Bit Hacks

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Binary Representation

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:

$x=(\sum _{k=1}^{w-2}{x}_k{2}^k)-x_{w-1}{2}^{w-1}$

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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