9. Binary arithmetic operations - Addition

Representation of signed binary numbers

There are three ways of representing a signed binary number (positive or negative).

1. Prefixing an extra sign bit to a binary number
2. Using ones compliment
3. Using two's compliment

Prefixing an extra sign bit to a binary number

In decimal numbers a sign bit for example: can be represented as ‘-1’ or ‘+1’ for negative and positive respectively. However in binary numbers, “1” and “0” are added as an extra bit on the left of a binary number to denote negative and positive numbers respectively. For example: ASCII is a 7-bit coding scheme, a number like 1₁₀ can be represented as 0000001₂ to show whether it’s a negative we add a “1” to the left of the binary digit thus: (1)0000001₂ and to show whether it’s a positive a “0” is added to the leftmost digit thus: (0)0000001_2­. The problem of using this method is that 0₁₀ can be represented in two ways thus: (0)0000000_2­ and (0)0000000_2­

Ones Compliment

The term complement refers to a part which together with another makes up a whole. For example: in geometry, two complimentary angles add up to one right angle (90⁰). Compliments are used to address the problem of signed numbers for instance; positive and negative numbers.

In decimal numbers (0-9), we talk of nine’s compliment. E.g. the nines compliment of 9 is 0, and that of 5 is 4 while that of 8 is 1. However, in binary numbers, the ones complement is the bitwise NOT applied to the number, Bitwise NOT is a unary operator (operates on only one operand) and performs logical negation on each bit. For example the bitwise NOT of 1100₂ is 0011₂ i.e. 0s are negated to 1s while 1s are negated to 0s. Likewise the bitwise NOT of 00101101₂ is 11010010₂ which represents -45₁₀ Looking at the two numbers, the most significant digit shows that the number has a sign bit “0” for “+0” and “1” for “-0”. Like the method of using an extra sign bit, in ones complement, there are two ways of representing a zero.

Twos compliment

Twos compliment, equivalent to tens compliment in decimal numbers, is the most popular way of representing negative numbers n computer systems.

Advantages of using twos compliment

1. There are no two ways of representing a zero as is the case with the 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 extra circuitry 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 complement of a decimal number 45₁₀ first convert it to its binary equivalent then find its ones compliment. Add a 1 to the ones complement. Example:

45₁₀ = 00101101₂

NOT (00101101) = 11010010

Twos’ complement = 11010010₂ + 1₂

= 11010011₂

Binary Addition

The five possible additions in binary are:

1. 0₂+0₂=0₂

2. 0₂+1₂=1₂

3. 1₂+0₂=1₂

4. 1₂+1₂=10₂

5. 1₂+1₂+1₂=11₂

The following examples explain binary addition.

Add these binary digits