Two’s Complement of a Signed Binary Number

Posted on | In

What

Used for representing signed binary numbers. Invert bits then add 1. For example, 12 in binary representation:

1
0000 0000 0000 0000 0000 0000 0000 1100

Invert the digits.

1
1111 1111 1111 1111 1111 1111 1111 0011

And add one.

1
1111 1111 1111 1111 1111 1111 1111 0100

Compared with One’s Complement

For One’s Complement, to get the negative counterpart, we only need to invert the digits. Compared to this method, two’s complement has the following pros:

  • In One’s Complement 0 has two representations(+0 and -0). For example in two bits, 00->+0, 01->+1, 10->-1, 11->-0. With n bits it can represent 2^n-1 numbers, with 0 being representated twice. In Two’s Complement, 00->+0, 01->+1, 10->-2, 11->-1. 0 is only representated once, and n bits can represent 2^n numbers. This is because for 0(taking 2 bits as an example): 00 invert digits-> 11 add one-> 100 ignore the overflow and just take 2 bits -> 00.
  • When doing arithmetic operations, One’s Complement requires an “add 1” at the end when dealing with negative numbers. For example, 2-1=1 in One’s Complement: 10+10=100 taking 2 bits->00, and we need to “add 1” so that 00+01=01. But for Two’s Complement, since there is only one representation of 0, we can skip the “add 1” procedure. In this example, 10+11=101 taking 2 bits->01 is exactly the answer we want.

Why Inversion and Adding One Works

Take number “75”(1001011) as an example.

To get the negative counterpart, we subtract 75 from 0 and get:

1
2
3
4
  00000000
- 01001011
-----------
  10110101

Which is the same as taking the last 8 bits in:

1
2
3
4
  100000000
- 001001011
------------
  010110101

Since 100000000 = 11111111 + 1:

1
2
3
4
5
         1
+ 11111111
- 01001011
-----------
  10110101

Note that in binary, when we subtract a number A from a number of all 1 bits, what we’re doing is inverting(~) the bits of A. Thus this operation is equivalent to inverting the number and then add 1.

References