Is there an easy way to figure this out? What is the lowest (most negative) number that can be represented by 7-bit two's complement? Show how to convert the number into its two's complement representation.
The lowest number is -2^6. To find a negative number's inverse in 2's complement (aka its absolute value) flip the bits and add one. So (-1)*1000001 = 0111110+1 = 0111111 = 1000000 - 1 = 2^6-1. As you can see there is a number lower than 1000001 and it is one less than it: 1000000. Finding it's absolute value we get:
(-1)*(100000) = (-1)*(100001-1) = (-1)*(100001) + 1 = (2^6-1)+1 = 2^6.