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In signed 2's complement representation of 4-bit positive binary numbers are:

  • 0000 -> +0
  • 0001 -> +1
  • 0010 -> +2
    ....
  • 0111 -> +7

Negative binary numbers(from -1 to -7) are obtained by taking 2's complement of this positive binary numbers(from +1 to +7). But there is -8 in this system. Where did -8 come from? Because there is no +8 in 4-bit system so that we can take 2's complement of it to result this -8.

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This is why:

0000 -> 0111 = 0 -> 7

1111 -> 1000 = -1 -> -8

both series can represent 8 numbers, the positive series already include the 0 so the negative serie doesn't have to do this.

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