It can actually represent all integers just fine. It works like any other base, except whereas for example in base 10, positions are multiplied by 10^0 = 1, 10^1 = 10, 10^2 = 100 and so on, in base 1 they are multiplied by 1^0 = 1, 1^1 = 1, 1^2 = 1, and so on. The result is that each digit adds 1 to the number, like tally marks. So we have, for example

111 = 3

11 = 2

1 = 1

= 0

-1 = -1

-11 = -2

-111 = -3

That is wierd. Base k includes numbers 0,...,k-1. For example 101 is a valid number in base 2 which is 2^2 + 2^1.

Using that logic, base 1 would only contain 0, hence it should not be able to expresss more values that 0.

Is your example actually a valid base 1?