Why is the carry of any n base system equal to the radix in arithmetic?

Firstly, in binary the carry is 2, in hex the carry is 16. What’s the reason for this?

3 4 A F 1
2 F B C 3

For the second question here’s an example in an attempt to formulate the question. In subtracting the above hex numbers:, The first operation will be to borrow 16 from F onto 1, which will result in (16+1)-3, which is E=14.The F which lent us 16 becomes E, which makes sense because it is in the 16^1 position. So far so good, but when we reach the third operation where we have to subtract B from A, we have to borrow 16 from 4, but 4 is in the 16^3 position, so decrementing 4 to 3, not only removes 16 from the whole number being subtracted, but it removes 1*16^3. Obviously, there is something wrong with conceptualizing it this way. So what is the right way to conceptualize it, what is going on when a 16 is borrowed from the nth position?