Signed / Unsigned

Signed binary is binary that can be positive or negative, so it can be two’s complement for a negative number or anything else for a positive number. The values of these fall in the range $ -2 ^ {n - 1} \leq x \leq 2 ^ {n - 1} $ where $ n $ is the number of bits.

Unsigned binary is binary that can only be positive, so it has double the range of signed binary. The values of these fall in the range $ 0 \leq x \leq 2 ^ n - 1 $ where $ n $ is the number of bits.

Negation

To get the negative of a binary number, invert the bits and add one. For example:

* 0101 0001 ( 81)
~ 0101 0001 =
  1010 1110 (-82)

  1010 1110 (-82)
- 0000 0001 (1)
  ---------
  1010 1111 (-81)

Addition

Addition is done the same as decimal just with 2 digit states instead of 10. For example:

  0111 0101 (117)
- 1000 0001 (129)
  ---------
          0 (carry the one)
         1
        1
       0
     1
    1
   1
  1
  ---------
= 1111 0110 (246)

Subtraction

Subtraction is just addition of the negative of the second number. For example:

  1000 0001 ( 129)
* 0111 0101 ( 117)

* 0111 0101 ( 117)
= 1000 1011 (-117)

  1000 0001 ( 129)
- 1000 1011 (-117)
  ---------
= 0000 1100 (  12)

Shifting

Shifting is simply moving the bits a certain amount to the left or right. Note that:

• Shifting an integer left by $ n $ is the same as multiplying it by $ 2 ^ n $.
• Shifting an integer right by $ n $ is the same as dividing by $ 2 ^ n $.

For example:

  0010 1010 (  42)
<< 2 (shifted left by 2 bits)
= 1010 1000 ( 168)

  1010 0100 ( 164)
>> 2 (shifted right by 2 bits)
= 0010 1001 (  41)