__Positive and Negative Integers__

The ones complement of a binary string is formed by replacing 0s by 1s and 1s by 0s.

The twos complement of a binary integer is formed by finding the 1s complement of the number and adding 1 to it.

__Worked questions __

1- Find the ones complement and twos complement of 000101101

To find the ones complement replace 0 by 1, 1 by 0

Ones complement of 000101101 = 111010010

Twos complement of 000101101 = Ones complement + 1

= 111010010+

1

__111010011 __

Answer: Ones complement is 111010010; twos complement is 111010011

2- Find the eight-bit twos complement of 110100

Make up to eight bits by adding Os on the left: 00110100

Ones complement = 11001011

Twos complement = Ones complement + 1

= 11001011 +

1

Answer: Twos complement = 11001100

__11001100__

__METHODS OF STORING INTEGERS __

There are several methods of storing integers in a computer so that positive and negative numbers are represented. Often the bit at the left of the string indicates whether the number is positive or negative. This bit is called the sign bit:

__Sign-and-magnitude method __

The sign bit is I for negative, 0 for positive. The size of the number is shown in the remaining bits.

__Twos complement method __

The sign bit is 1 for negative, 0 for positive. For positive numbers the remaining bits show the size of the number. For negative numbers the twos complement of the number is shown.

__Worked question __

Express the binary integers (a) 1011 (b) -11011 in sign-and-magnitude and in twos complement form as they would appear in an eight-bit location.

(a) In sign-and-magnitude form 1011 would be 00001011

In twos complement form 1011 would be 00001011

(b) In sign-and-magnitude form -11011--110011011

In twos complement form:

Extend to eight bits 11011_00011011

Find ones complement _11100100

Add 1 _11100101

** Note**: Positive numbers are the same for both methods but negative numbers are different.

__CONVERTING TWOS COMPLEMENT NEGATIVE INTEGERS TO DECIMAL __

__Method __

Use the same method as that for positive integers . However, treat the sign bit as a negative number.

1- Write powers of 2 above the digits of the number. Start at the right-hand end. That is write ... 64 32 16 8 4 2 1 (as many as necessary).

2- Write down the powers of 2 which are above a 1 in the number. Add them up counting the one over the sign bit as negative.

__Worked question __

111011 is a negative number in twos complement form. Find its value as a decimal number.

- 32 16 8 4 2 1

1 1 1 0 1 1

Number = -32+ 16+8+2+1

= -32+27

Answer: Number represented is -5