What is “2'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.

Hence, I wanted to start this community wiki post to define what Two's Complement is, how to use it and how it can affect numbers during operations like casts (from signed to unsigned and vice versa), bit-wise operations and bit-shift operations.

What I'm hoping for is a clear and concise definition that is easily understood by a programmer.

Solution 1:

Two's complement is a clever way of storing integers so that common math problems are very simple to implement.

To understand, you have to think of the numbers in binary.

It basically says,

for zero, use all 0's.
for positive integers, start counting up, with a maximum of 2^{(number of bits - 1)}-1.
for negative integers, do exactly the same thing, but switch the role of 0's and 1's and count down (so instead of starting with 0000, start with 1111 - that's the "complement" part).

Let's try it with a mini-byte of 4 bits (we'll call it a nibble - 1/2 a byte).

0000 - zero
0001 - one
0010 - two
0011 - three
0100 to 0111 - four to seven

That's as far as we can go in positives. 2³-1 = 7.

For negatives:

1111 - negative one
1110 - negative two
1101 - negative three
1100 to 1000 - negative four to negative eight

Note that you get one extra value for negatives (1000 = -8) that you don't for positives. This is because 0000 is used for zero. This can be considered as Number Line of computers.

Distinguishing between positive and negative numbers

Doing this, the first bit gets the role of the "sign" bit, as it can be used to distinguish between nonnegative and negative decimal values. If the most significant bit is 1, then the binary can be said to be negative, where as if the most significant bit (the leftmost) is 0, you can say the decimal value is nonnegative.

"Sign-magnitude" negative numbers just have the sign bit flipped of their positive counterparts, but this approach has to deal with interpreting 1000 (one 1 followed by all 0s) as "negative zero" which is confusing.

"Ones' complement" negative numbers are just the bit-complement of their positive counterparts, which also leads to a confusing "negative zero" with 1111 (all ones).

You will likely not have to deal with Ones' Complement or Sign-Magnitude integer representations unless you are working very close to the hardware.

Solution 2:

I wonder if it could be explained any better than the Wikipedia article.

The basic problem that you are trying to solve with two's complement representation is the problem of storing negative integers.

First, consider an unsigned integer stored in 4 bits. You can have the following

These are unsigned because there is no indication of whether they are negative or positive.

Sign Magnitude and Excess Notation

To store negative numbers you can try a number of things. First, you can use sign magnitude notation which assigns the first bit as a sign bit to represent +/- and the remaining bits to represent the magnitude. So using 4 bits again and assuming that 1 means - and 0 means + then you have

0000 = +0
0001 = +1
0010 = +2
...
1000 = -0
1001 = -1
1111 = -7

So, you see the problem there? We have positive and negative 0. The bigger problem is adding and subtracting binary numbers. The circuits to add and subtract using sign magnitude will be very complex.

What is

0010
1001 +
----

Another system is excess notation. You can store negative numbers, you get rid of the two zeros problem but addition and subtraction remains difficult.

So along comes two's complement. Now you can store positive and negative integers and perform arithmetic with relative ease. There are a number of methods to convert a number into two's complement. Here's one.

Convert Decimal to Two's Complement

Convert the number to binary (ignore the sign for now) e.g. 5 is 0101 and -5 is 0101
If the number is a positive number then you are done. e.g. 5 is 0101 in binary using two's complement notation.
If the number is negative then

3.1 find the complement (invert 0's and 1's) e.g. -5 is 0101 so finding the complement is 1010

3.2 Add 1 to the complement 1010 + 1 = 1011. Therefore, -5 in two's complement is 1011.

So, what if you wanted to do 2 + (-3) in binary? 2 + (-3) is -1. What would you have to do if you were using sign magnitude to add these numbers? 0010 + 1101 = ?

Using two's complement consider how easy it would be.

 2  =  0010
 -3 =  1101 +
 -------------
 -1 =  1111

Converting Two's Complement to Decimal

Converting 1111 to decimal:

The number starts with 1, so it's negative, so we find the complement of 1111, which is 0000.
Add 1 to 0000, and we obtain 0001.
Convert 0001 to decimal, which is 1.
Apply the sign = -1.

Tada!