What you've done is define your own number system, I do not believe you are representing what "base 1" might be.
If a base means, you can write numbers of this form:
abc
Where the value is:
aB^2 + bB^1 + cB^0
and you can only have B symbols, then your one symbol for base 1 would need to be 1, still useless, but not completely:
1 = 11^0 = 1
11 = 11^1 + 11^0 = 2
...
Note that you cannot represent 0 in base 1 with this method.
If you choose 0 as your one symbol for base 1, then the only number you can represent is 0. I assert this is even more useless than selecting 1 as the symbol.
0 = 01^0 = 0
00 = 01^1 + 01^0 = 0
...
As far as I can tell, the example was trying to interpret "10" as base 1. 10 has two different symbols which truly does not make sense in base 1.
I assert that base 1 is not a valid base, because you cannot represent all integers in it.
Additionally, the "unary point" or whatever it would be called, would serve no purpose:
1.1 = 11^0 + 1*1^-1 = 2
I'm not sure how that might disqualify something for a "base" but it certainly doesn't help :) Probably the strongest argument is the inability to represent 0.
Of course, this is not a practical notation, since 0 would hard to distinguish from actual absence, but it would be viable in some situations: a table of numbers say, where you know that no cells are empty. So theoretically, it is possible to represent 0 in base 1, regardless of its obvious impracticalities.
If a base means, you can write numbers of this form:
abc
Where the value is:
aB^2 + bB^1 + cB^0
and you can only have B symbols, then your one symbol for base 1 would need to be 1, still useless, but not completely:
1 = 11^0 = 1 11 = 11^1 + 11^0 = 2 ...
Note that you cannot represent 0 in base 1 with this method.
If you choose 0 as your one symbol for base 1, then the only number you can represent is 0. I assert this is even more useless than selecting 1 as the symbol.
0 = 01^0 = 0 00 = 01^1 + 01^0 = 0 ...
As far as I can tell, the example was trying to interpret "10" as base 1. 10 has two different symbols which truly does not make sense in base 1.
I assert that base 1 is not a valid base, because you cannot represent all integers in it.
Additionally, the "unary point" or whatever it would be called, would serve no purpose:
1.1 = 11^0 + 1*1^-1 = 2
I'm not sure how that might disqualify something for a "base" but it certainly doesn't help :) Probably the strongest argument is the inability to represent 0.