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Let \(x = \{x_{w-1}, x_{w-2}, \dots, x_{0}\}\) be a \(w\)-bit computer word.
So: 0b10010110 would represent 128 + 16 + 4 + 2 or 150
if it was unsigned.
The signed integer (two’s complement) would be:
\[x = (\displaystyle \sum_{k=1}^{w-2}x_{k}2^k) - x_{w-1}2^{w-1}\]
So: 0b10010110 would be -128 + 16 + 4 + 2 or -106.
Thus, 0b00000000 would be 0.
And 0b11111111 would be -1.
Thus, x + ~x = -1, or -x = ~x + 1.
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