# Nine

wolfgang59
Posers and Puzzles 23 Jun '15 02:36
1. wolfgang59
23 Jun '15 02:36
What percentage of integers contain the digit 9 (at least once)?
2. forkedknight
Defend the Universe
23 Jun '15 16:03
Assuming you don't use leading zeroes, the chances that a given integer has a '9' in it is:

1 - (8/9)(9/10)^n

Where 'n' is the length of the integer in digits.

I guess you might take the limit of that as 'n' approaches infinity? But that would make it very close to "all of them" I think
3. forkedknight
Defend the Universe
23 Jun '15 21:24

https://www.wolframalpha.com/input/?i=summation+from+n%3D1+to+infinity+of+%28%281-%288%2F9%29%289%2F10%29%5En%29%2F%2810%5En%29%29
4. wolfgang59
23 Jun '15 22:15
Originally posted by forkedknight
Assuming you don't use leading zeroes, the chances that a given integer has a '9' in it is:

1 - (8/9)(9/10)^n

Where 'n' is the length of the integer in digits.

I guess you might take the limit of that as 'n' approaches infinity? But that would make it very close to "all of them" I think
Yes.
100% of the integers have at least one occurrence of the digit "9".

ðŸ˜€
5. AThousandYoung
All My Soldiers...
29 Jun '15 00:51
Originally posted by forkedknight
Assuming you don't use leading zeroes, the chances that a given integer has a '9' in it is:

1 - (8/9)(9/10)^n

Where 'n' is the length of the integer in digits.

I guess you might take the limit of that as 'n' approaches infinity? But that would make it very close to "all of them" I think
If N = 1

1 - (8/9)(9/10) = 1 - (8/10) = 1/5

One fifth of all the one digit integers is not 9. Fail formula.
6. forkedknight
Defend the Universe
02 Jul '15 20:371 edit
Originally posted by AThousandYoung
If N = 1

1 - (8/9)(9/10) = 1 - (8/10) = 1/5

One fifth of all the one digit integers is not 9. Fail formula.
Sorry, I guess it's an off-by-one error:

p = 1 - (8/9)(9/10)^(n-1)
7. 30 Jul '15 11:03
Originally posted by wolfgang59
Yes.
100% of the integers have at least one occurrence of the digit "9".

ðŸ˜€
Do you say that all integers has at least one nine? Then I can give you a counterexample - 8.
8. forkedknight
Defend the Universe
30 Jul '15 19:452 edits
All, except for an infinite number of them
9. 30 Jul '15 21:03
But - and this is not a joke:

Among all integers there are exactly as many numbers with a nine in them as they are numbers without a nine in them. No approximately but exactly! Not as n reach infinity, but when n *is* infinity!

This is provable. Cantor showed us how.
10. wolfgang59
05 Aug '15 02:26
Originally posted by FabianFnas
But - and this is not a joke:

Among all integers there are exactly as many numbers with a nine in them as they are numbers without a nine in them. No approximately but exactly! Not as n reach infinity, but when n *is* infinity!

This is provable. Cantor showed us how.
Cantor cant count.
11. forkedknight
Defend the Universe
05 Aug '15 18:19
Originally posted by FabianFnas
But - and this is not a joke:

Among all integers there are exactly as many numbers with a nine in them as they are numbers without a nine in them. No approximately but exactly! Not as n reach infinity, but when n *is* infinity!

This is provable. Cantor showed us how.
Well the the answer should be 50%, and I don't think that's true...
12. 05 Aug '15 19:37
Originally posted by forkedknight
Well the the answer should be 50%, and I don't think that's true...
The number of all integers is infinitely many.
How much is 50% of infinity?
13. wolfgang59