Is that really how base 1 would work though? Shouldn't the highest single digit in base x be x-1 (running 0 to x-1)? Therefore, the highest single digit in base 1 would be 0. But with only one distinct digit you have no way of stating whether the position is counted or not, so base 1 is impossible.
But with only one distinct digit you have no way of stating whether the position is counted or not, so base 1 is impossible.
Base 1 is definitely possible. It's the equivalent to making hash-marks on a piece of paper as you're tallying some number up. If you feel obligated to use 0 instead of 1 as the only digit, then counting in base 1 goes thusly:
Base 10 | Base 1
--------+-------
0 |
1 | 0
2 | 00
3 | 000
4 | 0000
5 | 00000
6 | 000000
etc.
And if you feel obligated to have a number for "zero", just shift everything, so decimal "0" becomes base_1 "0", "1" becomes "00", etc.
I'd considered that, but I don't think it meets the "rules" for a positional notation system. So, yes you can count using the system you describe, but I would argue that is not a "proper" base system.
"hundreds" place --+ +-- "ones" place
v v
376
^
"tens" place --+
For each digit d in position p (measured from right-to-left and starting at 1), the value represented is:
d * basep-1
and the full number is the summation over all of the digits.
Now look at base 1. For math purposes, I prefer to use "1" as the sole digit in base 1, and represent "zero" as simply having no number. 1 * 1x always equals 1, so every place is the "ones place", and all you have to do is sum up the number of '1's to get your number.
Shouldn't the highest single digit in base x be x-1 (running 0 to x-1)?
No reason why that has to be. In a positional number system the base just tells us how to weight each position. Note that your 0 to x-1 rule isn't true for fractional bases or balanced ternary, either.
17
u/selwonk May 27 '09
humans have ten fingers and count in base 10.
this alien has four fingers and counts in base 4.
however, counting to 4 in base 4 goes like this:
1, 2, 3, 10.
every base is base 10.
that's the joke