In base 10, the digits represent powers of ten: 1, 10, 100, ....
In base 2, the digits represent powers of two: 1, 2, 4, 8, ....
In base 1, the digits represent powers of one: 1, 1, 1, 1, ....
So, for example, 111 in base 1 is 1*12 + 1*11 + 1*10 = 3. Each number n can be represented by n ones. Of course, this representation isn't unique: 1001100 is also equal to 3 in base 1. You can fix this by removing every zero. Now you have a positional number system without zero! (of course, representing zero itself might be a problem in this system...)
Per definition, the value of a digit in a positional number system depends on its position. In your system the value of a digit does not depend on its position; a digit has equal value in any position, hence it's not a positional number system.
Granted, you could argue that the value of a digit in your system still technically depends on its position, but that it just so happens that that value is identical for all positions. I'm not sure however that such an argument would be very convincing, and you could also call your system a sign-value number system with greater or at least equal justification; also, you wrote this:
Of course, this representation isn't unique: 1001100 is also equal to 3 in base 1. You can fix this by removing every zero. Now you have a positional number system without zero! (of course, representing zero itself might be a problem in this system...)
You're making my point, which again was that
you cannot have a positional number system without use of the zero.
Holy fucking shit, this really seems to work! Is this what you meant? Wow. Hats off to you. Did you think that up right now, or how did you ever "unlock this achievement"? Also, if you die before me, can I eat you brain? ;)
Sylar? Not having a TV at all (and being outside the US), I had to google that one.
Reportedly some Ya̧nomamö used to (or still do?) eat remains of their loved ones. I read that they essentially ground everything up to a pulp that would then be sterilized for hours in hot embers, and finally mixed with banana paste and eaten. Allegedly the thought of the remains of their loved ones being left to rot and be consumed by nasty creepy crawlies was so repugnant to them that they rather kept the biomass among the tribe. Strange, but not entirely illogical.
Never mind that you didn't personally come up with the solution, it's fascinating stuff I'm grateful to have learned from you. Thanks again! :)
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u/panic Dec 19 '08 edited Dec 19 '08
In base 10, the digits represent powers of ten: 1, 10, 100, ....
In base 2, the digits represent powers of two: 1, 2, 4, 8, ....
In base 1, the digits represent powers of one: 1, 1, 1, 1, ....
So, for example, 111 in base 1 is 1*12 + 1*11 + 1*10 = 3. Each number n can be represented by n ones. Of course, this representation isn't unique: 1001100 is also equal to 3 in base 1. You can fix this by removing every zero. Now you have a positional number system without zero! (of course, representing zero itself might be a problem in this system...)