Two's complement ($\mathbb{Z}/2^w\mathbb{Z}$)

For a fixed word width $w$, signed integers have the same bit patterns as residues modulo $2^w$. The leading bit has negative positional weight in the signed interpretation. Reading the same bits as unsigned gives $2^w+x$ for a negative signed integer $x$.

// Destination: computer-science.bigb

To widen an $m$-bit two's complement integer to $w>m$ bits, preserve the original low bits and fill the new high positions with the old sign bit $s$. The change in signed positional weights is $-s{2}^{w-1}+s{\sum }_{i=m-1}^{w-2}{2}^i=-s{2}^{m-1}$, so the represented signed value is unchanged. Zero extension instead preserves the unsigned interpretation, and changes the signed value of negative inputs.