In the first answer on a circuit uniformity question here: Definition of uniform boolean circuit

I have 2 questions:

1. Kaveh mentions that the input is in unary encoding. In the definition it says the input is $1^n$, afaik $1^n$ is a sequence of 1's repeated n times, but unary encoding is a sequence of 1's and ends with 0 there is no power of 1 that gives 0... So how is $1^n$ a unary encoding?
2. Why do we use a unary encoding and not the binary encoding of n? what happens if we use binary instead?

Thanks in advance

I have two questions on Kaveh's answer to Definition of uniform boolean circuit  :

1. Kaveh mentions that the input is in unary encoding. In the definition it says the input is $1^n$, afaik $1^n$ is a sequence of 1's repeated n times, but unary encoding is a sequence of 1's and ends with 0 there is no power of 1 that gives 0... So how is $1^n$ a unary encoding?
2. Why do we use a unary encoding and not the binary encoding of n? what happens if we use binary instead?