• The term compliment refers to a part which together with another makes up a whole. For example in geometry two complimentary angle (90⁰).
• The idea of compliment is used to address the problem of signed numbers i.e. positive and negative.
• In decimal numbers (0 to 9), we talk of nine’s compliment. For example the nines compliment
• Of 9 is 0, that of 5 is 4 while 3 is 6.
• However, in binary numbers, the ones compliment is the bitwise NOT applied to the number. Bitwise NOT is a unary operator (operation on only one operand) that performs logical negation on each bit. For example the bitwise NOT of 1100₂ is 0011₂e.
• 0s are negated to 1s while 1s are negated to 0s.

Twos compliment

• Twos compliment, equivalent to tens compliment in decimal numbers, is the most popular way of representing negative numbers in computer systems. The advantages of using this method are:

1. There are no two ways of representing a zero as in the case with other two methods.
2. Effective addition and subtraction can be done even with numbers that are represented with a sign bit without a need for circuitries to examine the sign of an operand.

• The twos compliment of a number is obtained by getting the ones compliment then adding a 1. For example, to get the twos compliment of a decimal number 45_10,
• First convert it to its binary equivalent then find its ones compliment. Add a 1 to ones compliment i.e.

45₁₀=00101101₂

Bitwise NOT (00101101) =11010010

Two’s compliment = 11010010₂+1₂

= 11010011₂

Binary addition

The five possible additions in binary are

1. 0 + 0 = 0
2. 0+ 1₂ = 1₂
3. 1₂ + 0 = 1₂
4. 1₂ + 1₂ = 10₂ (read as 0, carry 1)
5. 1₂ + 1₂ + 1₂ = 11₂ (read as 1, carry 1)

   Example 1

   Find the sum of 111₂ + 011₂

   Solution

   Arrange the bits vertically. 111

   Working from the right to the left, we proceed as follows: + 011

   Step 1 1₂ + 1₂ = 10₂, (write down 0 and carry 1) 1010₂

   Step 2 1₂ + 1₂ + 1₂ = 11₂, (add and carry over digit to 1 + 1 in order to get 1 + 1

   +1. From the sum, write down digit one the carry

   Forward)

   Step 3 1₂ + 1₂ + 0₂ = 10₂, (add the carry over digit to 1 + 0 in order to get

   1 + 1 + 0.since this is the last step, write down 10)

   Therefore 111₂ + 011₂ = 1010₂

   This can be summarized in the table

   1st number	1	1	1
   2nd number	0	1	1
   Carry digit	-	1	1
   sum	10	1	0

   Example 2

   Add the following binary number

   10110₂

   1011₂

   + 111₂

   Solution

   Add the first two numbers and then add the sum to the third number a follows:

   Step 1 Step 2

   10110₂ 100001₂

   + 1011₂ + 111₂

   100001₂ 101000₂