Computer Arithmetic - Fundamentals of Computer Organization and Architecture - page 63

to obtain (1)(001110). If the carry bit is added to the least significant bit of the result,
we obtain (001111), that is, ( 15), which is a correct result.

Table 4.1 shows a comparison between the 2's complement and the 1's comp-
lement in the representation of an 8-bit number, x.

þ

4.2. INTEGER ARITHMETIC

In this section, we introduce a number of techniques used to perform integer arith-
metic using the radix complement representation of numbers. Our discussion will
focus on the base “2” binary representation.

4.2.1. Two's Complement (2's) Representation

In order to represent a number in 2's complement, we perform the following two steps.

1. Perform the Boolean complement of each bit (including the sign bit);
2. Add 1 to the least signi fic ant bit (treating the number as an unsigned binary
integer), that is,

A ¼ A þ 1

Example Consider the representation of ( 22) using 2's complement.

2

22 ¼ 00010110

+
11101001 (1's complement)

þ 1

11101010 ( 22)

4.2.2. Two's Complement Arithmetic

Addition Addition of two n-bit numbers in 2's complement can be performed

using an n-bit adder. Any carry-out bit can be ignored without affecting the correct-
ness of the results, as long as the results of the addition is in the range

2 n 1 2 n 1 to þ 1.

Example Consider the addition of the two 2's complement numbers ( 7) and

2

( þ 4). This addition can be carried out as ( 2 7) þ ( þ 4) ¼ 2 3, that is, 1001 þ

(0100) ¼ 1101, a ( 2 3) in 2's complement.

2 n 1

þ

2 n 1

1is
important because a result outside this range will lead to an overflow and hence a
wrong result. In simple terms, an overflow will occur if the result produced by a

The condition that the result should be in the range to