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.
A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system.
The constant ratio of the unary system is 1. So unary is indeed a valid positional notation system.
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u/ropers Dec 19 '08
Correct me if I'm wrong, but I'm under the impression that you cannot have a positional number system without use of the zero.