I don't understand why an n-bit 2C system number can be extended to an (n+1)-bit 2C system number by making bit bn = bn−1, that is, extending to (n+1) bits by replicating the sign bit.
1 Answer
This works because of the way we calculate the value of a binary integer.
Working right to left, the sum of each bit_i * 2 ^ i, where i is the range 0 to n n is the number of bits Because each subsequent 0 bit will not increase the magnitude of the sum, it is the appropriate value to pad a smaller value into a wider bit field.
For example, using the number 5:
4 bit: 0101 5 bit: 00101 6 bit: 000101 7 bit 0000101 8 bit: 00000101 The opposite is true for negative numbers in a two's compliment system. Remember you calculate two's compliment by first calculating the one's compliment and then adding 1.
Invert the value from the previous example to get -5:
4 bit: 0101 (invert)-> 1010 + 1 -> 1011 5 bit: 00101 (invert)-> 11010 + 1 -> 11011 6 bit: 000101 (invert)-> 111010 + 1 -> 111011 7 bit: 0000101 (invert)-> 1111010 + 1 -> 1111011 8 bit: 00000101 (invert)-> 11111010 + 1 -> 11111011 