Why 2's complement binary subtraction is preffered over 1's complement?

Answer 1

1

Answer 2

That is just how 2's complement works. Put another way, you are asking:

Why, in two's complement representation, is the arithmetic negation of a number represented by flipping all the bits in the number and adding one?

One way to think about the answer, is that otherwise, there would be two different representations of 0; one for 0 (all 0 bits) and one for -0 (all 1 bits). In order that there will be only one representation of 0, we add one to the all-1-bits representation of -0, and because that 1 overflows the result, we end up with all 0 bits again.

Another advantage of this representation is that when you add a number and its binary negation, you get all 1 bits. If you add one to that, you end up with 0, which means that x + (-x) = 0.

Answer 3

They are different things, don't interchange them.

Example in C:

int x= 19;

printf ("1's complement of %d (%xh) is %d (%xH)\n", x, x, ~x, ~x);

printf ("2's complement of %d (%xh) is %d (%xH)\n", x, x, -x, -x);

result:

1's complement of 19 (13h) is -20 (ffffffecH)

2's complement of 19 (13h) is -19 (ffffffedH)

Answer 4

One's and two's complement are different ways of representing integer values. The one's complement of any value simply inverts all the bits in the value. The two's complement does the same but also adds one to the inverted value.

One's complement was originally used to negate a signed value, however its usage for this purpose is not widespread. To understand why, take the decimal value 7 which is represented by 00000111 in binary. The one's complement is therefore 11111000. Although this physically represents decimal 248, in signed notation the most significant bit denotes the sign. Thus 11111000 signifies -7 in one's complement; in other words the complete inversion of +7. Since the most significant bit denotes the sign, the range of signed 8-bit values is -127 to +127. However, the one's complement of 00000000 is 11111111 which is erroneous; the value zero is neither positive nor negative and yet one's complement treats +0 and -0 as begin two separate values.

Two's complement resolves the problem by adding one to the one's complement. Taking the example 7 once more, 00000111 becomes 11111000 + 1, which is 11111001 or -7 in two's complement notation. To invert the sign again, 11111001 becomes 00000110 + 1 which is 00000111, the original value. More importantly, the two's complement of 00000000 is 11111111 + 1 which is 00000000. Thus we replace the two +/- zero values with a single unsigned zero value, and the range of possible signed values increases by one: from -128 to +127.

Although it may seem more logical to simply switch the most-significant bit alone in order to switch the sign, this is no better than one's complement because 10000000 would represent -0. By completely inverting and adding one we actually simplify the logic of signed values by counting upwards from 10000000 (-128) to 11111111 (-1), followed by 00000000 (zero), followed by 00000001 (+1) to 01111111 (+127).