A simple question.

[Tiberius]

By the same logic I can say that 0.(1) = 0.11...111 is finite because it ends with "1".

A few things wrong with this statement:

1) 0.11...111 would be finite even if it didn't end with a 1, since I can name a number higher than it (0.2), and hence it is not infinite. What you mean is that it is finitely long, which is perfectly true if it ends with a 1, and the way you have written it (0.11...111) shows this.

2) 0.(1) is not the same as 0.11...111, since by definition 0.(1) doesn't have an end, and 0.11...111 has a very visible end. The correct notation for 0.(1) is 0.111... (the ellipsis indicating that the 1 repeats endlessly).

3) Whilst 0.11...111 does end with a very visible 1, 0.(1) or 0.111... does not. The ellipsis makes it endless, and so whilst it is a finite value, it is not of finite length.

In other words, my example hasn't an end by definition, either. If this "0.00...001" does not look as you'd like, then use this representation:

"1/10 * 1/100 * 1/1000 ..." - because this is clearly not an "invalid" thing.

Your example does have an end: 0.00...001 <-- there it is.

Whilst your calculation is a valid one, the answer to it diverges to 0 as I pointed out, and as such it is not a valid answer to the question "what is the first number that comes after 0?" because either (a) it eventually becomes 0, or (b) it diverges to a value which is impossible to calculate (since we would need a infinite amount of time in order to do it).

I don't care if infinite geometric series use or not the infinity in the actual calculation. I also don't care how this thing should be called. But, if you insist that "1/10 * 1/100 * 1/1000 ..." must not be used, perhaps you can show where it's written that this thing is forbidden to be used.

I never insisted that your calculation must not be used, I said that the calculation diverges to 0, which in mathematics either means it becomes equal to 0, or becomes infinitely small and cannot be calculated.

I asked you what the first number to come after 0 was, and you have not given me a valid answer.

[Zenith]

1) 0.11...111 would be finite even if it didn't end with a 1, since I can name a number higher than it (0.2), and hence it is not infinite. What you mean is that it is finitely long, which is perfectly true if it ends with a 1, and the way you have written it (0.11...111) shows this.

Show me a definition of "infinite". I thought that infinite means something that does not end.
This is what I've found about finite:

1. having bounds or limits; not infinite; measurable.
2. Mathematics .
a. (of a set of elements) capable of being completely counted.
b. not infinite or infinitesimal.
c. not zero.
3. subject to limitations or conditions, as of space, time, circumstances, or the laws of nature: man's finite existence on earth.

And infinite:

3. unlimited or unmeasurable in extent of space, duration of time, etc.: the infinite nature of outer space.
4. unbounded or unlimited; boundless; endless: God's infinite mercy.
5. Mathematics .
a. not finite.
b. (of a set) having elements that can be put into one-to-one correspondence with a subset that is not the given set.

So by this notion 0.(1) is not a finite number and 0.(0)1 is not a finite number.

Also, you said:
" 0.11...111 would be finite even if it didn't end with a 1, since I can name a number higher than it (0.2), and hence it is not infinite"
So I guess that -infinity is a finite number because even -10^10000000000000 is greater than it.

2) 0.(1) is not the same as 0.11...111, since by definition 0.(1) doesn't have an end, and 0.11...111 has a very visible end. The correct notation for 0.(1) is 0.111... (the ellipsis indicating that the 1 repeats endlessly).

I guess it's just a matter of notation. If I say that 0.11...111 has 234 decimals then yes, it is finite. But if I specify that it never ends, you can't put a real end to it. I know the notation "0.111..." is used to show an infinite number of decimals. But that's just a notation.

3) Whilst 0.11...111 does end with a very visible 1, 0.(1) or 0.111... does not. The ellipsis makes it endless, and so whilst it is a finite value, it is not of finite length.

Now it's a funny thing: if 0.111... does not end with a very visible 1, it doesn't mean that it can be other digit. So it's clearly "1", only that it's not shown (visible).

Your example does have an end: 0.00...001 <-- there it is.

Show me an end to this number: 1/10 * 1/100 * 1/1000 ...

Whilst your calculation is a valid one, the answer to it diverges to 0 as I pointed out, and as such it is not a valid answer to the question "what is the first number that comes after 0?" because either (a) it eventually becomes 0, or (b) it diverges to a value which is impossible to calculate (since we would need a infinite amount of time in order to do it).

Well, as far as I know, not even PI is possible to be calculated (with all decimals), but that doesn't make it an invalid number.

I never insisted that your calculation must not be used, I said that the calculation diverges to 0, which in mathematics either means it becomes equal to 0, or becomes infinitely small and cannot be calculated.

Not even PI can be calculated (to have the exact number). As about my number, it can practically become 0 only if you calculate the lim() of it.

I asked you what the first number to come after 0 was, and you have not given me a valid answer.

Do you have a better answer?

[Tiberius]

Show me a definition of "infinite". I thought that infinite means something that does not end.

