Question about natural numbers.

cragar

Are there an [itex] \aleph_0 [/itex] # of natural numbers with an
[itex] \aleph_0 [/itex] # of digits?

pwsnafu

Every natural number has a finite number of digits.

Number Nine

Adding to the above (which is correct), the set of infinite digit strings is uncountable.

cragar

ok I understand what you guys are saying but it still seems strange to me.
I feel like that is saying the natural numbers are not bounded but they have a finite number of digits. I mean you couldn't put a bound on the number of digits.

pwsnafu

Each individual natural number has a finite number of digits.
The entire set is unbounded.

Number Nine

You can't. This doesn't change the fact that every natural number has a finite number of digits.

Diffy

For any natural number you pick, I can pick one with more digits. For example, if you picked x I could pick 10x, or 100,000,000,000,000,000x.

However all three of those numbers have a finite number of digits.

As the natural numbers get larger and larger so do the number of digits.

Say you have f(x) = # of digits x has for all natural numbers.

Then it is certainly true that as x approaches infinity, so does f(x).

Millennial

To make the above a bit more rigorous, the number of digits in a natural number [itex]n[/itex] is given by [itex]\lfloor \log_{10}(n) \rfloor[/itex] and this obviously goes to infinity.

cragar

I could see the problem with saying that there are natural numbers with an
[itex] \aleph_0 [/itex] of digits because then I would have 10 choices for each number in the slot and I would have [itex] 10^{\aleph_0} [/itex] numbers which would be uncountable and a contradiction because the set of naturals is countable. Could I use this as a proof by contradiction to verify it?

jgens

No. The contradiction does not verify that every natural number has a base 10 representation with only finitely many digits.

Bacle2

If I understood you correctly, you want to compose all strings of finite length

with terms in {0,1,..,9} . If you write those strings as

Ʃ_i=0^Na_i10ⁱ

and let N→∞ , then(a) problem is that your sum will diverge much of the time, so that

many of those strings are not natural numbers.

cragar

ya thats what i am kinda saying