if I convert the binary number 000000 into 2's compliment I will get 1's compliment (invert) = 111111 2's compliment (add +1) = here I run into a problem, does this return 000000 and the 1 gets discarded or does this return 1000000?
Thanks in advance!
It discards the 1. The condition is similar to the Arithmetic Overflow.
Strength of two's complement is that it helps us retain a binary representation when signed numbers are to be considered, because in mathematics, the value of 0 is same as that of -0. If we have to give up one entire bit just for sign, then, in a 4-bit word, 0000 would denote 0 and 1000 would denote -0, wasting one representation. The two's complement helps get rid of this. If we assume 4-bit words:
val -val bits of val two's complement bits of -val (1's complement + 1)
0 0 0000 0000 (1111+0001)
1 -1 0001 1111 (1110+0001)
2 -2 0010 1110 (1101+0001)
3 -3 0011 1101 (1100+0001)
...
7 -7 0111 1001 (1000+0001)
8 -8 (no rep) 1000 (0111+0001)
(note that for -8 you have one's complement of 8 in an unsigned manner, i.e.8 = 1000 and hence its one's complement is 0111).
Thus you gain a representation for -8 by making 0 and -0 have the same bit pattern, i.e. 0000. Using this, for n bits, we can represent all integer values between -2^(n-1) to 2^(n-1)-1.