On average, how many more digits do octal numbers have than decimal numbers?

Asked Feb 14 '24 at 04:36

Active Feb 14 '24 at 04:36

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My question is, how could I calculate the expected value of the number of digits of an octal representation of an $n$-digit decimal number?

How can this question be answered in general for two different bases? E.g., how many more digits would base-9 numbers have than base-12 numbers on average?

asked Feb 14 '24 at 04:36

To clarify, are you counting the set of $2$-digit decimal numbers as $10, 11, \ldots, 98, 99$, or are you including $0, 1, \ldots, 9$? – Brian Moehring Feb 14 '24 at 04:45
3

$\log_b b'$---the proof is essentially the same as in https://math.stackexchange.com/a/936445/155629 – Travis Willse Feb 14 '24 at 05:05
Yeah, the discussion Travis Willse linked is the answer to this question. – Jonathan Beer Feb 14 '24 at 05:39
@BrianMoehring all natural numbers – Feb 14 '24 at 07:02
@TravisWillse thank you, I couldn't find that – Feb 14 '24 at 07:05

On average, how many more digits do octal numbers have than decimal numbers?

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