The Sum

The One's-Complement Sum

One's-complement arithmetic keeps overflow by folding it back into the low bits. The table shows the raw sum and the folded sum side by side.

highlighted = computed this step

Why the carry wraps

One's-complement addition folds overflow back into the low bits. The table exposes the raw sum before that end-around carry is applied.

raw=0x00024e17\text{raw}=0x00024e17

Add the words

With the checksum word zeroed, the raw sum is 0x00024e17. This is wider than a 16-bit checksum field.

raw=0x00024e17\text{raw}=0x00024e17
One's-complement sumThe checksum word is zeroed and the displayed sums are recomputed.One's-complement sumcompute zero word 5rowindexvalueword00x4500word10x003cword20x1c46word30x4000word40x4006word50x0000 (zeroed)word60xac10word70x0a63word80xac10word90x0a0craw sum0x00024e17folded0x4e19result0xb1e6

End-around carry fold

The high carry 0x02 is added back to the low bits, producing 0x4e19.

0x02+0x4e17=0x4e190x02+0x4e17=0x4e19
One's-complement sumThe checksum word is zeroed and the displayed sums are recomputed.One's-complement sumcompute zero word 5rowindexvalueword00x4500word10x003cword20x1c46word30x4000word40x4006word50x0000 (zeroed)word60xac10word70x0a63word80xac10word90x0a0craw sum0x00024e17folded0x4e19result0xb1e6

Summary

The one's-complement sum is not ordinary truncation; overflow is folded back into the result. Error detection over the exact bytes; timing/throughput is not modeled here.

folded=0x4e19\text{folded}=0x4e19
One's-complement sumThe checksum word is zeroed and the displayed sums are recomputed.One's-complement sumcompute zero word 5rowindexvalueword00x4500word10x003cword20x1c46word30x4000word40x4006word50x0000 (zeroed)word60xac10word70x0a63word80xac10word90x0a0craw sum0x00024e17folded0x4e19result0xb1e6