1's and 2's complements of binary numbers

[PG1995]

Hi

Most computers (I'm not sure if all) do most of the arithmetic using addition operation - i.e. subtraction, multiplication, and division all use addition operation. This makes circuit design simple and with only one circuit you get to do different things.

What I guess 1's and 2's complements let us do all operations, subtraction, division, and multiplication using addition method.

complement (noun)
2 c : a number that when added to another number of the same sign yields zero if the significant digit farthest to the left is discarded —used especially in assembly language programming
[M-W's Col. Dic.]

In base 10 the compliment of 3 would be 7 because 3+7 = 10 and after discarding the 1, we are left with 0

Thus 4 is the compliment of 6, and 8 of 2, etc.

The compliment of 55 would be 45, because 55+45=100 and removing the 1 leaves you with 0 again.

With binary, the compliment of 1 is 1 because 1+1 =10

The compliment of 1010 would be 0110 (which is 2's complement of 1010) because 1010 + 0110 = 10000 in binary.

1: Let's say we have a binary number 110 (6). It's 1's complement would be: 001 (1), and 2's complement would be: 010 (2). I don't see what 1's and 2's complements of the number tells us. Please help me with it.

2: Perhaps, using a particular example could help us a bit. We have binary number 1010 (10) and we want to subtract 110 (6) from it - i.e. 1010 - 110. In base 10 arithmetic it would be 10 - 6 = 4. So, how do complements help us here?

1's complement of 110 = 001 (1)
2's complement of 110 = 010 (2)

I do have more related queries on this topic but I would ask them after clearing this up. Thanks.

Scanned pages which can be useful:
1: **broken link removed**
2: https://img836.imageshack.us/img836/2465/floyddigitalfundamental.jpg

 

[john blue]

This should give you some idea:


Online CS Modules: Subtraction with One's Complement

 

[enjunear]

The "compliment" comes from the usage of splitting the binary range to give you a complimenting pair of positive and negative values around zero. The MSB being a "1" indicates a negative value, when the binary value is viewed as a signed number.

Binary  Unsigned
111      7
110      6
101      5
100      4
011      3
010      2
001      1
000      0

Binary  Signed
011      3
010      2
001      1
000      0
111     -1
110     -2
101     -3
100     -4

Wikipedia has a BIG write-up on the usage and mathematical rationale behind two's compliment (and one's compliment, since it's buried in the two's compliment process).
Two's complement - Wikipedia, the free encyclopedia