> Date: Sun, 17 Nov 2002 18:51:05 -0800
> From: Dave Storrs <[EMAIL PROTECTED]>
>
> Therefore, in base 1, you can only use the digit 0. (Actually, I
> think base 1 is a corner case--you only get one digit, but that digit
> is 1, so you can represent any number N by making N tally marks.)
Well, if you want to be ≪useful≫, yes. But to be consistent, take how
you compute the magnitude of base n:
----
\ i
/ d n
---- i
i
Where d_i ∈ [0, n) (or any set of n symbols representing those
magnitudes). Thus, our digits in base 1 are d_i ∈ [0, 1), or just 0.
So, any number represented in base 1 is 0.
OTOH, nothing can be represented in base 0, as there are no valid
symbols with which to work. If there were, all numbers would still be
0, because n^i is always zero.
But for usefulness sake, you can define 0 and 1 as digits, in which
case the number of 1's is the number represented. But that's no
fun...
As for semantics, base zero is of course an error, and I'd say base
one should be as well. Just because I prefer consistency over
almost-useless exceptions.
Luke
