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What is "two's complement"?

I'm in a computer systems course and have been struggling, in part, with two's complement . I want to understand it, but everything I've read hasn't brought the picture together for me. I've read the Wikipedia article and various other articles, including my text book.

What is two's complement, how can we use it and how can it affect numbers during operations like casts (from signed to unsigned and vice versa), bit-wise operations and bit-shift operations?

Answer*

I read a fantastic explanation [on Reddit][1] by jng, using the odometer as an analogy. 

[![enter image description here][2]][2]

> It is a useful convention. The same circuits and logic operations that
> add / subtract positive numbers in binary still work on both positive
> and negative numbers if using the convention, that's why it's so
> useful and omnipresent.
> 
> Imagine the odometer of a car, it rolls around at (say) 99999. If you
> increment 00000 you get 00001. If you decrement 00000, you get 99999
> (due to the roll-around). If you add one back to 99999 it goes back to
> 00000. So it's useful to decide that 99999 represents -1. Likewise, it is very useful to decide that 99998 represents -2, and so on. You have
> to stop somewhere, and also by convention, the top half of the numbers
> are deemed to be negative (50000-99999), and the bottom half positive
> just stand for themselves (00000-49999). As a result, the top digit
> being 5-9 means the represented number is negative, and it being 0-4
> means the represented is positive - exactly the same as the top bit
> representing sign in a two's complement binary number.
> 
> Understanding this was hard for me too. Once I got it and went back to
> re-read the books articles and explanations (there was no internet
> back then), it turned out a lot of those describing it didn't really
> understand it. I did write a book teaching assembly language after
> that (which did sell quite well for 10 years).


 [1]: https://www.reddit.com/r/compsci/comments/26jnqu/im_having_trouble_understanding_twos_compliment/chrpu5e
 [2]: https://i.sstatic.net/SKsRX.png

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Wow, it's been a long time since I've seen a speedo with both mph and kph. Australia switched over before I'd hit 10yo and I still remember having to remind the old man (slang: father) of the basic conversions when he tried to do 100mph in a 100kph zone :-)

paxdiablo
– paxdiablo

2021年06月12日 02:44:35 +00:00
Commented Jun 12, 2021 at 2:44
In any case, I think they stopped allowing the odo to roll back at some point. Disconnecting it from the car and using a drill to roll it back was a favourite trick of (some rather dodgy) people when trying to sell their cars with a lower mileage (funny how we still use that term, guess kilometerage never caught on).

paxdiablo
– paxdiablo

2021年06月12日 02:46:57 +00:00
Commented Jun 12, 2021 at 2:46
Related: What do we do with answers that are entirely copied and improperly attributed (only a "reference" link or similar is included)?

Peter Mortensen
– Peter Mortensen

2023年01月25日 04:21:27 +00:00
Commented Jan 25, 2023 at 4:21

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What is "two's complement"?

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