What is “2's Complement”?

Question

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 p

Answer 1

Like most explanations I've seen, the ones above are clear about how to work with 2's complement, but don't really explain what they are mathematically. I'll try to do that, for integers at least, and I'll cover some background that's probably familiar first.

Recall how it works for decimal:
  2345
is a way of writing
  2 × 10³ + 3 × 10² + 4 × 10¹ + 5 × 10⁰.

In the same way, binary is a way of writing numbers using just 0 and 1 following the same general idea, but replacing those 10s above with 2s. Then in binary,
  1111
is a way of writing
  1 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰
and if you work it out, that turns out to equal 15 (base 10). That's because it is
  8+4+2+1 = 15.

This is all well and good for positive numbers. It even works for negative numbers if you're willing to just stick a minus sign in front of them, as humans do with decimal numbers. That can even be done in computers, sort of, but I haven't seen such a computer since the early 1970's. I'll leave the reasons for a different discussion.

For computers it turns out to be more efficient to use a complement representation for negative numbers. And here's something that is often overlooked. Complement notations involve some kind of reversal of the digits of the number, even the implied zeroes that come before a normal positive number. That's awkward, because the question arises: all of them? That could be an infinite number of digits to be considered.

Fortunately, computers don't represent infinities. Numbers are constrained to a particular length (or width, if you prefer). So let's return to positive binary numbers, but with a particular size. I'll use 8 digits ("bits") for these examples. So our binary number would really be
  00001111
or
  0 × 2⁷ + 0 × 2⁶ + 0 × 2⁵ + 0 × 2⁴ + 1 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰

To form the 2's complement negative, we first complement all the (binary) digits to form
  11110000
and add 1 to form
  11110001
but how are we to understand that to mean -15?

The answer is that we change the meaning of the high-order bit (the leftmost one). This bit will be a 1 for all negative numbers. The change will be to change the sign of its contribution to the value of the number it appears in. So now our 11110001 is understood to represent
  -1 × 2⁷ + 1 × 2⁶ + 1 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰
Notice that "-" in front of that expression? It means that the sign bit carries the weight -2⁷, that is -128 (base 10). All the other positions retain the same weight they had in unsigned binary numbers.

Working out our -15, it is
  -128 + 64 + 32 + 16 + 1
Try it on your calculator. it's -15.

Of the three main ways that I've seen negative numbers represented in computers, 2's complement wins hands down for convenience in general use. It has an oddity, though. Since it's binary, there have to be an even number of possible bit combinations. Each positive number can be paired with its negative, but there's only one zero. Negating a zero gets you zero. So there's one more combination, the number with 1 in the sign bit and 0 everywhere else. The corresponding positive number would not fit in the number of bits being used.

What's even more odd about this number is that if you try to form its positive by complementing and adding one, you get the same negative number back. It seems natural that zero would do this, but this is unexpected and not at all the behavior we're used to because computers aside, we generally think of an unlimited supply of digits, not this fixed-length arithmetic.

This is like the tip of an iceberg of oddities. There's more lying in wait below the surface, but that's enough for this discussion. You could probably find more if you research "overflow" for fixed-point arithmetic. If you really want to get into it, you might also research "modular arithmetic".