[thaumasiotes]

Unary representation differs from better bases in one very important regard: a base-n representation of a number uses n symbols, and you write the number as the coefficients of a power series in n. The n symbols you use are the natural numbers m such that 0 <= m < n.

But in a unary representation, the only number you could represent that way is 0. Instead, a unary system uses as coefficients the natural numbers m such that 0 < m <= 1. An immediate consequence is that there's no way to represent zero.

(In "unary-style" binary, you'd count like this: 1, 2, 11, 12, 21, 22, 111, 112... . The constraint that none of your coefficients can be zero (because it doesn't exist) makes the representation unique. If you're willing to use infinite representations along with finite ones, you could then represent zero in two's complement as ...111112, that is, one more than -1.)

[mturmon]

Yes, thanks for the qualification regarding zero...the extension to the "base-1 case" is not as simple as I made it sound. Point taken.