About to read computer science, I have just stumbled accross the concept of "Two's complement". I understand how to apply the "algorithm" to calculate these on paper, but I have not yet obtained an understanding of why it works. I think this site: https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html provides an explanaition why "flipping the digits" and adding one produces the compliment. What I do not understand is why adding the complement is equivalent to substracting the original number. Could somebody please give an explanation (maybe with a decimal example of the same concept as well?)?
Many thanks!
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I'll stick to 8-bit quantities, but the same applies in general.
The key to understanding two's complement is to note that we have a set of finitely many (in particular, 28=256.
Intuitively, arithmetic modulo 12.
For example, if it is 2:00, since
and similarly, if it is 9 since
Notice that subtracting 12−4=8 hours. In particular, we could have computed the above as follows:
That is: when performing arithmetic modulo n−x.
Now, let's apply this idea modulo 00000011 is the same as adding
and there's your two's complement.
Notably, we don't think of 128.
Note: an interesting infinite analog to the two's complement system of subtraction is that of infinite series 2-adic numbers. In particular, we can say something strange like