematics makes sense to you.

Originally posted by adam warlock
Either you say "at infinity the number the sequence [b]is is 1" or you say "as one approaches infinity the sums gets infinitely close to 1". By other words either you use a language that is static in both ends of the sentence or you use a language that is sentence at both ends of the sentence. Mixing static and dynamic language is just plain worng. ...[text shortened]... rst I want to make sure that the language that we usually use at mathematics makes sense to you.[/b]

No, I don't think it's semantics at all, it's a good point. I concur that that sentence I wrote was not correct.

However, I would still not say "at infinity the number is 1", I would say "at infinity the number is infinitely close to 1".

And since I'm being stubborn 🙂 and we're trying to use the correct mathematical rules:

You say at x = infinity:

0 = 1/f(infinity)

If I say that:

2 = 6/3, then 3 = 6/2

And thus

f(infinity) = 1/0

Here we're clearly breaking rules, because we may not divide by zero. Again, this appears to me is a case of breaking mathematical rules, because it is attempted to use infinity in a calculation.

Originally posted by Great King Rat
Yes. In statistical analysis there can be a very clear difference between 1.0 and 1.00.

There is no difference between 007 and 7.

In programming there is a very significant difference between 007 and 7.

Originally posted by twhitehead
In programming there is a very significant difference between 007 and 7.

I should have definitely written "as far as I am aware, there is no difference" 😉

Originally posted by Great King Rat
I still don't see why we therefore should say "at infinity the number the sequence is 1", when we could also say "at infinity the number the sequence gets infinity close to is 1".

Sorry, I just don't get it.

There is no such number as infinity, and it is incorrect to say 'at infinity'. Mathematicians typically avoid doing so. (I'll avoid saying 'us' since it seems to upset Duchess64 ).

Originally posted by Great King Rat
No, I don't think it's semantics at all, it's a good point. I concur that that sentence I wrote was not correct.

However, I would still not say "at infinity the number is 1", I would say "at infinity the number is infinitely close to 1".

And since I'm being stubborn 🙂 and we're trying to use the correct mathematical rules:

You say at x = inf ...[text shortened]... a case of breaking mathematical rules, because it is attempted to use infinity in a calculation.

however, I would still not say "at infinity the number is 1", I would say "at infinity the number is infinitely close to 1".

In that case I'd ask you what is the difference between a number "being 1" and a number "being infinitely close to 1".?

And since I'm being stubborn 🙂 and we're trying to use the correct mathematical rules:

You say at x = infinity:

0 = 1/f(infinity)

If I say that:

2 = 6/3, then 3 = 6/2

And thus

f(infinity) = 1/0

Here we're clearly breaking rules, because we may not divide by zero. Again, this appears to me is a case of breaking mathematical rules, because it is attempted to use infinity in a calculation.

Here there a couple of main issues and maybe the main issue of note is that we are abusing notation and making a few semantic mistakes.

If we are dealing with real numbers then x=infinity isn't a well defined expression. What that notation suggests really is x->infinity (read x tends to infinity).

With that in mind let us get back to f(x)=1/x. When we say that f(infinity)=1/infinity=0 we really mean that we are taking limits and making x take on larger and larger values and that while we do that f(x) gets closer and closer to 0.
In the limit we say f(x)->0 as x->infinity.
Thus, again we get that dichotomy between a dynamic language and a static language. We say that the limit is 0 or we say that f(x)->0 as x->infinity but mixing up the two things is just wrong.

When we say that 1+1/2+1/4+1/8+...1/2^n+...=1 we mean to say two equivalent things:
1 - The limit of the partial sums (if by now you don't know what a partial sum is just tell me and I'll try to make it clear) is 1.
2 - The as the number of terms being considered increases the result of the partial sums gets closer and closer to 1.

And for me this the main issue that you have. The reconciliation of those two sentences just doesn't work for you because they seem to be telling different things, but really they are saying the same thing (if I'm mistaken here please correct me!)

One could way for you to see that these two sentences are actually equivalent is to think in terms of epsilon and deltas.
The naive notion that we have of what is the limiting procedure in mathematics can be made rigorous in a lot of different (but equivalent) ways. One such way is the epsilon-delta definition basically what it says in lay terms is:
If the limit of a function exists in a given point then as you approach the said point the difference between the function and its limit also decreases.
Take a look at these two links to see the definition and then a couple of simple exercises with the definition of limit (actually in these two links you'll see the definition of continuity but the gist is the same):
http://climbingthemountain.wordpress.com/2009/07/01/real-analysis-%E2%80%93-limits-and-continuity-iv/
http://climbingthemountain.wordpress.com/2011/04/23/real-analysis-limits-and-continuity-v/

Getting back to Earth we'll take a look at a trivial example: f(x)=4.
Let us take the limit x->infinity in the previous function.

It is f(infinity)=4 (since the righthandside doesn't depend on x it'll always be 4)

Let us at a second trivial example:
f(x)=2x

Let us take the limit x->2. It is f(2)=2*2=4.

If you were to use the epsilon-delta definition of a limit you'd see that you could always find an epsilon for each delta. And that means that the limit indeed is the values that we presented. The thing with Math is that if you accept a definition and see that it works for cases that your intuition agrees with you'll also have to agree with the definition even if it runs against your intuition. It just a matter of being consistent. Now the thing is that you can use the epsilon-delta condition to show that 1+1/2+1/4+1/8+...1/2^n+... indeed is equal to 1. And if you accept the validity of the epsilon-delta definition for trivial cases you have to accept it for non trivial cases too.

Just to sum things up and say my last words on this already long post:

1+1/2+1/4+1/8+...1/2^n+...=1 basically means what you say what it means. The result of the sum gets infinitely close to 1 as we take more terms into account. But this last sentence also means that in the limit (which is to say that you consider all terms) the sum equals 1.

Again did any of this make any sense to you?

PS: Sorry for the highly repetitive post
PPS: in the extended real number line you can define +infinity and -infinity and expressions like 1/0=infinity do make sense and very well defined. So your saying that you can't divide by 0 is totally true, but only when your dealing with numbers that are on the real number line.
PPPS: If you want to get really academical you can also check out non-standard analysis.

Originally posted by twhitehead
There is no such number as infinity, and it is incorrect to say 'at infinity'. Mathematicians typically avoid doing so. (I'll avoid saying 'us' since it seems to upset Duchess64 ).

There is no such number as infinity

Actually there is such a number.

Mathematicians typically avoid doing so

Actually they don't.

Originally posted by adam warlock

There is no such number as infinity

Actually there is such a number.

Mathematicians typically avoid doing so

Actually they don't.

I am sure you are more knowledgeable than me in this area, so I won't dispute that. But do you have any references along those lines?

I did come across this:
http://en.wikipedia.org/wiki/Point_at_infinity

But its not quite the same thing.