Let’s say we are in base 3. Typically the symbols used to construct representations for numbers using the standard positional notation would be the numerals {0, 1, 2}. We could use any 3 symbols but these three work well since most people are comfortable with standard base 10 notation. When one says 0 is 0 in base 3 (or any other base) what the mean is precisely this:

To represent the number 0 in base 3 we just write: 0.

To represent the number 1 in base 3 we just write: 1.

To represent the number 3 in base 3 we must write: 10.

In general, for any positive number n the symbols used to construct representations for number will be {0, 1, 2,. . ., 9,. . ., n-1}. In base n the number 0 is represented as: 0. In base n the number 1 is represented as 1. The number n is represented as: 10.

Thus it makes sense to always use the symbol 0 and 1 as part of the set of numerals in whatever base you are in and if one does this then 0 is 0 and 1 is 1.