How to Represent Signed Integers?

Example of things getting wrong: Adding the $6$-bit numbers $18_{10} = 010010_2$ and $23_{10} = 010111_2$:

1    11   <-- carry bits
0   10010
0 + 10111
    _____
1   01001

Look what happened: we got a negative number, which, obviously, isn't the correct sum.

The reason for this mishap is because we got a carry bit that was carried over to the sign bit, which mustn't happen: if this happens, we get an overflow situation, in which, the received result is incorrect. The $6$-bit format is simply too small to represent the sum $18 + 23$: a format with more than 6 bits is needed to avoid the overflow.