A simple question.

[Tiberius]

You say that 0.00...0001 is finite.
How can you call 0.00...0001 and 0.11...1111 infinite??
the former is the representation of product(i=1,infinite) of 1/(10^i) = 1/10 * 1/100 * 1/1000 * ...
and the later is the representation of sum (i = 1, infinity) of 1/10^i = 1/10 + 1/100 + 1/1000 + ... = 0.(1)

0.00...0001 is finitely long because it ends with a 1 (i.e. there is nothing after the last 1). 0.111... is infinitely long because it doesn't have an end by definition.

As I've said before, using an infinite geometric series doesn't use infinity in the actual calculation. The series tends to infinity, and the answer it calculates is derived via a specific geometric formula instead: https://secure.wikimedia.org/wikipedia/e...ric_series

For your sum example:

1/10 + 1/100 + 1/1000 ...

Where a = 1/10, and r = 1/10, the sum of the infinite series is a / (1 - r), which is:

(1/10) / (1 - 1/10)

= 0.1 / 0.9

= 0.111...

For the product example:

1/10 * 1/100 * 1/1000, since the r value is less than 1, the product is said to diverge to zero as the calculation progresses. Whether the actual product of the calculation is in fact 0 is still debated by mathematicians, but it doesn't matter in this case because if it isn't, we are unable to write the actual product in mathematical terms.

Anyway, you can't find a number smaller than it because there are an infinite number of decimals (which mean, an infinite number of "0" decimals, all before that "1").

There can't be an infinite number of "0" digits before a "1". In order for there to be a "1" on the end, there needs to be an end. If there are an infinite number of "0" digits then there cannot possibly be an end for the "1" to go on. So either your string of "0" digits has and end for the "1" to go on (therefore making the number finitely long), or there isn't such a number that you speak of.