So by this notion 0.(1) is not a finite number and 0.(0)1 is not a finite number.

An "infinite" number is a number that has infinite value, whilst you are talking about numbers that have a finite value, but are infinitely long. That was the difference. 0.(1) is a finite number, as it has a finite value, but it is infinitely long.

As I've said before, 0.(0)1 is an invalid number; it can't exist. You cannot have an infinite number of 0's and then have a 1 on the end, because there can't be an end for the 1 to go on.

Also, you said:
" 0.11...111 would be finite even if it didn't end with a 1, since I can name a number higher than it (0.2), and hence it is not infinite"
So I guess that -infinity is a finite number because even -10^10000000000000 is greater than it.

-infinity is a totally different ball park. Both infinity and -infinity are limits of the number line. All numbers have to lie between them. Infinity is positively infinite, whilst -infinity is negatively infinite. I assumed you were talking about positive infinity, since you were talking about lengths of numbers (all of which are positive). I don't see where -infinity comes into it here.

If you want, I'll restate what I said above. 0.11...111 is not positively infinite because I can name a number higher than it (0.2), and it is not negatively infinite because I can name a number lower than it (0).

I guess it's just a matter of notation. If I say that 0.11...111 has 234 decimals then yes, it is finite. But if I specify that it never ends, you can't put a real end to it. I know the notation "0.111..." is used to show an infinite number of decimals. But that's just a notation.

The problem is you do specify that it ends, because after the ellipsis, you've put 111. That is the end. Notation in this case matters, because if you insist on using the notation 0.11...111, and then saying that the ... means there are an infinite number of 1's there, you are causing a contradiction, because you have an infinite number of 1's followed by "111". This cannot happen.

Now it's a funny thing: if 0.111... does not end with a very visible 1, it doesn't mean that it can be other digit. So it's clearly "1", only that it's not shown (visible).

No, it does not end with a 1. It doesn't end at all. The ... notation implies that the 1's repeat forever, and that the number is infinitely long. As such, it does not have an end.

Show me an end to this number: 1/10 * 1/100 * 1/1000 ...

There is no end to this number, it continues for an infinite length, with an infinite number of 0's. In mathematics, it is either equal to 0, or is undefined.

Well, as far as I know, not even PI is possible to be calculated (with all decimals), but that doesn't make it an invalid number.

No calculation uses all the digits of pi; we simply use the first few of them in most cases. In this case however, where we want the first actual number to follow 0, your number doesn't fit the bill, since we require every digit.

Not even PI can be calculated (to have the exact number). As about my number, it can practically become 0 only if you calculate the lim() of it.

...and in this case, we want the limit of it, since you gave me a sequence calculation that diverge to 0.

Do you have a better answer?

Yes. The first number that comes after 0 is impossible to calculate, and hence is unknown.

[Zenith]

An "infinite" number is a number that has infinite value, whilst you are talking about numbers that have a finite value, but are infinitely long. That was the difference. 0.(1) is a finite number, as it has a finite value, but it is infinitely long.

As I've said before, 0.(0)1 is an invalid number; it can't exist. You cannot have an infinite number of 0's and then have a 1 on the end, because there can't be an end for the 1 to go on.

You know, I've been thinking of something...
if we've got a meter long segment, and we want to split it in 3 equal parts, the dimension of each part would be 0.(3) m.

That is:
we get a segment of a line, and want to get the first 1/3 of it. We call the new segment, that would be extracted, a:
We calculate 0.3 m from left and draw a line to it. here, a = "0.3 m";
We zoom and we add 0.03 to it. Now, a = "0.03 m". We've noticed that a has increased;
We zoom further and we add 0.003 to it. Now, a = "0.003 m". the length of a has increased again.
And this process would continue forever.

So, wouldn't this mean that the resulting segment a would be infinitely long?
And this would mean that a stick of 1 meter length can never be splat equally, because the stick is formed of atoms that have specific sizes (and you cannot split their protons, etc.). Right?

[Tiberius]

Yes, it would be infinitely long (equal to 0.333...)

Whether a metre long segment can be split into three equal parts is probably not something that this sort of maths can sort out. Firstly, although we are splitting 1 segment, that 1 segment is made up of trillions upon trillions of atoms. If those atoms were arranged in a perfect order, and were of an amount divisible by three, then you could theoretically split the segment into three (each piece having the same amount of atoms).

Of course, practically this is probably impossible to do. In mathematics, this is just another quirk of number theory, not really applicable to the real world (since in the real world, infinity may not even exist).

[Zenith]

Yes, it would be infinitely long (equal to 0.333...)

Whether a metre long segment can be split into three equal parts is probably not something that this sort of maths can sort out. Firstly, although we are splitting 1 segment, that 1 segment is made up of trillions upon trillions of atoms. If those atoms were arranged in a perfect order, and were of an amount divisible by three, then you could theoretically split the segment into three (each piece having the same amount of atoms).

Yeah, except if the number of atoms that form the stick is not divisible by 3