Advantage of 2's complement over 1's complement?

Question

What is the advantage of 2's complement over 1's complement in negative number representation in binary number system? How does it affect the range of values stored in a certain bit representation of number in binary system?

Answer 1

The primary advantage of two's complement over one's complement is that two's complement only has one value for zero. One's complement has a "positive" zero and a "negative" zero.

Next, to add numbers using one's complement you have to first do binary addition, then add in an end-around carry value.

Two's complement has only one value for zero, and doesn't require carry values.

You also asked how the range of values stored are affected. Consider an eight-bit integer value, the following are your minimum and maximum values:

Notation     Min   Max ==========  ====  ==== Unsigned:      0   255 One's Comp: -127  +127 Two's Comp: -128  +127 

References:

• http://en.wikipedia.org/wiki/Signed_number_representations
• http://en.wikipedia.org/wiki/Ones%27_complement
• http://en.wikipedia.org/wiki/Two%27s_complement

Answer 2

The major advantages are:

1. In 1's there is a -0 (11111111) and a +0 (00000000), i.e two value for the same 0. On the other hand, in 2's complement, there is only one value for 0 (00000000). This is because

   +0 --> 00000000 

   and

    -0 --> 00000000 --> 11111111 + 1 --> 00000000 

2. While doing arithmetic operations like addition or subtraction using 1's, we have to add an extra carry bit, i.e 1 to the result to get the correct answer, e.g.:

          +1(00000001)      +        -1(11111110)      -----------------      = (11111111) 

but the correct answer is 0. In order to get 0 we have to add a carry bit 1 to the result (11111111 + 1 = 00000000).

In 2's complement, the result doesn't have to be modified:

               +1(00000001)               +                -1(11111111)          -----------------               = 1 00000000